Final Answers
by Gérard P. Michon, Ph.D.

(unless otherwise stated)

© 2000 - 2014  by Gérard Michon.  All Rights Reserved.  All texts and illustrations are copyrighted; short excerpts may only be quoted according to applicable copyright laws.

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Table of Contents
See also:  Dates of creation of all indexed pages

Approach your problems from the right end and begin with the answers.
Then one day, perhaps, you will find the final question.

"The Chinese Maze Murders"   by   Robert Hans van Gulik  (1910-1967) 
It's better to know some of the questions than all of the answers.
James Grover Thurber  (1894-1961) 
Whoever answers before pondering the question is foolish and confused.
Proverbs 18:13

Mathematical Proofs

  1. Only a negative deserves a proof  (no counterexamples).
  2. Proving by induction  the truth of infinitely many things.
  3. Stochastic proofs  leave only a  vanishing  uncertainty.
  4. Heuristic arguments  state the likelihood of a conjecture.
  5. Computer-aided proofs:  The  4-color theorem  (1976).
  6. Lack of a good approach  doesn't invalidate a statement.

    Measurements and Units

  7. All metric prefixes: Current SI prefixes, obsolete prefixes, bogus prefixes...
  8. Prefixes for units of information  (multiples of the bit).  Brontobyte hoax.
  9. Density one. Relative and absolute density precisely defined.
  10. Acids yielding a mole of H+ per liter are normal (1N) solutions.
  11. Calories: Thermochemical calorie, gram-calorie, IST calorie (and Btu).
  12. Horsepowers: hp, metric horsepower and boiler horsepower.
  13. The standard acceleration of gravity has been 9.80665 m/s2 since 1901. 
  14. Tiny durations; zeptosecond (zs, 10-21s) & yoctosecond (ys, 10-24s).
  15. A jiffy is either a light-cm or 10 ms (tempons and chronons are shorter).
  16. The length of a second. Solar time, ephemeris time, atomic time.
  17. The length of a day. Solar day, atomic day, sidereal or Galilean day.
  18. Scientific year = 31557600 SI seconds  (» Julian year of  365.25 solar days).
  19. The International inch (1959) is 999998/1000000 of a US Survey inch.
  20. The typographer's point  is  exactly  0.013837" = 0.3514598 mm.
  21. How far is a league?  Land league, nautical league.
  22. Radius of the Earth and circumference at the Equator.
  23. Extreme units of length. The very large and the very small. 
    Surface Area:
  24. Acres, furlongs, chains and square inches... 
    Volume, Capacity:
  25. Capitalization of units. You only have a choice for the liter (or litre ).
  26. Drops or minims: Winchester, Imperial or metric. Teaspoons and ounces.
  27. Fluid ounces: American ounces (fl oz) are about 4% larger than British ones.
  28. Gallons galore: Winchester (US) vs. Imperial gallon (UK), dry gallon, etc.
  29. US bushel and Winchester units of capacity (dry = bushel, fluid = gallon).
  30. Kegs and barrels: A keg of beer is half a barrel, but not just any "barrel". 
    Mass, "Weight":
  31. Tiny units of mass.  A hydrogen atom is about 1.66 yg.
  32. Solar mass:  The unit of mass in the  astronomical system of units.
  33. Technical units of mass. The slug and the hyl.
  34. Customary units of mass  which survive in the electronic age.
  35. The  poids de marc  system:   18827.15 French grains  to the kilogram.
  36. A talent was the mass of a cubic foot of water.
  37. Tons: short ton, long ton (displacement ton), metric ton (tonne), assay ton...
    • Other tons: Energy (kiloton, toe, tce), cooling power, thrust, speed.

    The Art of Rounding Numbers:

  38. Scientific notation:  Nonzero numbers given as multiples of  powers of ten.
  39. So many "significant" digits  imply a result of limited precision.
  40. Standard deviation  specifies the  uncertainty  in the  precision  of a result.
  41. Engineering notation  reduces a number to a multiple of a power of 1000.
  42. The quadratic formula  is numerically inadequate in common cases.
  43. Devising robust formulas  which feature a stable floating-point precision.

    Scales and Ratings: Measuring without Units

  44. The Beaufort scale  is now defined in terms of wind speed.
  45. The Saffir / Simpson scale  for hurricanes.
  46. The Fujita scale  for tornadoes.
  47. Decibels:  A general-purpose logarithmic scale for relative power ratios.
  48. Sound intensity level (SIL)  is well-defined;  SPL is an  approximation.
  49. Apparent and absolute magnitudes  of stars.
  50. Acidity.  The pH scale was invented by  Søren Sørensen  in 1909.
  51. The Richter scale  for earthquakes and other sudden energy releases.

    Scaling and Scale Invariance

  52. The scale of animals  according to Galileo Galilei.
  53. Jumping fleas...  compared to jumping athletes...
  54. Drag coefficient  of a sphere as a function of the Reynolds number  R.

    Model Trains:  Scales, gauges & other mathematical aspects.

  55. Lexicon of railroad jargon:  North-American, British and French usage.
  56. A short history of the scales of model trains...
  57. Gauges of worldwide lines.  60% are standard gauge  (56.5'' = 1.4351 m).
  58. Gauges of model tracks.  HO and OO models can share the same tracks.
  59. Types of model trains.  Naming usual combinations of a scale and a gauge.
  60. Large model trains  are often used outdoors.  Some are ridable.
  61. Gauges are  almost  in geometric progression:  Z,N,TT, HO, S,O,1,3...
  62. The height of a miniature rail, in thousands of an inch, is called its  code.
  63. Composition of modern miniature railsNickel-silver  contains no silver.
  64. Travelage  is the number of crossties per unit of distance.
  65. Luminous power of sources at scale 1/s  should be  1/s that of prototypes.
  66. Scenery scale.  81 is the geometric mean between 87 (HO) and 76 (OO).
  67. Mirrors and shadows.  Overlooked aspects of scenery in confined spaces.
  68. Couplers.
  69. Loading gauge.
  70. Radius of curved track available from different manufacturers.
  71. Least separation between curved parallel tracks, for  specific  rolling stock.
  72. Turnouts.  The  frog number  is the tangent of the  frog angle.
  73. Sectional tracks.  The need for compensator and/or correctors.
  74. DC control.  Traditional control of model trains, by  Direct current.
  75. DCCDigital command control  of several locomotives on the same track.
  76. Blocks.  A layout can be divided into blocks powered separately.
  77. Detection & transponding.  Locomotives located by the power they use.
  78. OpenLCB (NMRAnet).  Local control bus  =  Control area network (CAN).
  79. Famous trains and locomotives  and their miniature counterparts.
  80. Dream lines.  Legendary railroad services, past and present.
  81. A quick survey of a few grand layouts.  Great modeling achievements.

    Numerical Constants: Mathematical & Physical Constants

    Physical Constants:
  82. For the utmost in precision, physical constants are derived in a certain order.
  83. Primary conversion factors  between customary systems of units.
    6+1 Basic Dimensionful Physical Constants  (Proleptic SI)
  84. Speed of Light in a Vacuum (Einstein's Constant):   c = 299792458 m/s.
  85. Magnetic Permeability of the Vacuum: An exact value defining the ampere.
  86. Planck's constant:  The ratio of a photon's energy to its frequency.
  87. Boltzmann's constant:  Relating temperature to energy.
  88. Avogadro's number:  The number of things per mole of stuff.
  89. Mechanical Equivalent of Light (683 lm/W at 540 THz) defines the candela.
  90. Newton's constant of gravitation  and a futuristic definition of the second.
    Fundamental Mathematical Constants: 
  91. 0:  Zero is the most fundamental and most misunderstood of all numbers.
  92. 1 and -1:  The unit numbers.
  93. p ("Pi"): The ratio of the circumference of a circle to its diameter.
  94. Ö2: The diagonal of a square of unit side.  Pythagoras' Constant.
  95. Ö3: Diameter of a cube of unit side.  Constant of Theodorus.
  96. f: The diagonal of a regular pentagon of unit side.  The Golden Number.
  97. Euler's  e:  Base of the exponential function which equals its own derivative.
  98. ln(2):  The alternating sum of the reciprocals of the integers.
  99. An engineering favorite:  The decimal logarithm of 2.
  100. Euler-Mascheroni Constant  g :  Limit of   [1 + 1/2 + 1/3 +...+ 1/n] - ln(n).
  101. Catalan's Constant  G :  The alternating sum of the reciprocal odd squares.
  102. Apéry's Constant  z(3) :  The sum of the reciprocals of the perfect cubes.
  103. Imaginary  i:  If "+1" is a step forward, "+ i" is a step sideways to the left.
    Exotic Mathematical Constants: 
  104. Delian constant:  21/3  is the solution to the classical  duplication of the cube.
  105. Gauss's constant:  Reciprocal of the arithmetic-geometric mean of 1 and Ö2.
  106. Mertens constant:  The limit of   [1/2 + 1/3 + 1/5 +...+ 1/p] - ln(ln p)
  107. Artins's constant  is the proportion of long primes in decimal or binary.
  108. Ramanujan-Soldner constant (m):  Positive root of the logarithmic integral.
  109. The Omega constant:  W(1) is the solution of the equation   x exp(x) = 1.
  110. Feigenbaum constant (d) and the related reduction parameter (a).
    Some Third-Tier Mathematical Constants: 
  111. Brun's Constant:  A standard uncertainty  (s)  means a 99% level of  ±3s
  112. Prévost's Constant:  The sum of the reciprocals of the Fibonacci numbers.
  113. Grossman's Constant:  One recurrence converges for only one initial point.
  114. Ramanujan's Number:   exp(p Ö163)   is almost an integer.
  115. Viswanath's Constant:  Mean growth in random additions and subtractions.
  116. Copeland-Erdös Number:  Almost all numbers are  normal,  it's one of them!

    Counting, Combinatorics, Probability

  117. Always change your first guess if you're always told another choice is bad.
  118. The Three Prisoner Problem predated Monty Hall and Marilyn by decades.
  119. Seating N children at a round table in (N-1)! different ways.
  120. How many Bachet squares?  A 1624 puzzle using the 16 court cards.
  121. Choice Numbers:  C(n,p) is the number of ways to choose p items among n.
  122. Multichoice Numbers:  Putting  n  objects into distinct boxes of fixed sizes.
  123. C(n+2,3) three-scoop sundaes. Several ways to count them (n flavors).
  124. C(n+p-1,p) choices of p items among n different types, allowing duplicates.
  125. How many new intersections of straight lines defined by n random points.
  126. Face cards. The probability of getting a pair of face cards is less than 5%.
  127. Homework Central: Aces in 4 piles, bad ICs, airline overbooking.
  128. Binomial distribution. Defective units in a sample of 200.
  129. Siblings with the same birthday. What are the odds in a family of 5?
  130. Covariance:  A generic example helps illustrate the concept.
  131. Variance of a binomial distribution, derived from general principles.
  132. Standard deviation. Two standard formulas to estimate it.
  133. Markov's inequality  is used to prove the  Bienaymé-Chebyshev inequality.
  134. Bienaymé-Chebyshev inequality:  Valid for  any  probability distribution.
  135. Inclusion-Exclusion: One approach to the probability of a union of 3 events.
  136. The "odds in favor" of poker hands: A popular way to express probabilities.
  137. Probabilities of a straight flush in 7-card stud. Generalizing to "q-card stud".
  138. Probabilities of a straight flush among 26 cards  (or any other number).
  139. The exact probabilities in 5-card, 6-card, 7-card, 8-card and 9-card stud.
  140. Rearrangements of  CONSTANTINOPLE  so no two vowels are adjacent...
  141. Four-letter words from  POSSESSES:  Counting with generating functions.
  142. How many positive integers below 1000000  have their digits add up to 19?
  143. Polynacci Numbers:  Flipping a coin n times without  p  tails in a row.
  144. Winning in finitely many flips or losing endlessly...
  145. 252 decreasing sequences of 5 digits  (2002 nonincreasing ones).
  146. How many ways are there to make change for a dollar?  Closed formulas.
  147. Partitioniong an amount into the parts minted in a certain currency.
  148. The number of rectangles in an N by N chessboard-type grid.
  149. The number of squares in an N by N grid:  0, 1, 5, 14, 30, 55, 91, 140, 204...
  150. Screaming Circles:  How many tries until there's no eye contact?
  151. Average distance between two random points on a segment, a disk, a cube...
  152. Average distance between two points on the surface of a sphere.

    Diamonds Hearts Spades Clubs Playing Cards

  153. Short history of playing cards:  From China to Europe, to the New World.
  154. Sizes of playing cards:  French, Bridge, Poker, French tarot, etc.
  155. How playing cards are made:  Either 2 layers of paper or "100% plastic".
  156. Suits:  Spades, hearts, diamonds & clubs  (swords, cups, coins & wands).
  157. The four court cards:  Ace, king, queen, jack  (king, queen, cavalier, page).
  158. The  Mameluke  52-card standard deck  with 3 figure cards per suit.
  159. 78-card tarot deck:  21 trumps, 1 fool, 4 suits of 14  (incl. 4 court cards).
  160. The Major Arcana:  Trumps and fool of the tarot deck, in occult parlance.
  161. Names of the court cards  in the French tradition.  Hundred Years' War.
  162. 48-card  Aluette  deck:  Latin suits, mimicks and names of special cards.
  163. Jokers  from Euchre  (1857)  found their way into Poker in the 1870's.
  164. The 40-card Spanish  baraja  deck  lacks  8, 9 & 10.
  165. The 32-card  piquet  deck  lacks 2-6.  French  Belote  and German  Skat.
  166. Skat:  The most popular German card game  (32-card deck).
  167. 24-card deck  for Euchre (single deck) and Pinochle (double deck).
  168. Happy Families:  44-card British deck of 11 families of 4  (1851).
  169. Jeu des 7 familles:  42=card French deck of 7 families of 6  (1876).
  170. 1000 Bornes:  106 cards for a boardless car-racing game  (1954).
  171. Set® cards:  Combinatorics of a modern 81-card ternary deck  (1974).

    The Card Games Played in Casinos  (Banked Games)

  172. Gaming chips:  Color coding, shapes & sizes, designs.  Jetons & plaques.
  173. Casino edge:  Gambling beyond the cost of entertainement is  foolish.
  174. Faro  was the most popular banking game from 1825 to 1915.
  175. Baccarat  (Punto-Banco).
  176. EZ Baccarat(™).  The original  Dragon-7  and newer  Panda-8  side bets.
  177. Carribean Stud.
  178. Three-Card Poker.

    The Game of Blackjack  (Twenty-One)

  179. Glossary:  A few specialized term used in blackjack.
  180. Casino rules  for Blackjack.
  181. Basic strategy  against an  infinite shoe.
  182. Pair of aces  (soft 12).  What to do if you're not allowed to split it?
  183. Blackjack enumerations  using polynomials.
  184. History of Blackjack counting.

    Poker  101 :  Rules, Odds & Glossary

  185. 5-card draw:  The simplest form of poker is the basis for  video poker.
  186. The 2598960 poker hands  come in 9 or 10 types, rarest ones first.
  187. Kickers  may help break ties between hands bearing the same name.
  188. Perfect Poker:  "Deuces or better" have 1 to 1 odds with the  full-wheel rule.
  189. Poker chips:  Color, size and weight.
  190. Poker chip sets:  Practical repartitions into various denominations.
  191. Handling chips:  Counting them, stacking them, betting with them.
  192. Poker chip tricks.
  193. 7-card stud  was the most popular variant of poker before NLHE and PLO.
  194. 7-card combos:  Odds of best 5-card hands extracted from 7 random cards.
  195. Betting rules:  Antes & blinds, checking, opening, calling and raising.
  196. Glossary:  The jargon of poker.

    Poker  102 :  No-Limit Hold 'em  &  Pot-Limit Omaha

  197. Texas hold 'em:  Two hole cards (hand) and five community cards (board).
  198. Preflop probabilities  (win or tie a showdown)  for all 169 starting hands.
  199. If you have kings in an m-handed game,  how often do you run into aces?
  200. Trips or quads  from a lone board pair.
  201. With 3 clubs on the board:  If a player has a flush, does someone else?
  202. Nontransitivity of matches:  Pairwise showdowns can be  inconsistent.
  203. How much to bet  depends on the goal  (make money or avoid elimination).
  204. Omaha Hold 'em:  Use 2 hole cards (out of 4) and 3 board cards (out of 5).

    Stochastic Processes & Stochastic Models

  205. Poisson Processes: Random arrivals happening at a constant rate (in Bq).
  206. Simulating a poisson process  with a  uniform  random number generator.
  207. Markov Processes: When only the present influences the future...
  208. The Erlang B Formula assumes callers don't try again after a busy signal.
  209. Markov-Modulated Poisson Processes may look like Poisson processes.

    "Utility" and Decision Analysis

  210. The Utility Function: A dollar earned is usually worth less than a dollar lost.
  211. St. Petersburg's Paradox: What would you pay to play the Petersburg game?

    Social Choice Theory

  212. Condorcet's Paradox:  A group of rational voters need not behave rationally.

    Elementary Geometry

  213. Center of an arc determined with straightedge and compass.
  214. Surface areas: Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
  215. Power of a point  with respect to a circle.
  216. Euler's line  goes through the orthocenter, the centroid and the circumcenter.
  217. Euler's circle  is tangent to the incircle and the excircles  (Feuerbach, 1822).
  218. Barycentric coordinates & trilinears  examplify  homogeneous coordinates.
  219. Elliptic arc: Length of the arc of an ellipse between two points.
  220. Perimeter of an ellipse. Exact formulas and simple ones.
  221. Surface of an ellipse.
  222. Volume of an ellipsoid  (either a spheroid or a scalene ellipsoid).
  223. Surface area of a spheroid  (oblate or prolate ellipsoid of revolution).
  224. Quadratic equations in the plane describe ellipses, parabolas, or hyperbolas.
  225. Centroid of a circular segment. Find it with Guldin's (Pappus) theorem.
  226. Parabolic arc of given extremities  with a prescribed apex between them.
  227. Focal point of a parabola. y = x 2 / 4f (where f is the focal distance).
  228. Parabolic telescope: The path from infinity to focus is constant.
  229. Make a cube go through a hole in a smaller cube.
  230. Octagon: The relation between side and diameter.
  231. Constructible regular polygons  and constructible angles (Gauss).
  232. Areas of regular polygons of unit side: General formula & special cases.
  233. For a regular polygon of given perimeter, the more sides the larger the area.
  234. Curves of constant width: Reuleaux Triangle and generalizations.
  235. Irregular curves of constant width. With or without any circular arcs.
  236. Solids of constant width. The three-dimensional case.
  237. Constant width in higher dimensions.
  238. Fourth dimension. Difficult to visualize, but easy to consider.
  239. Volume of a hypersphere and hyper-surface area, in any dimensionality.
  240. Hexahedra. The cube is not the only polyhedron with 6 faces.
  241. Descartes-Euler Formula: F-E+V=2 but restrictions apply.

    Projective Geometry

  242. Projective duality:  Points are incident to lines.  Lines are incident to points.
  243. Pappus' theorem.
  244. Pascal's theorem  was proven by  Blaise Pascal  when he was 16.
  245. Brianchon's theorem:  The dual of Pascal's theorem.
  246. Desargues' theorem.


  247. Metric spaces:  The motivation behind more general  topological  spaces.
  248. Abstract topological spaces  are defined by calling some subsets  open.
  249. Basis of a topology:  A set is open  iff  it's a union of sets from the base.
  250. Closed sets  are sets (of a topological space) whose complements are open.
  251. Subspace F of E:  Its open sets are the intersections with F of open sets of E.
  252. Separation axioms:  Flavors of topological spaces, according to  Trennung.
  253. Compactness of a topological space:  Any open cover has a  finite subcover.
  254. Real-valued continuous functions on compact sets  attain their extremes.
  255. Borel setsTribes  form the topological foundation for  measure theory.
  256. Locally compact sets  contain a  compact neighborhood  of every point.
  257. General properties of sequences  characterize topological properties.
  258. Continuous functions  let the  inverse image  of any open set be open.
  259. Restrictions remain continuous.  Continuous extensions may be impossible.
  260. The product topology  makes projections continuous on a cartesian product.
  261. Connected sets  can't be split by open sets.  The empty set  is  connected.
  262. Intermediate-value theorem.
  263. Path-connected sets  are a special case of  connected sets.
  264. Homeomorphic sets.  An  homeomorphism  is a  bicontinuous function.
  265. Arc-connected spaces are path-connected.  The converse need not be true.
  266. Homotopy:  A progessive transformation of a  function  into another.
  267. The fundamental group:  The homotopy classes of all loops through a point.
  268. Homology and Cohomology.  Poincaré duality.
  269. Descartes-Euler Formula:  F-E+V = 2, but restrictions apply.
  270. Euler Characteristic:   c   (chi)  extended beyond its traditional definition...
  271. Winding number  of a continuous planar curve about an outside point.
  272. Fixed-point theorems  by  BrouwerShauder  and  Tychonoff.
  273. Turning number  of a planar curve with a well-defined oriented tangent.
  274. Real projective plane  and Boy's surface.
  275. Hadwiger's  additive continuous functions of d-dimensional rigid bodies.
  276. Eversion of the sphere.  An homotopy  can  turn a sphere inside out.
  277. Classification of surfaces:  "Zero Irrelevancy Proof" (ZIP) by J.H. Conway.
  278. Braid groups:  Strands, braids and pure braids.


  279. Complete metric space:  A space in which all Cauchy sequences converge.
  280. Flawed alternatives to completeness.
  281. Banach spaces  are complete normed vector spaces.
  282. Fréchet spaces  are generalized Banach spaces.

    Fractal Geometry:

  283. Fractional exponents  were first conceived by Nicole d'Oresme (c. 1360).
  284. The von Koch curve (and  snowflake):  Dimension of self-similar objects.
  285. Hausdorff dimension is revealed by a covering with balls of radius  < e.
  286. The Julia set and the Fatou set of an analytic function  are complementary.
  287. The Mandelbrot set was so named by  Adrien Douady  &  John H. Hubbard.

    Angles and Solid Angles:

  288. Planar angles  (from one direction to another)  are  signed  quantities.
  289. Bearing:  Unless otherwise specified, this is the angle  west of north.
  290. Solid angles  are to spherical patches what planar angles are to circular arcs.
  291. Circular measures:  Angles and solid angles aren't quite dimensionless...
  292. Solid angle formed by a trihedron :   Van Oosterom & Strackee  (1983).
  293. Solid angle subtended by a rhombus.  Apex of a right  rhombic pyramid.
  294. Formulas for solid angles  subtended by patches with simple shapes.
  295. Right ascension and declination.  Precession of celestial coordinates  (a,d).

    Curvature and Torsion:

  296. Curvature of a planar curve:  Variation of inclination with distance  dj/ds.
  297. Curvature and torsion  of a three-dimensional curve.
  298. Distinct curvatures and  geodesic  torsion  of a curve drawn on a surface.
  299. The two fundamental quadratic forms  at a point of a parametrized surface.
  300. Lines of curvature:  Their  normal  curvature is extremal at every point.
  301. Geodesic lines.  Least length is achieved with  zero geodesic curvature.
  302. Meusnier's theorem:  Tangent lines have the same  normal curvature.
  303. Gaussian curvature of a surface.  The  Gauss-Bonnet theorem.
  304. Parallel-transport of a vector around a loop.  Holonomic angle of a loop.
  305. Total curvature of a curve.  The Fary-Milnor theorem for knotted curves.
  306. Linearly independent components  of the  Riemann curvature tensor.

    Planar Curves:

  307. Cartesian equation of a straight line:  passing through two given points.
  308. Confocal Conics:  Ellipses and hyperbolae sharing the same pair of  foci.
  309. Spiral of Archimedes:  Paper on a roll, or groove on a vinyl record.
  310. Hyperbolic spiral:  The inverse of the  Archimedean spiral.
  311. Catenary:  The shape of a thin chain under its own weight.
  312. Witch of Agnesi.  How the versiera (Agnesi's cubic) got a weird name.
  313. Folium of Descartes.
  314. Lemniscate of Bernoulli:  A quartic curve shaped like the  infinity symbol.
  315. Along a Cassini oval, the product of the distances to the two foci is constant.
  316. Limaçons of Pascal:  The cardioid  (unit epicycloid) is a special case.
  317. On a Cartesian oval, the weighted average distance to two poles is constant.
  318. The envelope of a family of curves  is everywhere tangent to one of them.
  319. The evolute of a curve  is the locus of its centers of curvature.
  320. Involute of a curve:  Trajectory of a point of a line  rolling  on that curve.
  321. Parallel curves  share their normals, along which their distance is constant.
  322. The nephroid  (or  two-cusped epicycloid )  is a  catacaustic  of a circle.
  323. Freeth's nephroid:  A special  strophoid  of a circle.
  324. Bézier curves  are algebraic splines.  The cubic type is the most popular.
  325. Piecewise circular curves:  The traditional way to specify curved forms.
  326. Intrinsic equation  [curvature as a function of arc length]  may have  spikes.
  327. The quadratrix (trisectrix) of Hippias squares the circle and trisects angles.
  328. The parabola  is  constructible  with straightedge and compass.
  329. Mohr-Mascheroni constructions  use the compass alone  (no straightedge).


  330. Glossary  of terms related to gears.
  331. Gear ratio:  Ratio of the input rotation to the output rotation  (signed).
  332. Planar curves  rolling without slipping while rotating about two fixed points.
  333. Congruent ellipses  roll against each other while revolving around their foci.
  334. Elliptic gears:  A family of gears which include ellipses and sine curves.
  335. Watchmaker gearsOgival surfaces for pinions & radial planes for wheels.
  336. La Hire's theorem :  An hypocycloid of ratio  2  is a straight line.
  337. Cycloidal gear:  Epicycloidal addendum curve  & hypocycloidal dedendum.
  338. The law of conjugate action  was formulated by  Leonhard Euler  (c. 1754).
  339. Involute tooth profiles  provide a  uniform  rotational speed ratio.
  340. Harmonic Drive:  A flexspline  with 2 fewer teeth than the circular spline.
  341. Circular arc helical gears:   E. Wildhaber (1923)  &  M.L. Novikov (1956).
  342. Double circular arc helical gears were standardized by the Chinese in 1981.

    Polyhedra (3D), Polychora (4D), Polytopes (nD)

  343. Hexahedra.  The cube is not the only polyhedron with 6 faces.
  344. Fat tetragonal antiwedge:  Chiral hexahedron of unit volume and  least area.
  345. Duality:  To a face of a polyhedron corresponds a vertex of its dual.
  346. Enumeration of polyhedra: Convex polyhedra with n faces and k edges.
  347. The 5 Platonic solids: Cartesian coordinates of the vertices.
  348. Symmetries  may  equate all commensurate components of a polyhedron.
  349. Equimetric polyhedra  feature constant measures for all elements of a kind.
  350. The 13 Archimedean solids  and their  duals  (Catalan solids).
  351. Polyhedra in certain families are named after one prominent polygon.
  352. Some special polyhedra may have a traditional (mnemonic) name.
  353. Deltahedra have equilateral triangular faces. Only 8 deltahedra are convex.
  354. Johnson Polyhedra and the associated nomenclature.
  355. Polytopes are the n-dimensional counterparts of 3-D polyhedra.
  356. A simplex of touching unit spheres may allow a center sphere to bulge out.
  357. Regular Antiprism:  Height and volume of a regular n-gonal antiprism.
  358. The Szilassi polyhedron  features 7 pairwise adjacent hexagonal faces.
  359. Wooden buckyball:  Cutting 32 blocks to make a truncated icosahedron.
  360. Zonogons, zonohedra, zonotopes.  Zones and zonoids.
  361. Space-filling polyhedra:  Cuboctahedron, truncated octahedron, etc.


  362. 16 possible standard dice  (opposite faces add up to  7).  Two handedness.
  363. 30  labelings of a die:  For  16  of them, opposite faces  never  add up to  7.
  364. 3-sided spindle:  A 9-hedron with 6 unstable faces.
  365. Nontransitive dice:  Every die is dominated by another die from the set.
  366. Sicherman dice  yield any total with the same probability as a regular pair.
  367. Percentile dice  and other ways several dice can have equiprobable sums.
  368. Polyhedral dice  were popularized by  rôle playing games.
  369. Commonly available dice sizes:  Small, medium, large, jumbo and giant.
  370. Convex isohedra  are fair dice, by reason of  symmetry  between faces.
  371. Round dice:  Outer isohedral marks  &  steel ball in an  isogonal  cavity.
  372. Scalene  isogonal polyhedra.  Their duals are  scalene  isohedra.
  373. Juryeonggu:  Korean die with  8  hexagonal faces and  6  square ones.
  374. Fairness of a non-isohedral die  may depend on the way it's tossed.
  375. Are there any  intrinsically  fair dice  which aren't isohedral?
  376. Necessary conditions  an  absolutely fair  die must satisfy.
  377. Quasistatic  probability is proportional to the  solid angle  a face subtends.
  378. Thermal tossing  puts a face of minimal height at the  bottom.
  379. Balanced mesohedral dice  are fair for  both  quasistatic and thermal tossing.
  380. Mesodecahedron:  Mesohedron with  10  faces.
  381. Mesopentahedron:  The mesohedral proportions of a  rhombic pyramid.
  382. Mesoheptahedron:  Mesohedron with  7  faces.
  383. Statistical bias  of unfair dice.

    Isohedra.  The symmetry of fair dice.

  384. Classification of all convex isohedra.  Intrinsic fair dice.
  385. Disphenoids  are  tetrahedra  where opposing edges have equal length.
  386. The hexakis icosahedra  (120 faces)  include the  disdyakis triacontahedra.
  387. Two chiral fair dice:  No mirror symmetry,  24  or  60  pentagonal faces.
  388. A pseudo-isohedral die.  Its faces are congruent, but is it fair?

    Graph Theory

  389. The bridges of KönigsbergEulerian graphs  and the birth of  graph theory.
  390. Undirected graphs are digraphs with  symmetrical  adjacency matrix.
  391. Adjacency matrix of a directed graph  (digraph)  or of a  bipartite graph.
  392. The 3-utilities problem:  Providing 3 cottages with water, gas & electricity.
  393. Silent Circles:  An enumeration based on adjacency matrices  (Alekseyev).
  394. Silent Prisms:  Modifying the screaming game for short-sighted people.
  395. Tallying  markings of one edge per node where no edge is marked twice.
  396. Line graph:  Nodes of L(G) are edges of G  (connected  iff  adjacent in G).
  397. Transitivity:  Vertex-transitive and/or edge-transitive graphs.
  398. Desargues graph  and  distance transitivity.


  399. Factorial zero is 1, so is an empty product; an empty sum is 0.
  400. Anything to the power of 0 is equal to 1, including 0 to the power of 0.
  401. Idiot's Guide to Complex Numbers.
  402. Using the Golden Ratio (f) to express the 5 [complex] fifth roots of unity.
  403. "Multivalued" functions are functions defined over a Riemann surface.
  404. Square roots are inherently ambiguous for negative or complex numbers.
  405. The difference of two numbers, given their sum and their product.
  406. All symmetric polynomials of 3 variables are determined by the first three.
  407. Geometric progression of 6 terms. Sum is 14, sum of squares is 133.
  408. Quartic equation involved in the classic "Ladders in an Alley" problem.


  409. Chebyshev polynomials  give cos(nq) as a function of cos q
  410. Legendre polynomials  and  zonal harmonics.
  411. Laguerre polynomials.  Hypergeometric confluent function.
  412. Hermite polynomials.  Eigenstates of the quantum harmonic oscillator.

    Matrices and Determinants

  413. Permutation matrices  include the identity matrix and the exchange matrix.
  414. Operations on matrices  are conveniently defined using  Dirac's notation.
  415. The determinant  is proportional to any  completely antisymmetrical  form.
  416. MinorsFirst minors  are obtained by deleting one row and one column.
  417. Adjugate of a matrix:  Tranpose matrix of its cofactors  A adj(A) = det(A) I
  418. Eigenvectors and eigenvalues  of an operator or a matrix.
  419. The characteristic polynomial  of an operator doesn't depend on the basis.
  420. Cayley-Hamilton theorem:  A matrix vanishes its characteristic polynomial.
  421. Vandermonde matrix:  The successive powers of elements in its second row.
  422. Generalized Vandermonde matrix  involving  fractional  powers...
  423. Cholesky decomposition   L L*   of a positive-definite Hermitian matrix.
  424. Toeplitz matrix:  Constant diagonals.
  425. Circulant matrix:  Cyclic permutations of the first row.
  426. Wendt's Determinant:  The circulant of the binomial coefficients.
  427. Hankel matrix:  Constant skew-diagonals.
  428. Catberg matrix:  Hankel matrix of the reciprocal of Catalan numbers.
  429. Hadamard matrix:  Unit elements and orthogonal columns.
  430. Sylvester matrix  of two polynomials has their resultant for determinant.
  431. The discriminant of a polynomial is the resultant of itself and its derivative.

    Trigonometry, Elementary Functions, Special Functions

  432. Numerical functions: Polynomial, rational, algebraic, transcendental, special.
  433. Trigonometric functions:  Memorize a simple picture for 3 basic definitions.
  434. Solving triangles with the law of sines, law of cosines, and law of tangents.
  435. Spherical trigonometry:  Triangles drawn on the surface of a sphere.
  436. Sum of tangents of two half angles, in terms of sums of sines and cosines.
  437. The absolute value of the sine of a complex number.
  438. Exact solutions to transcendental equations.
  439. All positive rationals (& square roots) as trigonometric functions of zero!
  440. The sine function:  How to compute it numerically.
  441. Chebyshev economization saves billions of steps in common computations.
  442. The Gamma function:  Its definitions, properties and special values.
  443. Lambert's W function is used to solve practical transcendental equations.

    Hypergeometric Functions

  444. Pochhammer's symbolUpper factorial of k increasing factors:  x(x+1)...
  445. Gauss's hypergeometric function:  2+1 parameters (and one variable).
  446. Kummer's transformations relate different hypergeometric expressions.
  447. Sum of the reciprocal of Catalan numbers, in closed hypergeometric form.


  448. Derivative:  The slope of a function and/or something more abstract.
  449. Integration: The Fundamental Theorem of Calculus.
  450. Integration by parts:  Reducing an integral to another one.
  451. Length of a parabolic arc.
  452. Top height of a curved bridge  with a  5280 ft  span and a  5281 ft  length.
  453. Sagging:  A cable which spans 28 m and sags 30 cm is 28.00857 m long.
  454. The length of the arch of a cycloid is 4 times the diameter of the wheel.
  455. Integrating the cube root of the tangent function.
  456. Changing inclination to a particle moving along a parabola.
  457. Algebraic area of a "figure 8" may be the sum or the difference of its lobes.
  458. Area surrounded by an oriented planar loop  which  may  intersect itself.
  459. Linear differential equations of higher order and/or in several variables.
  460. Theory of Distributions:  Convolution products and their usage.
  461. Laplace Transforms: The Operational Calculus of Oliver Heaviside.
  462. Integrability of a function and of its absolute value.
  463. Analytic functions of a linear operator; defining  f (D) when D is d/dx...

    Differential Equations

  464. Ordinary differential equations.  Several examples.
  465. A singular change of variable  may not be valid over a maximal domain.
  466. Vertical fall  against fluid resistance  (valid for viscous and quadratic drag).
  467. Jet propulsion:  Expelling stuff at speed u makes  (u-v) m  remain constant.
  468. Riccati equation:  When  y'  is a quadratic function of  y...

    Differential Forms  &  Vector Calculus

  469. Generalizing the  fundamental theorem of calculus.
  470. Vectorial  surface  dotted into an observing direction gives  apparent  area.
  471. Practical identities  of vector calculus.

    Optimization:  Operations Research, Calculus of Variations

  472. Stationary points  (saddlepoints)  are where  all  partial derivatives vanish.
  473. Single-variable optimization:  Derivative vanishes at any interior extremum.
  474. Extrema of a function of two variables  obey a  second-order  inequality.
  475. Saddlepoints of a multivariate function.  One equation for each variable.
  476. Lagrange multipliers:  Constrained optimization of an  objective function.
  477. Minimizing the lateral surface area of a cone  of given base and volume.
  478. Euler-Lagrange equations  hold along the path of a  stationary  integral.
  479. Noether's theorem:  One conserved quantity for each Lagrangian symmetry.
  480. The brachistochrone curve  is a cycloid (in a uniform gravitational field).
  481. Isoperimetric Inequality:  The largest area enclosed by a loop of unit length.
  482. Plateau  extended the calculus of variations from paths to membranes.
  483. Embedded minimal surfaces: Plane, catenoid, helicoid, Costa's surface, etc.
  484. Connecting blue dots to red dots  in the plane, without any crossings...
  485. The shortest way to connect 3 dots  can be to join them to a  fourth  point.
  486. The Honeycomb Theorem:  A conjecture of old, proved by  Thomas Hales.
  487. Counterexamples to Kelvin's conjecture.  Unit spatial tiles of least area.

    Analysis, Convergence, Series, Complex Analysis

  488. Cauchy sequences help define real numbers rigorously.
  489. Permuting the terms of a series may change its sum arbitrarily.
  490. Two decreasing divergent series  may have a convergent minimum!
  491. Uniform convergence of continuous functions makes the limit continuous.  Augustin Cauchy 
 (1789-1857)  Joseph Fourier 
  492. Defining integrals: Cauchy, Riemann, Darboux, Lebesgue.
  493. Cauchy principal value of an integral.
  494. Fourier series. A simple example.
  495. Infinite sums evaluated with  Fourier series.
  496. A double sum is often the product of two sums  (possibly Fourier series).
  497. At a jump,  a Fourier series is the half-sum of its left and right limits.
  498. Gibbs phenomenon; 9% overshoot of partial Fourier series near a jump.
  499. Method of Frobenius about a regular singularity of a differential equation.
  500. Laurent series of a function about one of its poles.
  501. Cauchy's Residue Theorem is helpful to compute difficult definite integrals.
  502. Tame complex functions: Holomorphic and meromorphic functions.

    Complex Power Series  &  Analytic Continuations

  503. Taylor's expansion  of a differentiable function as a power series.
  504. Radius of convergence.  The convergence disk of a complex power series.
  505. The exponential series.  Proving that   exp (x)  exp (y)   =   exp (x+y)
  506. Analytic continuation:  Power series that converge on overlapping disks.
  507. Decimated power series are equal to finite sums involving  roots of unity.

    Summation of Divergent Series

  508. Summing geometric series:  Equating things that match over some domain.
  509. Desirable properties of summation methods  yield rules for handling them.
  510. Formal series are ketsLinear  summation methods are  bras.
  511. Summation by convergence  is compatible with  regular  summations.
  512. Functional analysis:  Using the  Hahn-Banach extension theorem.
  513. Euler summation  (1746).
  514. All  Nörlund summations  are linear, stable, regular and consistent  (1919).
  515. Abel summation .
  516. Lindelöf summation  (1903).
  517. Borel summation  (1899).
  518. Mittag-Leffler summation method  (1908).
  519. Weierstrass summation:  Summation by  analytic continuation  (1842).
  520. Zeta-function regularization  (1916).  Dubbed  heat-kernel regularization.
  521. The Mercator series  is the integral of the  geometric series  (1668).
  522. Ramanujan's irregular summation  (1913).  "Sum" of the  harmonic series.
  523. Summations of p-adic integers  for a special radix or for all of them.
  524. Moments, Stieltjes functions and Stieltjes series.
  525. Shanks' transformation  greatly accelerates an  alternating  convergence.
  526. Richardson extrapolation.
  527. Index-free acceleration of sequences with harmonic convergence.
  528. Parametrized acceleration  based on the expected type of convergence.
  529. Asymptotic analysis  and  asymptotic series.
  530. Stirling's approximation  and  Stirling's series.

    Fourier Transform  &  Tempered Distributions  Joseph Fourier 

  531. Convolution  as an inner operation among numerical functions.  Joseph Fourier 
  532. Duality:  The product of a  bra  by a  ket  is a (complex) scalar.
  533. distribution  associates a scalar to every  test function.
  534. Schwartz functions  are suitable  rapidly decreasing  test functions.
  535. Tempered distributions  are functionals over Schwartz functions.
  536. The Fourier Transform  associates a  tempered distribution  to another.
  537. Parseval's theorem  (1799).  The Fourier transform is unitary.
  538. Noteworthy distributions and their Fourier transforms Antoine Parseval 
    • Dirac's  d  and the  uniform  distribution  ( f (x) = 1).
    • The  signum  function  sgn(x)  and its transform:   i / ps
    • The Heaviside step function  H(x) = ½ (1+sgn(x))  and its transform.
    • The square function  P(x) = H(x+½)-H(x-½)   and   sinc ( ps )
    • The triangle function  L(x)   and   sinc2 ( ps )
    • The normalized Gaussian distribution is its own Fourier transform.
  539. Sampling formula:  The unit comb  (Shah function)  is its own Fourier transform.
  540. Far image of a picture on translucent film  is its Fourier transform.
  541. The Radon transform  (used in lateral tomography)  is easily inverted.
  542. Competing definitions of the Fourier transform.  For the record.

    Discrete Fourier Transforms  &  Fast Fourier Transform

  543. Discrete Fourier Transform,  defined as a  unitary involution.

    Set Theory and Logic

  544. The logic of Aristotle.  Syllogisms about  categorical propositions.
  545. The  Barber's Dilemma  is not a paradox, if analyzed properly.
  546. What is infinity?  There's more to it than a pretty symbol  (¥).
  547. Peano's axioms  provide a rigorous definition of the  set of natural integers.
  548. Ordered numbers:  From integers to rational, real and surreal numbers.
  549. There are more real than rational numbers.  Cantor's  diagonal argument.
  550. Cantor's ternary set.  A vanishing set of reals  equipollent  to the whole line.
  551. The axioms of set theory: Fundamental axioms and the Axiom of Choice.
  552. The Axiom of restriction  looks like a  sterile  addition to the ZFC axioms.
  553. Equivalents and alternatives  to the  axiom of choice.
  554. The existence of nonmeasurable sets  is guaranteed by the Axiom of Choice.
  555. Functions and applications  are special types of  binary relations.
  556. A set is smaller than its powerset:  A simple proof applies to all sets.
  557. Transfinite cardinals  describe the various sizes of  infinite  sets.
  558. The continuum hypothesis:  Is the continuum the smallest uncountable set?
  559. Transfinite ordinals:  Counting to infinity... and beyond.
  560. Surreal numbers  include reals, transfinite ordinals, infinitesimals & more.
  561. Multidimensional (hypercomplex) numbers:  To octonions and beyond.
  562. A set belongs to a  class  in NBG  (a  conservative extension  of ZFC).
  563. Tarski-Grothendieck theory (TG)  is a  nonconservative  extension of ZFC.

    Integer Arithmetic, Number Theory

  564. The number 1 is not prime.  Good definitions allow simple theorems.
  565. Composite numbers are not prime, but the converse need not be true...
  566. Two prime numbers whose sum is equal to their product.
  567. Gaussian integers:  Factoring into primes on a two-dimensional grid.
  568. The least common multiple may be obtained without factoring into primes.
  569. Standard Factorizations:   n4 + 4   is never prime for   n > 1   because...
  570. Euclid's algorithm  gives the GCD  and  the related Bézout coefficients.
  571. Bézout's Theorem:  The GCD of p and q is of the form  u p + v q.
  572. Greatest Common Divisor  (GCD)  defined for all commensurable numbers.
  573. Linear equation in integers  can be solved using  Bézout's theorem.
  574. Pythagorean Triples:  Right triangles whose sides are coprime integers.
  575. The number of divisors of an integer.
  576. Perfect squares are the only integers with an odd number of divisors.
  577. The product of all divisors  is  often  a perfect square.
  578. Perfect numbers and Mersenne primes.  Do odd perfect numbers exist?
  579. Multiperfect & hemiperfect numbers.  Whole or half-integral abundancies.
  580. Fast exponentiation by repeated squaring.
  581. Partition function. How many collections of positive integers add up to 15?
  582. A Lucas sequence whose oscillations never carry it back to -1.
  583. A bit sequence  with intriguing statistics.  Counting squares between cubes.
  584. Binet's formulas: N-th term of a sequence obeying a linear recurrence.
  585. The square of a Fibonacci number  is  almost  the product of its neighbors.
  586. D'Ocagne's identity  relates conjugates products of Fibonacci numbers.
  587. Catalans's identity  generalizes  Cassini's Identity.
  588. Faulhaber's formula gives the sum of the p-th powers of the first n integers.
  589. Multiplicative functions:  If a and b are  coprime, then  f(ab) =  f(a) f(b).
  590. Moebius function:  Getting  N  values with  O(N Log(Log N))  additions.
  591. Dirichlet convolution is especially interesting for  multiplicative  functions.
  592. Dirichlet powers of arithmetic functions  (especially, the Möbius function).
  593. Dirichlet powers of multiplicative functions  are given by a  superb formula.
  594. Totally multiplicative functions are the simplest multiplicative functions.
  595. Dirichlet characters are important  totally multiplicative functions.
  596. Euler products  and generalized zeta functions.

    Positional Numeration  &  Number Systems

  597. Modular arithmetic can tell the last digits of extremely large numbers.
  598. Leading digits of  insanely  large numbers can't be found by logarithms.
  599. Powers of ten expressed as products of two factors  without zero digits.
  600. Divisibility by 7, 13, and 91 (or by B2-B+1 in base B).
  601. Lucky 7's.  Any integer divides a number composed of only 7's and 0's.
  602. The decimal representation of rational numbers  is  ultimately periodic.
  603. Midy's theorem:  Properties of periods in radix-B numeration
  604. Numbers with two decimal expansions.  E.g.,  1  and  0.99999999999999...
  605. Binary and/or hexadecimal numeration for floating-point numbers as well.
  606. Extract a square root the old-fashioned way.
  607. Ternary system:  Is base 3  really  the best radix for positional numeration?
  608. Dozenal counting:  (Roman) ounce, as, dozen, gross, great-gross.

    Prime Numbers

  609. A prime number  is a positive integer with 2 distinct divisors (1 and itself).
  610. Euclid's proof:  There are infinitely many primes.
  611. Dirichlet's theorem:  There are infinitely many primes of the form  kN+a.
  612. Green-Tao theorem:  Arbitrarily long arithmetic progressions of primes.
  613. Von Mangoldt's function  is  Log p  for a power of a prime p,  0 otherwise.
  614. Prime Number Theorem:  The probability that N is prime is roughly 1/ln(N).
  615. Riemann's power-prime counting function (J).  Riemann's explicit formula.
  616. The average number of factors  of a large number  N  is  Log N.
  617. The average number of distinct prime factors  of  N  is  Log Log N.
  618. The largest known prime:  Historical records, old and new.
  619. The Lucas-Lehmer Test  checks the primality of a Mersenne number  fast.
  620. Formulas giving only primes  may not help with new primes.
  621. Ulam's Lucky Numbers  and other sequences generated by sieves.

    Modular Arithmetic

  622. Chinese remainder theorem:  Remainders define an integer, within limits.
  623. Modular arithmetic: The formal algebra of congruences, due to Gauss.
  624. Fermat's little theorem:   For any prime p not dividing aap-1 is 1 modulo p.
  625. Euler's totient functionf(n) counts the integers coprime to n, from 1 to n.
  626. Fermat-Euler theorem:  If  a is coprime to n,  a  to the  f(n)  is 1 modulo n.
  627. Carmichael's reduced totient function (l) :  A special divisor of the totient.
  628. 91 is a pseudoprime to half of the bases coprime to itself.
  629. Carmichael Numbers:  An absolute pseudoprime  n divides  an-a  for any a.
  630. Chernik's Carmichael numbers:  3 prime factors   (6k+1)(12k+1)(18k+1).
  631. Other products  that yield a Carmichael number  iff  every factor is prime.
  632. Large Carmichael numbers may be obtained in various ways.
  633. Conjecture: Any odd integer coprime to its totient has Carmichael multiples.

    Group Theory and Symmetries

  634. Monoids  feature an  associative  operation and a  neutral element.
  635. The inverse of an element  comes in 2 flavors that coincide when both exist.
  636. Free monoid:  All the finite strings (words) in a given  alphabet.
  637. Raising something to the power of an integer.
  638. Groups  are monoids in which  every element  is invertible.
  639. A subgroup is a group  contained in another group.
  640. Generators  of a group are not contained in any  proper  subgroup.
  641. Lagrange's theorem:  The order of a subgroup divides the order of the group.
  642. Normal subgroups  and their quotients in a group.
  643. Homomorphism:  The image of a product is the product of the images.
  644. The symmetric group  on  E  consists of all the bijections of  E  onto itself.
  645. Inner automorphisms:  Inn(G)  is isomorphic to  G  modulo its center.
  646. The conjugacy class formula  uses conjugacy to tally elements of a group.
  647. Simple groups  are groups without  nontrivial  normal subgroups.
  648. The derived subgroup  of a group is  generated  by its  commutators.
  649. Direct product of two groups  (called a  direct sum  for additive groups).
  650. Groups of small orders.  Basic families:  Cyclic groups, dihedral groups, etc.
  651. Enumeration  of "small" groups.  How many groups of order n?
  652. Classification of finite simple groups,  by Gorenstein and many others.
  653. Sporadic groupsTits Group, 20 relatives of Fischer's Monster, 6 pariahs.
  654. Classical groups:  Their elements depend on parameters from a  field.
  655. The Möbius group  consists of homographic transformations of  CÈ{¥}.
  656. Lorentz transformations  may  change spatial orientation or time direction.
  657. Symmetries of the laws of nature:  A short primer.

    Ring Theory

  658. Rings  are sets endowed with addition, subtraction and multiplication.
  659. Divisors of zeroidempotent  elements and  nilpotent  elements.
  660. Nonzero characteristic:  Least  p  for which all sums of  p  like terms vanish.
  661. Ideals  within a ring are  multiplicatively absorbent  additive subgroups.
  662. Quotient ring, modulo an ideal:  The residue classes modulo that ideal.
  663. Cauchy multiplication  is well-defined for "formal power series" over a ring.
  664. Ring of polynomials  whose coefficients are in a given ring.
  665. Galois rings.  Residues of modular polynomials,  modulo  one of them.

    Fields, Galois Fields and Skew Fields

  666. Vocabulary:  We consider  skew fields  to be  noncommutative.  Some don't.
  667. Fields  are commutative rings where every nonzero element has a reciprocal.
  668. Wedderburn's TheoremFinite  division rings are necessarily  commutative.
  669. Every  finite  integral domain is a field.
  670. Galois fields  are the  finite fields.  Their orders are powers of primes.
  671. The trivial field  is a singleton.  It's the only field where 0 is invertible.
  672. Splitting field  of P in F[x] :  Smallest extension of F where P fully factors.
  673. Conway's Nim-Field  is algebraically complete.  It contains infinite ordinals.
  674. Ternary multiplication  compatible with  ternary addition  (without "carry").

    Vector Spaces  (over a field)  and  Modules  (over a ring)

  675. Vectors  were originally just differences between points in ordinary space...
  676. Abstract vector spacesVectors can be added, subtracted and  scaled.
  677. Dimension of a vector space:  The number of its independent generators.
  678. Subspaces.  Intersection.  Sum.  Direct sums of supplementary subspaces.
  679. Linear maps  between vector spaces respect addition and scaling.
  680. Quotient of two vector spacesHyperplanes  have  codimension  1.
  681. Fundamental theorem of linear algebra  and  rank theorem.
  682. Modules  are vectorial structures over a  ring of scalars  (instead of a  field).
  683. Normed vector spaces.  The fundamental properties of a  norm.
  684. Dual space:  The set of all [continuous] linear functions with scalar values.
  685. Lebesgue spacesSequence spaces  exemplify more general types.
  686. Tensors:  Multilinear functions of vectors and covectors with scalar values.
  687. Algebra:  A vector space with a scalable and distributive internal product.
  688. Clifford algebra:  Unital associative algebra endowed with a quadratic form.
  689. Things that are not vectorial  because they're not defined  intrinsically.
  690. David Hestenes proposed  geometric calculus  as a denotational unification.

    Convex Geometry

  691. Convex sets  in a  real  vector space.
  692. Aspect ratio:  The  length  divided by the  height  (i.e., the smallest width).
  693. A norm is characterized  by a closed convex body, symmetric about  0.
  694. Convex hullConv (S)  is the smallest convex set containing the set  S.
  695. Closed halfspaces  generate all  closed convex sets  by intersection.
  696. Polar of a  closed  convex set.  Dot-product duality among convex bodies.
  697. Separating hyperplane  (in the loose sense)  between disjoint convex sets.
  698. A compact convex  can be  strictly  separated from a disjoint closed convex.
  699. Two disjoint open convexes  are separated by a nonintersecting hyperplane.

    Functional Analysis

  700. Functionals  assign scalar values to  some  functions over an infinite set I.
  701. Eduard Helluy (1912):  The space  C[a,b]  (continuous functions over [a,b]).
  702. Sublinear functionals  are merely  subadditive  and  positively homogeneous.
  703. Hahn-Banach extension theorem.  Extending a dominated linear functional.
  704. Hahn-Banach separation theorem.  A different view of the same result.
  705. Generalization of Hahn-Banach  to complex or quaternionic linear spaces.

    Ring of p-adic Integers, Field of p-adic Numbers

  706. The ring of p-adic integers.  Objects with infinitely many radix-p digits.
  707. Polyadic integers:  Greek naming of p-adic integers.
  708. What if p isn't prime?  Dealing with  divisors of zero.
  709. Decadic Integers:  The strange realm of 10-adic integers  (composite radix).
  710. Decadic puzzle:  A tribute to the late columnist  J.A.H. Hunter  (1902-1986).
  711. The field of p-adic numbersQuotient field  of the ring of p-adic integers.
  712. Dividing two p-adic numbers  looks like "long division", only backwards...
  713. Overbar notation,  for p-adic and rational numbers alike.
  714. The p-adic metric can be used to define p-adic numbers analytically.
  715. The reciprocal of a p-adic number  computed by successive approximations.
  716. Ratios of rational integers have two representations:  g-adic and radix-g.
  717. Solving algebraic equations  in p-adic integers.
  718. Q-linear maps  between  R  and  Qp  are discontinuous at every point.
  719. Hasse's local-global principle.  Established for the quadratic case in 1920.
  720. Rotating digit patterns  (in base g)  may double the corresponding values.
  721. Rotating digits one place to the left divides some integers by k.


  722. Pseudoprimes to base aPoulet numbers  are pseudoprimes to base 2.
  723. Weak pseudoprimes to base a :  Composite integers  n  which divide  (an-a).
  724. Counting the bases  to which a composite number is a pseudoprime.
  725. Strong pseudoprimes to base a  are less common than Euler pseudoprimes.
  726. The witnesses of a composite number:  At least  75%  of nontrivial bases.
  727. Rabin-Miller Test:  An efficient and trustworthy  stochastic  primality test.
  728. The product of 3 primes  is a pseudoprime when all  pairwise  products are.
  729. Wieferich primes  are scarce but there ought to be infinitely many of them.
  730. Super-pseudoprimesAll  their composite divisors are pseudoprimes.
  731. Maximal super-pseudoprimes  have no super-pseudoprime multiples.

    Factoring into Primes

  732. Jevons Number.  Factoring  8616460799  is now an  easy  task.
  733. Challenges  help tell  special-purpose  and  general-purpose  methods apart.
  734. Special cases  of  a priori  factorizations are helpful to number theorists.
  735. Trial division  may be used to weed out the small prime factors of a number.
  736. Ruling out factors  can speed up trial divison in special cases.
  737. Recursively defined sequences  (over a  finite  set)  are  ultimately periodic.
  738. Pollard's rho factoring method  is based on ultimately periodic sequences.
  739. Pollard's p-1 Method  finds prime factors  p  for which  p-1  is  smooth.
  740. Williams' p+1 Method  is based on the properties of Lucas sequences.
  741. Lenstra's Elliptic Curve Method  generalizes Pollard's p-1 approach.
  742. Dixon's method:  Combine small square residues into a solution of   x 2 º y 2

    Quadratic Reciprocity

  743. Motivation:  On the prime factors of some quadratic forms...
  744. Quadratic residues:  Half of the nonzero residues modulo an odd prime  p.
  745. Euler's criterion:  A quadratic residue raised to the power of  (p-1)/2  is 1.
  746. The Legendre symbol  (a|p)  extends to values of  p  besides  odd primes.
  747. The law of quadratic reciprocity  states a simple but surprising fact.
  748. Gauss' Lemma  expresses a  Legendre symbol  as a product of many  signs.
  749. Eisenstein's Lemma:  A variation of  Gauss's lemma  allows a simpler proof.
  750. One of many proofs of the  law of quadratic reciprocity.
  751. Artin's Reciprocity.

    Continued Fractions  (and related topics)

  752. What is a continued fraction?  Example:  The expansion of p.
  753. The convergents of a number are its best rational approximations.
  754. Large partial quotients allow very precise approximations.
  755. Regular patterns in the continued fractions of some irrational numbers.
  756. In almost all cases, partial quotients are ≥ k with probability  lg(1+1/k).
  757. Elementary operations on continued fractions.
  758. The Baire space:  Continued fraction expansions of  irrationals  in  [0,1].
  759. Expanding functions as continued fractions.
  760. Engel expansions of positive numbers are nondecreasing integer sequences.
  761. Pierce expansions of numbers from 0 to 1.  Strictly increasing sequences.

    Recreational Mathematics

  762. Counterfeit Coin: In 3 weighings, find an odd object among 12, 13 or 14.
  763. Counterfeit Penny Problem:  Find an odd object in the fewest weighings.
  764. Seven-Eleven: Four prices with a sum and product both equal to 7.11.
  765. Equating a right angle and an obtuse angle, with a clever false proof.
  766. Choosing a raise: Trust common sense, beware of  fallacious accounting.
  767. 3 men pay $30 for a $25 hotel room, the bellhop keeps $2... Is $1 missing?
  768. Chameleons: A situation is unreachable because of an invariant quantity.
  769. Sam Loyd's 14-15 puzzle also involves an invariant quantity (and 2 orbits).
  770. Einstein's riddle: 5 distinct colors, nationalities, drinks, smokes and pets.
  771. Numbering n pages of a book takes this many digits (formula).
  772. The Ferry Boat Problem (by Sam Loyd): To be or not to be ingenious?
  773. Hat overboard !   What's the speed of the river?
  774. All digits once and only once: 48 possible sums (or 22 products).
  775. 2-people bridge crossed by 4 people (U2).  Four paces, one flashlight!
  776. Managing supplies to travel 6 days while carrying enough for only 4 days.
  777. Go south, east, north and you're back... not necessarily to the North Pole!
  778. Icosapolis: Numbering a 5 by 4 grid so adjacent numbers differ by at least 4.
  779. Unusual mathematical boast for people born in 1806, 1892, or 1980.
  780. Puzzles for extra credit: From Chinese remainders to the Bookworm Classic.
  781. Simple geometrical dissection:  A proof of the Pythagorean theorem.
  782. Early bird saves time by walking to meet incoming chauffeur.
  783. Sharing a meal: A man has 2 loaves, the other has 3, a stranger has 5 coins.
  784. Fork in the road: Find the way to Heaven by asking only one question.
  785. Proverbial Numbers: Words commonly associated with some numbers.
  786. Riddles: The Riddle of the Sphinx and other classics, old and new.

    The Mathematical Games of Martin Gardner

  787. Martin Gardner (1914-2010)  described himself as "strictly a journalist".
  788. FlexagonsHexaflexagons  were popularized by Martin Gardner in 1956.
  789. Polyominoes:  The 12 pentominoes and other tiles invented by  Sol Golomb.
  790. Soma:  7 nonconvex solids consisting of  3 or 4  cubes make a larger cube.
  791. Tessellations by convex pentagons.  The contributions of  Marjorie Rice.
  792. Kites and Darts.  The  aperiodic  tilings of Roger Penrose.
  793. Ambigrams:  Calligraphic spellings which change when rotated or flipped.
  794. The Game of Life.  John Conway's  endearing  cellular automaton  (1970).
  795. Rubik's Cube:  Ernõ Rubik (1974)  Singmaster (1979)  Gardner (1981).
  796. It's impossible to tie a knot  without letting go of the ends of the string.
  797. On the limited knowledge of Man.  An Indian legend...

    Mathematical "Magic" Tricks

  798. 1089:  Subtract a 3-digit number and its reverse, then...
  799. Multiples of Nine:  A secret symbol is revealed.
  800. Casting Out Nines:  A missing digit is revealed.
  801. Triple threat  mind reading.
  802. Mass media mentalism  by  David Copperfield  (1992).
  803. Grey Elephants in Denmark:  Classroom  mental magic.
  804. Fitch Cheney's 5-card trick:  4 cards tell the fifth one.
  805. Generalizing the 5-card trick and  Devil's Poker...
    Clubs Hearts Spades Diamonds
  806. Kruskal's Count.
  807. Paths to God.
  808. Stacked Deck.
  809. Enigma Card Trick.
  810. Magic Age Cards.
  811. Ternary Cards.
  812. Magical 21  (or 27).
  813. The Final 3  are the chosen cards.
  814. Boolean Magic.
  815. Perfect Faro Shuffles.
  816. Equal Numbers of  Heads !

    Illusions and Deceit

  817. Deceit and lying.
  818. Misdirection.
  819. Find the Lady.
  820. Cups and balls.  One of the most ancient tricks.
  821. Chop cup.  Invented by "Chop-Chop" Wheatley in 1954.
  822. Invisible Thread Reel  (ITR)  by  James George  (1992).
  823. Force and Reveal:  A whole class of magic tricks.

    Mathematical Games (Strategies)

  824. Dots and Boxes: The "Boxer's Puzzle" position of Sam Loyd.
  825. The Game of Nim: Remove items from one of several rows. Don't play last.
  826. Sprague-Grundy numbers are defined for all positions in impartial games.
  827. Moore's Nim:  Remove something from at most (b-1) rows.  Play last.
  828. Normal Kayles:  Knocking down a pin or two adjacent pins may split a row.
  829. Grundy's Game:  Split a row into two unequal rows, if at all possible.
  830. Wythoff's Game:  Take either from one heap or equally from both heaps.

    The Game of Chess

  831. Nalimov Tables:  Perfect analysis of endgame situations.
  832. Evaluation function:  Estimating a quiescent position statically.
  833. Minimax search tree:  The basic paradigm for analyzing two-player games.
  834. Alpha-beta pruning.  Iin a minimax search, some alternatives can be ignored.
  835. Hash tables.  How to avoid analyzing the same position more than once.
  836. Short chess games.  Checkmates occuring during the opening moves.
  837. Classic traps:  Fishing pole, etc.

    Ramsey Theory

  838. The pigeonhole principle:  What's entailed by fewer holes than pigeons.
  839. Among 70 distinct integers between 1 and 200, two must differ by 4, 5 or 9.
  840. n+1 of the first 2n integers  always include two which are coprime.
  841. Largest sets of small numbers with at most  k  pairwise coprime integers.
  842. Ramsey's Theorem:  Monochromatic complete subgraphs of a large graph.
  843. Infinite alignment among infinitely many lattice points in the plane?  Nope.
  844. Infinite alignment in a lattice sequence with bounded gaps?  Almost...
  845. Large alignments in a lattice sequence with bounded gaps.  Yeah!
  846. Van der Waerden's theorem:  Long monochromatic arithmetic progressions.
  847. Happy-Ending Problem:  Unavoidable  convex n-gons  among  m  points.


  848. Ford circles:  Nonintersecting circles touching the real line at rational points.
  849. Farey series:  The rationals from 0 to 1, with a bounded denominator.
  850. The Stern-Brocot tree  features every positive rational once and only once.
  851. Any positive rational  is a unique ratio of two consecutive Stern numbers.
  852. Pick's formula gives the area of a lattice polygon by counting lattice points.

    History, Nomenclature, Vocabulary, etc.

    History :
  853. Earliest mathematics on record. Before Thales was Euphorbus...
  854. Indian numeration became a positional system with the introduction of zero.
  855. Roman numerals are awkward for larger numbers.
  856. The invention of logarithms: Napier, Bürgi, Briggs, St-Vincent, Euler.
  857. The earliest mechanical calculators.  W. Shickard (1623)  &  Pascal (1642).
  858. The Fahrenheit Scale: 100°F  was meant to be the normal body temperature.
  859. The revolutionary innovations  which brought about new civilizations.
    Nomenclature & Etymology :
  860. The origin of the word "algebra", and also that of "algorithm".
  861. The name of the avoirdupois system  is from a  pristine  form of French.
  862. Long Division:  Cultural differences in long division layouts.
  863. Is a parallelogram a trapezoid? In a mathematical context [only?], yes it is...
  864. Naming polygons. Greek only please; use hendecagon not "undecagon".
  865. Chemical nomenclature:  Sequential names are  systematic  or traditional.
  866. Fractional prefixes: hemi (1/2) sesqui (3/2) hemipenta (5/2) hemisesqui (3/4).
    • Matches, phosphorus, and phosphorus sesquisulphide.
  867. Zillion. Naming large numbers.
  868. Zillionplex. Naming huge numbers.

    Style and Usage

  869. Abbreviations:  Abbreviations of scholarly Latin expressions.
  870. "Resp."  is a  mathematical symbol  with its own syntax.
  871. Typography of long numbers.
  872. Intervals denoted with square brackets (outward for an excluded extremity).
  873. Dates  in the simplest ISO 8601 form  (with  customary  time stamps or not).
  874. The names of operands in common numerical operations.
  875. Spoken numbers.
  876. Pronouncing mathematical expressions, like native English speakers do.
  877. PEMDAS:  A mnemonic for a rule that  should not  be taught.
  878. Physical units:  Their products and their ratios.
  879. The word  respectively  doesn't have the same syntax as "resp."

    Setting the Record Straight

  880. The heliocentric system was known two millenia before Copernicus.
  881. The assistants of Galileo and the mythical experiment at the Tower of Pisa.
  882. Switching calendars: Newton was not born the year Galileo died.
  883. The Lorenz Gauge is due to Ludwig Lorenz (1829-1891) not H.A. Lorentz.
  884. Special Relativity was first formulated by Henri Poincaré.
  885. The Fletcher-Millikan "oil-drop" experiment isn't entirely due to Millikan.
  886. Collected errata  about customary physical units.
  887. Portrait of Legendre:  The  mathematician  was confused with a politician.
  888. The iconography used for Apollonius of Perga  was meant for another man.
  889. Dubious quotations:  Who  really  said that?

    Ancient Knowledge

  890. Classical geometry  describes an  homogeneous  space indifferent to scale.
  891. Obliquity of the ecliptic  in the time of Eratosthenes (276-194 BC).
  892. Vertical wells at Syene are completely sunlit only once a year, aren't they?
  893. Eratosthenes sizes up the Earth:  700 stadia per degree of latitude.
  894. The distance to the Moon  was computed by  Aristarchus  and  Hipparchus.
  895. Latitude and longitude:  The spherical grid of meridians and parallels.
  896. Itinerary units:  The  land league  and the  nautical league.
  897. Amber, compass and lightning:  Glimpses of electricity and magnetism.

    The Scientific Method

  898. On the nature of physical laws:  The example of gravitation.
  899. Controlled Experiment:  A concept attributed to Sir Francis Bacon (1590).
  900. History of the Scientific Method.
  901. Distinguishing between Science and  Pseudoscience.
  902. Faster-than-light neutrinos?  How the media butcher the  scientific method.

    The Arrow of Time

  903. What is time?  Why don't we remember the future?
  904. The beginning of time.  Was there anything before that?
  905. Time machines:  Unavoidable microscopically, impossible macroscopically.
  906. Determinism  precludes the arrow of time.
  907. GPS time  is now universally available very inexpensively.


  908. Introduction:  Geometry, statics, kinematics, dynamics and beyond...
  909. The notion of force.  Statics,  mechanical advantage  and  virtual work.
  910. Speed.  Allowing the division of  unlike  quantities  (distance and time).
  911. Mean-speed theorem.  The distance traveled at constant acceleration.
  912. The timing experiments of Galileo:  From the  pendulum  to falling bodies.
  913. The true period of a pendulum  is proportional to   1 / agm ( 1 , cos A/2 ).
  914. The  parabola  of a cannonball,  compared to Aristotle's  triangular  path.
  915. Conservation of momentum  is key to  Newton's three laws of motion.
  916. The  work done  to a point-mass  equals the change in its  kinetic energy.
  917. Relativistic work done  and the corresponding change in  relativistic energy.
  918. Relativistic thermodynamics:  A point-mass endowed with internal heat.
  919. Spacecraft speeds up upon reentry into the upper atmosphere.
  920. Lewis Carroll's monkey climbs a rope over a pulley, with a counterweight.
  921. Two-ball drop can make one ball bounce up to 9 times the dropping height.
  922. Normal acceleration  =  Square of speed divided by the radius of curvature.
  923. Roller-coasters  must rise more than half a radius above any  loop-the-loop.
  924. Conical pendulum:  A hanging bob whose trajectory is an horizontal circle.
  925. Conical pendulum constrained by a hemisphere:  The string tension.
  926. Ball in a BowlPure rolling  increases the period of oscillation by 18.3%.
  927. Hooke's LawSimple harmonic motion  of a mass suspended to a spring.
  928. Speed of an electron estimated with the Bohr model of the atom.
  929. Hardest Stuff:  Diamond is no longer the hardest material known to science.
  930. Hardness  is an elusive  nonelastic  property, distinct from  stiffness.
  931. Hot summers, hot equator! The distance to the Sun is not the explanation.
  932. Kelvin's Thunderstorm:  Using falling water drops to generate high voltages.
  933. The Coriolis effect:  A dropped object falls to the east of the plumb line.
  934. Terminal velocity  of an object falling in the air.
  935. Angular momentum and torque.  Spin and orbital angular momentum.

    Motion of Rigid Bodies  (Classical Mechanics)

  936. Rotation vector  of a moving rigid body (and/or "frame of reference").
  937. Angular momentum  equals  moment of inertia  times  angular velocity.
  938. Kinetic energy of a solid:  Sum of its translational and rotational energies.
  939. Moments about a point or a plane  are convenient mathematical fictions.
  940. Moment of inertia of a spherical distribution  or an  homogeneous ellipsoid.
  941. Moment of inertia of the Earth  is equal to  0.330695 M a 2.
  942. Perpendicular Axis Theorem:  Axis of rotation perpendicular to a  lamina.
  943. The Parallel Axis Theorem:  Moment of inertia about an off-center axis.
  944. Moment of inertia of a thick plate,  derived from the  parallel axis theorem.
  945. Moment of inertia of a right cone  or  conical frustum.
  946. Momenta of homogeneous bodies.  List of common examples.
  947. Rigid pendulum  moving under its own weight about a fixed horizontal axis.
  948. Reversible pendulum.  The same period around two distinct axes.  Isaac Newton 

    Newtonian Gravity

  949. All physical theories  have a limited range of validity.
  950. Gravity vs. Electrostatics:  Straight comparisons.
  951. Binet's formulas:  Deriving Kepler's laws for two orbiting bodies.
  952. Airy weighs the Earth  by timing a pendulum deep in a mine.
  953. Rigid equilateral triangle  formed by three gravitating bodies.
  954. The five Lagrange points  of two gravitating bodies in circular orbit.
  955. Geosynchronous Orbit:  Semimajor radius of 36000 km around the Earth.
  956. The gravitational self-energy  of a ball  (mass M, radius R)  is  -0.6 GM2/R
  957. Tides on Earth:  Dominant rôle of the Moon.  Lesser rôle of the Sun.
  958. Asteroid 99942 Apophis:  Near-Earth objects and  gravitational keyholes.
  959. Mass distributions of galaxies.  Evidence for the existence of  dark matter.

    Friction & Dissipative Mechanics

  960. Coefficients of friction: ds the kinetic one.
  961. Example  involving a nontrivial choice between static and kinetic regimes.
  962. Minimum inclination of a ladder  leaning against a frictionless wall.
  963. Spinning cylinder on an horizontal plane:  The skidding before pure roll.Drawing a dotted line on a blackboard.
  964. Coefficient of restitution  (e)  Ratio of initial to final closing speed.

    The Physics of Billiards  (Classical Mechanics)

  965. Billiard and pool tables:  Sizes, slate bed, cloth, rails and cushions.
  966. Billiard balls:  Phenolic resin binding a dense powder has replaced ivory.
  967. Cue sticks:  Butts and shafts.  Basic construction.  Anti-squirt technology.
  968. The contents of a cue case  reflect the player's basic choices.
  969. Cue tips.  Leather and phenolic tips.
  970. Two types of billiard chalk  to reduce hand friction or increase tip friction.
  971. Normal trajectory of a billiard ball:  A parabola followed by a straight line.
  972. Making the cue ball stop  after hitting the object ball.
  973. The impossible 90° cut-shot  made possible with extreme english.
  974. Squirt between cue and cue ball with extreme English  (vertical spin axis).
  975. Jump shots.  Legal and illegal ways to send the cue ball up in the air.

    Geometrical Optics

  976. Matrix methods:  Transformations of a ray's inclination and radial distance.
  977. A crystal ball  (index n and radius R)  has focal length  f = R / (2n-2).
  978. Lensmaker's formula  Focal lens as a function of signed curvatures.
  979. Concave mirrors  create enlarged virtual images of objects in front of them.
  980. Opposition effect  increases albedo by eliminating micro-shadows.


  981. Huygens' Principle.  A convenient fiction to describe wave propagation.
  982. Diffraction  occurs when when a wave emanates from a bounded source.
  983. Young's  double-slit  experiment  demonstrates the wavelike nature of light.
  984. Celerity  is the speed with which  phase  propagates.
  985. Standing waves  feature stationary nodes and antinodes.
  986. Chladni patterns:  The lines formed by nodes in an oscillating  plate.
  987. Snell's Law (1621)  gives the angle of refraction (if anything is refracted).
  988. Birefringence.  Discovery of  polarization  (Erasmus Bartholinus, 1669).
  989. Brewster's angle  is the incidence which yields a 100% polarized reflection.
  990. Fresnel equations:  Reflected or refracted intensities of polarized light.
  991. Stokes parameters:  A standard description of the  state of polarization.
  992. Transverse wave on a rope:  (celerity) 2 = (tension) / (linear mass density).

    Colors & Dispersion

  993. Dispersion relation: Pulsatance vs. wave number; frequency vs. wavelength.
  994. Group velocity  is the traveling speed of a beat phenomenon.
  995. Rayleigh scattering  makes the sky blue and sunsets red.
  996. Index of refraction of water  for light of different colors.
  997. A spherical drop  reflects light back (red up to 42.34° & violet up to 40.58°).
  998. The  length  of a rainbow:  Mathematical digression.

    Infrared Transmissions   (of remote control codes  &  data)

  999. Wavelength:  940 nm (319 THz) is the most common specification for IR.
  1000. Modulation:  38 kHz (38.4 kHz).  Also:  30, 33, 36, 36.7, 40, 56, 455 kHz.
  1001. Remote shutter release for cameras.  The simplest type of infrared control.
  1002. On/off patterns  to encode data bits and the start/stop of data frames.
  1003. Unexplained datasheet mysteries.  Why are Rohm's specs slightly off?
  1004. Discrete IR control codes  provide critical functions for automated control.
  1005. RC-5 and RC-6.  Philips and the well-documented  European protocol.
  1006. NEC Protocol.  The Japanese format.
  1007. SIRC Protocol  by Sony.
  1008. RCA Protocol:  64 ms to send a 4-bit address and 8-bit data at 56 kHz.
  1009. HP 82240B:  The standard printer for HP scientific calculators, since 1989.
  1010. Serial protocol:  2400 Bd, 1 start bit, 8 data bits, 1 stop bit, odd parity.
  1011. Resurrrecting 455 kHz modulation  to transmit at high-speed  (19200 Bd).

    Lasers :  From masers to laser beams

  1012. Stimulated emission  is crucial to blackbody equilibrium  (Einstein, 1916).
  1013. Bose-Einstein Statistics  is what explains stimulated emission of bosons.
  1014. Population inversion :  When energetic states are abnormally abundant.
  1015. LASER Cavity  "Light Amplification by Stimulated Emission of Radiation".
  1016. Gaussian beam.  The shape of an ideal laser beam.

    Analytical Mechanics  &  Classical Field Theory

  1017. Fermat's principle  (least time)  for light (c.1655) predates Newton.
  1018. Maupertuis principle  of  least action  (1744).
  1019. Virtual Work:  A substitute for Newton's laws that cancels constraint forces.
  1020. Phase Space:  A   phase  describes completely the state of a classical system.
  1021. Either velocities or momenta  are added to configuration to specify a  phase.
  1022. Relativistic point-mass:  Lagrangian, Hamiltonian and free momentum.
  1023. Charge in a magnetic field:  The canonical momentum isn't the linear one.
  1024. The Lagrangian  is a function of positions and velocities.
  1025. The Hamiltonian  depends on positions and momenta.
  1026. Poisson brackets:  An abstract synthetic view of analytical mechanics.
  1027. Liouville's theorem:  The Hamiltonian  phase volume  doesn't change.
  1028. Noether's theorem:  Conservation laws express the symmetries of physics.
  1029. Field theory:  Lagrangian function of a continuum of values and velocities.

    Electromagnetism  (Maxwell's Equations)

  1030. Clarifications:  Vector calculus (Heaviside) & microscopic view (Lorentz).
  1031. The vexing problem of units  is a thing of the past if you stick to SI units.
  1032. The Lorentz force  on a test particle defines the local electromagnetic fields.
  1033. Electrostatics (1785):  The study of the electric field due to static charges.
  1034. Electric capacity  is an electrostatic concept  (adequate at low frequencies).
  1035. Electrostatic multipoles:  The multipole expansion of an electrostatic field.
  1036. Birth of electromagnetism (1820):  Electric currents create magnetic fields.
  1037. Biot-Savart Law:  The  static  magnetic induction due to steady currents.
  1038. Magnetic scalar potential:  A  multivalued  static scalar field.
  1039. Magnetic monopoles do not exist :  A law stating a fact not yet disproved.
  1040. Ampère's law (1825):  The law of static electromagnetism.
  1041. Faraday's law (1831):  Electric circulation induced by magnetic flux change.
  1042. Self-induction  received by a circuit from the magnetic field it produces.
  1043. Ampère-Maxwell law:  Dynamic generalization (1861) of  Ampère's law.
  1044. Putting it all together:  Historical paths to Maxwell's  electromagnetism.
  1045. Maxwell's equations  unify electricity and magnetism dynamically  (1864).
  1046. Continuity equation:  Maxwell's equations imply  conservation of charge.
  1047. Waves anticipated by Faraday, Maxwell & FitzGerald.  Observed by Hertz.
  1048. Electromagnetic energy density  and the flux of the Poynting vector.
  1049. Planar electromagnetic waves:  The simplest type of electromagnetic waves.
  1050. Maxwell-Bartoli radiation pressure.  First detected by  P. Lebedev  in 1899.
  1051. Electromagnetic potentials  are postulated to obey the  Lorenz gauge.
  1052. Solutions to Maxwell's equations,  as  retarded  or  advanced  potentials.
  1053. Electrodynamic fields  corresponding to  retarded  potentials.
  1054. Electrodynamic fields  corresponding to  advanced  potentials.
  1055. The gauge of retarded potentials:  is it  really  the Lorenz gauge?
  1056. Power radiated by an accelerated charge:  The Larmor formula (1897).
  1057. Lorentz-Dirac equation  for the motion of a point charge is of  third  order.

    Electromagnetic Dipoles

  1058. Molecular electric dipole moments.  First studied by Peter Debye in 1912.
  1059. Force exerted on a dipole  by a  nonuniform  field.
  1060. Torque on a dipole  is proportional to its cross-product into the field.
  1061. Electric and magnetic dipoles:  Dipolar solutions of Maxwell's equations.
  1062. Static distributions of magnetic dipoles  can be emulated by steady currents.
  1063. Static distributions of electric dipoles  are equivalent to charge distributions.
  1064. Field at the center of a uniformly magnetized or polarized sphere of any size.
  1065. Sign reversal  in magnetic and electric fields from matching dipoles.
  1066. Relativistic dipoles:  A moving magnet develops an electric moment.

    Magnetism,  Electromagnetic Properties of Matter

  1067. Magnetization and polarization  describe densities of  bound  dipoles.
  1068. Distinct magnetization and polarization  gauges  may yield the same field.
  1069. Maxwell's equations in matter:  Electric displacement & magnetic strength.
  1070. Electric susceptibility  is the propensity to be polarized by an  electric field.
  1071. Electric permittivity and magnetic permeability.  Related to susceptibilities.
  1072. Paramagnetism:  Weak  positive  susceptibility.
  1073. DiamagnetismLorentz force turns orbital moments against an external B.
  1074. Magnetic levitation:  How to skirt the theorem of Samuel Earnshaw (1842).
  1075. Pyrolytic carbon:  The most diamagnetic substance, at room temperature.
  1076. Bohr-van Leeuwen Theorem:  Diamagnetism and paramagnetism cancel ?!
  1077. Thermodynamics of dielectric matter:  dU = E.dD + ...
  1078. Ferromagnetism:  Permanent magnetization without an external field.
  1079. Antiferromagnetism:  When adjacent dipoles tend to oppose each other...
  1080. Ferrimagnetism:  With two kinds of dipoles, partial cancellation may occur.
  1081. Magneto-optical effect  discovered by Faraday on September 13, 1845.
  1082. Ohm's Law:  Current density is proportional to electric field:  j = s E.

    Motors and Generators

  1083. Homopolar motor:  The first electric motor, by  Michael Faraday  (1831).
  1084. Faraday's disk  can generate huge currents at a low voltage.
  1085. Magic wheels:  Two repelling ring magnets mounted on the same axle.
  1086. Beakman's motor.  Current switches on and off as the coil spins.
  1087. Tesla turbine.  Stack of spinning disks with outer intake and inner outflow.

    The Vacuum

  1088. Aristotle's plenism.  Downfall of the  Horror Vacui  doctrine  (17th century).
  1089. Vacuum tubes.  Heated filaments, grids and electrons moving in a vacuum.
  1090. Dirac's equation  predicted  positrons as holes in a bizarre vacuum.
  1091. The Quantum Vacuum.  The vacuum isn't empty.  Structure of the vacuum.

    Special Relativity

  1092. Observers in motion:  An elementary derivation of the Lorentz Transform.
  1093. Adding up parallel velocities:  The combined speed can never exceed c.
  1094. Combining velocities  using an angle measured in a moving frame.
  1095. The headlight effect:  An isotropic source will radiate forward if it moves.
  1096. Closing speed:  The distance between objects may decrease faster than c.
  1097. Fizeau's empirical relation  between refractive index  (n) and  Fresnel drag.
  1098. The Harress-Sagnac effect  used to measure rotation with fiber optic cable.
  1099. Combining relativistic speeds:  Using rapidity, the rule is transparent.
  1100. Relative velocity of two photons: Undefined if they have the same direction
  1101. Minkowski spacetime:  Lorentz transform applies to 4-vector coodinates.
  1102. The Lorentz transform expressed vectorially  for a  boost  of speed  V.
  1103. Wave vector:  The 4-dimensional gradient of the phase describes a wave.
  1104. Doppler shift:  The relativistic effect is not purely radial.
  1105. Relativistic momentum  and Einstein's relation between mass and energy.
  1106. Kinetic energy:  At low speed, the relativistic energy varies like  ½ mv 2.
  1107. Photons and other massless particles:  Finite energy at speed  c.
  1108. The de Broglie celerity  (u)  is inversely proportional to a particle's speed.
  1109. Compton diffusion:  The result of collisions between photons and electrons.
  1110. The Klein-Nishina formula:  gives the  cross-section  in Compton scattering.
  1111. Compton effect is suppressed  for visible light and bound electrons.
  1112. Elastic shock:  Energy transfer is  v.dp.  (None is seen from the barycenter.)
  1113. Photon-photon scattering  is like an  elastic collision of two photons.
  1114. Cherenkov effect:  When an electron exceeds the celerity of light...
  1115. Constant acceleration  over an entire lifetime will take you  pretty far.


  1116. Photons  are quanta of light that are  both  wavelike and corpuscular.
  1117. The photoelectric effect  was explained by Albert Einstein in 1905.

    Nuclear Physics

  1118. Henri Becquerel  and the discovery of natural radioactivity (1896).
  1119. Pierre & Marie Curie:  The discovery of new radioactive elements (1898).
  1120. Geiger-Marsden experiment:  There's a tiny dense nucleus inside the atom!
  1121. Alpha-decay:  Polonium (Po-210, Z=84) decays into Lead (Pb-206, Z=82).
  1122. Mass Defect:  In a nuclear reaction, the Q-value balances the mass change.
  1123. The  standard  decay modesa, b-, 2b-, b+, e (electron capture)  or  IT.
  1124. The 4 radioactive series:  Thorium, Neptunium, Uranium and Actinium.
  1125. Other decay modes:  Proton or neutron emission, fission and spallation.
  1126. The Geiger counter  measures the  activity flux  of ionizing radiation.
  1127. Scintillation  allows quantitive measurements of a gamma spectrum.
  1128. Cross-section:  The target's size depends on the projectile's speed.
  1129. Artificial radioactivity:  Neutron bombardment creates unstable nuclides.
  1130. Chain reactions:  When neutron-induced decays produce more neutrons...
  1131. Critical mass:  The smallest mass that will allow runaway chain reactions.
  1132. Thermonuclear bombs.  Nuclear fusion ignited by fission devices.
  1133. Carbon-dating:  Radiocarbon ratio starts decaying when an organism dies.
  1134. Fusion of deuterons:  Helium is formed with liberation of energy.
  1135. The Proton-Proton chain fusion  powers all stars less than 1.5 solar masses.
  1136. Tokamak reactors:  Deuterium-Tritium fusion  (DT)  is the easiest to ignite.
  1137. Farnsworth-Hirsch fusor:  Controlled fusion on a desktop.  Neutron source.
  1138. Polywell reactor:  The design advocated by the late Robert Bussard.
  1139. Amateur nuclear physics:  Demystifying nuclear energy and radioactivity.
  1140. The Radioactive Boyscout  and other misguided experimenters.

    General Relativity

  1141. The Harress-Sagnac effect seen by an observer rotating with an optical loop.
  1142. Relativistic rigid motion  is an  equilibrium  modified at the speed of  sound.
  1143. In the Euclidean plane:  Contravariance and covariance.
  1144. In the Lorentzian plane:  Contravariance and covariance revisited.
  1145. Tensors of rank n+1  are linear maps that send a vector to a tensor of rank n.
  1146. Signature  of the quadratic form defined by a given metric tensor.
  1147. Covariant and contravariant coordinates  of rank-n tensors, in 4 dimensions.
  1148. The metric tensor and its inverse.  Lowering and raising indices.
  1149. Partial derivatives  along  contravariant  or  covariant  coordinates.
  1150. Christoffel symbols:  Coordinates of the partial derivatives of basis vectors.
  1151. Covariant derivativesAbsolute differentiation.  The  nabla  operator  Ñ.
  1152. Contravariant derivatives:  The lesser-known flavor of absolute derivatives.
  1153. The antisymmetric part of Christoffels symbols  form a fundamental  tensor.
  1154. Totally antisymmetric spacetime torsion  is described by a  vector field.
  1155. Levi-Civita symbols:  Antisymmetric with respect to any pair of indices.
  1156. Einstein's equivalence principle  implies  vanishing  spacetime torsion.
  1157. Ricci's theorem:  The covariant derivative of the metric tensor vanishes.
  1158. Curvature:  The  Ricci tensor  is a contraction of the  Riemann tensor.
  1159. The Bianchi identity  shows that the  Einstein tensor  is divergence free.
  1160. Stress tensor:  Flow of energy density is density of [conserved] momentum.
  1161. Einstein's Field Equations:  16 equations in covariant form (Einstein, 1915).
  1162. Free-falling bodies:  Their trajectories are  geodesics  in curved spacetime.
  1163. The "anomalous" precession of Mercury's perihelion  is entirely  relativistic.
  1164. Schwarzschild metric:  The earliest exact solution to Einstein's equations.
  1165. What is mass?
  1166. Unruh temperature experienced by an accelerating observer.
  1167. Electromagnetism:  Covariant expressions, using tensors.
  1168. Kaluza-Klein theory of electromagnetism  involves a  fifth dimension.
  1169. Harvard Tower Experiment:  The slow clock at the bottom of the tower.
  1170. Shapiro time delay:  Effect on radar signals of gravitational time dilation.

    String Theory and other "Theories of Everything"

  1171. Unification:  Consistency is required.  Actual high-energy unification is not.
  1172. Kaluza-Klein Theory:  Postulating an extra dimension for electromagnetism.
  1173. 1960's hadron physicsRegge trajectories  begat constant-tension strings.
  1174. Gabriele Veneziano:  The magic of Euler's  beta and gamma functions.
  1175. Leonard Susskind (1940-):  The basic idea of a fundamental string.
  1176. Joël Scherk (1946-1979) & John Schwarz:  Rediscovering  gravity.
  1177. Michael Green & John Schwarz:  Hoping for a  Theory of Everything.
  1178. String QuintetFive  different consistent string theories!
  1179. M-Theory:  Ed Witten's 11-dimensional brainchild, unveiled at  String '95.
  1180. The brane world scenarios  of  Lisa Randall  and  Burt Ovrut.

    Physics of Gases and Fluids

  1181. The Magdeburg hemispheres  are held together by more than a ton of force.
  1182. The ideal gas laws  of Boyle, Mariotte, Charles, Gay-Lussac, and Avogadro.
  1183. Joule's law:  Internal energy of an ideal gas depends only on temperature.
  1184. Deflating a tire:  Releasing pressurized gas into the atmosphere.
  1185. The Van der Waals equation and other interesting equations of state.
  1186. Virial equation of state.  Virial expansion coefficients.  Boyle's temperature.
  1187. Viscosity is the ratio of a shear stress to the shear strain rate it induces.
  1188. Permeability and permeance: Vapor barriers and porous materials.
  1189. Resonant frequencies of air in a box.
  1190. The Earth's atmosphere. Pressure at sea-level and total mass above.
  1191. The first hot-air balloon  (Montgolfière)  was demonstrated on June 4, 1783.
  1192. Sulfur hexafluoride  is a very heavy gas and a good electrical insulator.

    Transport Propertities of Matter

  1193. Viscosity:  The transport of microscopic momentum.
  1194. Brownian motion  and  Einstein's estimate of molecular sizes.
  1195. Thermal Conductivity:  The transport of microscopic energy.
  1196. Diffusivity:  The transport of  chemical concentration.
  1197. Speed of Sound:  Reversible transport of a pressure disturbance in a fluid.

    Filters and Feedback

  1198. Complex pulsatance:  s = s+iw  (damping constant + imaginary pulsatance)
  1199. Complex impedance:  Resistance and reactance.
  1200. Quality Factor (Q).  Ratio of maximal stored energy to dissipated power.
  1201. Nullators and norators:  Strange dipoles for analog electronic design.
  1202. Corner frequency  of a simple  first-order  low-pass filter.  -3 dB bandwidth.
  1203. Second-order  passive low-pass filter, with inductor and capacitor.
  1204. Two cascaded RC low-pass filters  can  almost  achieve critical damping.
  1205. Sallen-Key filters:  Active filters do not require inductors.
  1206. Lowpass Butterworth filter of order n :  The flattest low-frequency response.
  1207. Linkwitz-Riley crossover filters  used in modern active audio crossovers.
  1208. Chebyshev filters:  Ripples in either the passband or the stopband.
  1209. Elliptic (Cauer) filters  encompass all Butterworth and Chebyshev types.
  1210. Legendre filters  maximal roll-off rate for monotonous frequency response.
  1211. Gegenbauer filters:  From Butterworth to Chebyshev, via Legendre.
  1212. Phase response  of a filter.
  1213. Bessel-Thomson filters:  Phase linearity and group delay.
  1214. Gaussian filters:  Focusing on time-domain communication pulses.
  1215. Linear phase equiripple:  Ripples in group delay to go beyond Bessel filters.
  1216. DSL filters  allow POTS below 3400 Hz & block digital data above 25 kHz.
  1217. Switched capacitor:  Faking a resistor with a capacitor and a SPDT switch.

    Fantasy Engineering: Just for fun.

  1218. Raising the Titanic, with (a lot of) hydrogen.
  1219. Gravitational Subway:  From here to anywhere on Earth, in 42 minutes.
  1220. In a vacuum tube, a drop to the center of the Earth would take 21 minutes.

    Steam Engines, Heat Engines

  1221. The aeolipile:  This ancient steam engine demonstrates jet propulsion.
  1222. Edward Somerset of Worcester (1601-1667):  Steam fountain blueprint.
  1223. Denis Papin (1647-1714):  Pressure cooking and the first piston engine.
  1224. Thomas Savery (c.1650-1715):  Two pistons and an independent boiler.
  1225. Thomas Newcomen (1663-1729) & John Calley:  Atmospheric engine.
  1226. Nicolas-Joseph Cugnot (1725-1804):  The first automobile  (October 1769).
  1227. James Watt (1736-1819):  Steam condenser and  Watt governor.
  1228. Richard Trevithick (1771-1833) and the first railroad locomotives.
  1229. Sadi Carnot (1796-1832):  Carnot's cycle.  The theoretical  efficiency limit.
  1230. Sir Charles Parsons (1854-1931):  The modern  steam turbine  (1884).
  1231. Drinking Bird:  Room-temperature engine based on evaporative cooling.


  1232. Elementary concept of temperature.  The zeroth law of thermodynamics.  Lord Kelvin 
1824-1907  Hermann von Helmholtz 
  1233. Conservation of energy:  The first law of thermodynamics.
  1234. Increase of Entropy:  The second law of thermodynamics.
  1235. State variablesExtensive  and  intensive  quantities.
  1236. Entropy  is  missing information, a measure of  disorder.
  1237. Nernst Principle  (third law):  Entropy is zero at zero temperature.
  1238. Thermodynamic potentials  are convenient alternatives to  internal energy.
  1239. Calorimetric coefficients, adiabatic coefficient  (g)  heat capacities, etc.
  1240. Relations between  isothermal  and  isentropic  coefficients
  1241. The thermal Grüneisen parameter.
  1242. Entropy of a Van der Waals fluid  as derived from its equation of state.
  1243. Dulong-Petit Law (1819).  The molar heat capacity of a metal is about  3 R.
  1244. Thermal effects of molecular vibrations  at moderate temperatures.
  1245. Latent heat  (L)  is the heat transferred in a change of  phase.
  1246. Cryogenic coefficients:  Lower temperature with an  isenthalpic  expansion.
  1247. Relativistic Thermodynamics:  A moving body appears  cooler.
  1248. Inertia of energy  for an object at nonzero temperature.
  1249. Stefan's Law:  A black body radiates as the fourth power of its temperature.
  1250. The "Fourth Law":  Is there really an upper bound to temperature?
  1251. Hawking radiation:  On the entropy and temperature of a black hole.
  1252. Partition function:  The cornerstone of the statistical approach.

    Thermodynamics and Elasticity

  1253. Elastic properties  Reversible deformations in perfectly resilient materials.
  1254. Hysteresis and resilience.  Stored elastic energy is never fully recovered.
  1255. Elastomers.  Unsaturated rubbers are cured by  sulfur vulcanization.
  1256. Thermal expansion coefficients:  Cubical  scalar  and linear  tensor.
  1257. Invar  anomaly:  The low thermal expansion of 36% Ni / 86% Fe alloy.
  1258. Waves in a solid: P-waves (fastest), S-waves, E-waves (thin rod), SAW...
  1259. Thermodynamics of acoustics:  Dynamic coefficients and isothermal ones.
  1260. Rayleigh Wave: The quintessential surface acoustic wave (SAW).

    Demons of Classical Physics

  1261. Laplace's Demon:  Deducing past and future from a detailed snapshot.
  1262. Maxwell's Demon:  Trading information for entropy.
  1263. Shockley's Ideal Diode Equation:  Diodes don't violate the Second Law.
  1264. Szilard's engine & Landauer's Principle: Thermodynamic cost of  forgetting.

    Statistical Physics

  1265. Lagrange multipliers.  One multiplier for each constraint of an optimization.
  1266. Microcanonical equilibrium.  Isolated system:  All states are equiprobable.
  1267. Equipartition of energy.  Every degree of freedom gets an equal share.
  1268. Canonical equilibriumBoltzmann factor  in a heat bath.
  1269. Grand-canonical equilibrium  when  chemical  exchanges are possible.
  1270. Bose-Einstein statistics:  One state may be occupied by  many  particles.
  1271. Fermi-Dirac statistics:  One state is occupied by  at most one  particle.
  1272. Boltzmann statistics:  The  low-occupancy limit  (most states unoccupied).
  1273. Maxwell-Boltzmann distribution  of molecular speeds in an  ideal gas.
  1274. Partition function:  The cornerstone of the statistical approach.

    Quantum Mechanics

  1275. Quantum Logic:  The surprising way quantum probabilities are obtained.
  1276. Swapping particles either negates the quantum state or leaves it unchanged.
  1277. The Measurement Dilemma:  What makes  Schrödinger's cat  so special?
  1278. Matrix Mechanics:  Like measurements, matrices don't commute.
  1279. Schrödinger's Equation:  Nonrelativistic quantum particle in a classical field.
  1280. Noether's Theorem:  Conservation laws express the symmetries of physics.
  1281. Kets  are Hilbert vectors (duals of bras) on which observables operate.
  1282. Observables  are operators explicitely associated with physical quantities.
  1283. Commutators are the quantities which determine  uncertainty relations.
  1284. Uncertainty relations  hold whenever the commutator does not vanish.
  1285. Spin  is a form of angular momentum without a classical equivalent.
  1286. Pauli matrices:  Three 2 by 2 matrices with  eigenvalues  +1 and -1.
  1287. Quantum Entanglement:  The  singlet  and  triplet  states of two electrons.
  1288. Bell's inequality  is violated for the  singlet  state of two electron spins.
  1289. Generalizations of Pauli matrices  beyond spin ½.
  1290. Density operators  are quantum representations of imperfectly known states.

    The Schrödinger Equation

  1291. Hamilton's analogy equates the principles of Fermat and Maupertuis.
  1292. Box confinement by a finite potential  in one dimension and 3 dimensions.
  1293. Rotator:  Quantization of the angular momentum.
  1294. Harmonic oscillator.
  1295. Coulomb potential:  Classification of chemical orbitals.

    Quantum Field Theory

  1296. Elementary particles:  Quarks and leptons.  Electroweak bosons.  Graviton?
  1297. Second Quantization:  Particles are modes of a quantized field.
  1298. Bethe-Salpeter Equation:  A relativistic equation for bound-state problems.

    Ancient Recipes and Modern Chemistry

  1299. Measuring chemical stuff in  moles  (mol)  makes  stoichiometry  obvious.
  1300. Modern distillation  (alembic = still-head)  is due to  Mary the Jewess.
  1301. The retort  was a prominent tool of alchemists and chemists for centuries.
  1302. Production and distillation of alcohol.  Its origins and limitations.
  1303. Black powder:  An ancient explosive, still used as a propellant (gunpowder).
  1304. Predicting explosive reactions:  A useful but oversimplified rule of thumb.
  1305. Thermite  generates temperatures hot enough to weld iron.
  1306. Enthalpy of Formation:  The tabulated data which gives energy balances.
  1307. Exothermic crystallization  of  sodium acetate trihydrate  ("hot ice").
  1308. Gibbs Function  (free energy):  Its sign tells the direction of spontaneity.
  1309. Berthollet's Law of Mass Action  governs every chemical equilibrium.
  1310. Labile  is not quite the same as  unstable.
  1311. Inks:  India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.
  1312. Traditional pigments:  Carbon black, vermillion, brazilin, malachite, etc.
  1313. Beeswax  is dominated by a long-chain ester  (a "wax")  called  mycerin.
  1314. Pine pitch & cedar pitch:  Two similar products with different properties.
  1315. Gum Arabic:  The  magic bullet  of ancient chemistry.
  1316. Ancient acids:  From vinegar and lemon juice to vitriolic acid and more.
  1317. Gold ChemistryAqua regia ("Royal Water") dissolves gold and platinum.
  1318. Who was the "father" of modern chemistry?

    Organic Chemistry

  1319. Birth of organic chemistryUrea  was first made  chemically  in 1828.
  1320. Aliphatic saturated hydrocarbons  are called  alkanes.
  1321. Unsaturated hydrocarbons  feature some carbons tied by multiple bonds.
  1322. Functional groups  determine the basic reactions of  organic chemistry.

    Ionization, Oxidation-Reduction and Electrochemistry

  1323. The oxidation number  increases by  oxidation  and decreases by  reduction.
  1324. Salt bridges  put solutions in electrical contact but prevent transfers of ions.
  1325. Nernst equation:  The voltage induced by different concentrations.
  1326. Redox Reactions:  Oxidizers are  reduced  by accepting electrons...

    Practical Chemistry

  1327. Basic glassware:  Flasks, funnels, tubes, bulbs, condensers, etc.
  1328. PTFE  =  Polytetrafluoroethylene  =  Teflon®.
  1329. Ground-glass joints:  Standard glass-to-glass conical joints have a 1:10 taper.
  1330. Titration.  Measuring the volume of a reactant of known concentration.
  1331. Chemistry set  from a bygone era  (if memory serves).
  1332. Waterlock:  1 g of  sodium polyacetate  can hold  825 mL  of water.
  1333. Negative-X:  Water ignites a mixture of zinc and  ammonium nitrate.
  1334. Nitrogen triiodide  Is an extremely unstable explosive when dry.

    Medicine by the Numbers

  1335. The normal body temperature  is  37°C  (98.6°C)  or is it?
  1336. Normal blood pressureSystolic  (max.)  and  diastolic  (min.) pressures.
  1337. Normal pulse.  1 Hz  (one hertz)  is  60  beats per minute.
  1338. Blood circulation  (1628).  Discovered by  William Harvey  (1578-1657).
  1339. Respiration  is a form of combustion  (Lavoisier  and  Laplace, 1780).
  1340. Normal caloric intake.  100 W  of power is about  2065 kcal/day.
  1341. International Unit  (IU) is an arbitrarily-defined rating of  biological activity.
  1342. Concentration  is an amount (either mass or moles) per volume.
  1343. Glycosylated hemoglobin  (HbA1c) relates to  average  blood glucose (bG).
  1344. Medical abbreviations  commonly used in prescriptions and elsewhere.

    Cosmology 101

  1345. Kant's Island Universes:  The Universe is filled with  separate  galaxies.
  1346. The Cosmological Principle: The Universe is homogeneous and isotropic.
  1347. The Big Bang:  An idea of Georges Lemaître  mocked by Fred Hoyle.
  1348. The Cosmic Microwave Background (CMB): Its spectrum and density.
  1349. Cosmic redshift (z) of light from a Universe (1+z) times smaller than now.
  1350. Multiple choices and misguided explanations for cosmic redshifts.
  1351. Hubble Law  relates  redshift  and  distance  for comoving points.
  1352. Omega (W): The ratio of the density of the Universe to the critical density.
  1353. Look-Back Time:  The time ellapsed since observed light was emitted.
  1354. Distance:  In a cosmological context, there are several flavors of distances.
  1355. Comoving points follow the expansion of the Universe.
  1356. The Anthropic Principle:  An unsatisfactory type of absolute constraint.
  1357. Dark matter & dark energy: Gravity betrays the existence of  dark  stuff.
  1358. The Pioneer Effect: The anomalous escape of the Pioneer spaceprobe.

    Galaxies and large-scale structure of the Universe

  1359. The Milky Way  is the name given to the star system that harbors our Sun.
  1360. The local group  is dominated by the Milky Way & Andromeda galaxies.
  1361. The virgo cluster  dominates our corner of the Universe.
  1362. Superclusters  are the largest objects in the Universe.

    Stars and Stellar Objects

  1363. Nuclear fusion  is what powers the stars.
  1364. Brown dwarves  glow from gravitational contraction.  Fusion isn't ignited.
  1365. Red dwarves  can burn hydrogen for  trillions  of years.
  1366. The Jeans mass  above which gases at temperature T collapse by gravitation.
  1367. Main sequence:  The evolution of a typical star.
  1368. Metallicity  (Z)  measures the abundance of all elements  beyond helium.
  1369. Eta Carinae  and  hypergiants.  The most massive stars possible.
  1370. Betelgeuse  and red supergiants.
  1371. Rigel  and blue supergiants.
  1372. Planetary nebulae:  Aftermaths of stellar explosions.
  1373. White dwarfs:  The ultimate fate of our Sun and other small stars.
  1374. Neutron stars:  Remnants from the supernova collapse of medium stars.
  1375. Stellar black holes  form when supermassive stars run out of nuclear fuel.
  1376. Binary stars:  Pairs of unlike stars often gravitate around each other.
  1377. Binary X-ray source:  A small  accretor  in tight orbit around a  donor  star.

    The Solar System

  1378. Astronomical unit  (au).  Successive definitions of a standard unit of length.
  1379. Mean distance between the Sun and the Earth,  A tad above  1 au.
  1380. Parsec:  Triangulating interstellar distances, using the motion of the Earth.
  1381. The solar corona  is a very hot region of rarefied gas.
  1382. Solar radiation:  The Sun has radiated away about 0.03% of its mass.
  1383. The Titius-Bode Law: A numerical pattern in solar orbits?
  1384. The 4 inner rocky planets:  Mercury, Venus, Earth, Mars.
  1385. Earth and MoonThis  is home.
  1386. The asteroid belt:  Planetoids and bolids between Mars and Jupiter.
  1387. The 4 outer giant gaseous planets: Jupiter, Saturn, Uranus, Neptune.
  1388. Discovery of NeptuneUrbain Le Verrier  scooped John Couch Adams.
  1389. Pluto  and other  Kuiper Belt Objects  (KBO).
  1390. Sedna  and other planetoids beyond the  Kuiper Belt.
  1391. What's a planet?  The latest definition excludes  Pluto.
  1392. Heliosphere and Heliopause:  The region affected by solar wind.
  1393. Oort's Cloud  is a cometary reservoir at the fringe of the Solar System.

    Practical Formulas

  1394. Easy conversion between Fahrenheit and Celsius scales:  F+40 = 1.8 (C+40)
    Automotive :
  1395. Car speed is proportional to tire size & engine rpm, divided by gear ratio.
  1396. 0 to 60 mph in 4.59 s, may not always mean 201.96 feet.
  1397. Car acceleration. Guessing the curve from standard data.
  1398. "0 to 60 mph" time, obtained from vehicle mass and actual average power.
  1399. Thrust  is the ratio of  power to speed  [measured along direction of thrust].
  1400. Power as a function of chamber size  for internal combustion engines.
  1401. Optimal gear ratio  to maximize top speed on a flat road  (no wind).
    Surface Areas :
  1402. Heron's formula (for the area of a triangle) is related to the Law of Cosines.
  1403. Brahmagupta's formula gives the area of a quadrilateral  (cyclic  or not).
  1404. Bretschneider's formula  for a quadrilateral of given sides and diagonals.
  1405. Vectorial area of a quadrilateralHalf  the cross-product of its diagonals.
  1406. Parabolic segment:  2/3 the area of circumscribed parallelogram or triangle.
    Volumes :
  1407. Content of a cylindrical tank (horizontal axis), given the height of the liquid.
  1408. Volume of a spherical cap, or content of an elliptical vessel, at given level.
  1409. Content of a cistern (cylindrical with elliptical ends), at given fluid level.
  1410. Volume of a cylinder or prism, possibly with tilted [nonparallel] bases.
  1411. Volume of a conical frustum:  Formerly a staple of elementary education...
  1412. Volume of a sphere...  obtained by subtracting a cone from a cylinder !
  1413. The volume of a tetrahedron  is the determinant of three edges, divided by 6.
  1414. Volume of a wedge of a cone.
    Averages :
  1415. Splitting a job evenly between two unlike workers.
  1416. Splitting a job unevenly between two unlike workers.
  1417. Mixing solutions to obtain a predetermined intermediate rating.
  1418. Alcohol solutions are rated by volume not by mass.
  1419. Mixing alcohol solutions to obtain an exact rating by volume (ABV).
  1420. Special averages: harmonic (for speeds), geometric (for rates), etc.
  1421. Mean Gregorian month: either  30.436875 days, or  30.4587294742534...
  1422. The arithmetic-geometric mean  is related to a  complete elliptic integral.
    Geodesy and Astronomy :
  1423. Distance to the horizon is proportional to the square root of your altitude.
  1424. Distance between two points on a great circle at the surface of the Earth.
  1425. Euclidean distance between two cities, along a line  through  the Earth.
  1426. Geodetic coordinates:  Point of elevation h at latititude  j  and longitude  q.
  1427. The figure of the Earth. Geodetic and geocentric latitudes.
  1428. Area of a spherical polygon.  How to apply  Girard's formula.
  1429. Kepler's Third Law: The relation between orbital period and orbit size.

    Philosophy and Science

  1430. Creation and Discovery in Science.
  1431. Search for Extraterrestrial Intelligence.  If we listen, we  must  talk.
  1432. The Anthropic Principle:  The laws of physics must allow human life.
  1433. Science and Politics:  Political support for Science makes a society worthy.
  1434. What's Mathematics anyway?  The groundwork of scientific knowledge.
    Below are topics not yet integrated with the rest of this site's navigation.

    Perimeter of an Ellipse

  1435. Circumference of an ellipse:  Exact series and approximate formulas.
  1436. Ramanujan I and Lindner formulas:  The journey begins...
  1437. Ramanujan II:  An awesome approximation from a mathematical genius.
  1438. Hudson's Formula and other  Padé approximations.
  1439. Peano's Formula:  The sum of two approximations with cancelling errors.
  1440. The YNOT formula  (Maertens, 2000.  Tasdelen, 1959).
  1441. Euler's formula is the first step in an exact expansion.
  1442. Naive formulap  ( a + b )  features a  -21.5% error for elongated ellipses.
  1443. Cantrell's Formula:  A modern attempt with an overall accuracy of 83 ppm.
  1444. From Kepler to Muir.  Lower bounds and other approximations.
  1445. Relative error cancellations in symmetrical approximative formulas.
  1446. Complementary convergences of two series.  A nice foolproof algorithm.
  1447. Elliptic integrals & elliptic functions.  Traditional symbols vs. computerese.
  1448. Padé approximants  are used in a whole family of approximations...
  1449. Improving Ramanujan II  over the whole range of eccentricities.
  1450. The Arctangent Function as a component of several approximate formulas.
  1451. Abed's formula uses a parametric exponentA>.  Improved looks for a brainchild of  Shahram Zafary.
  1452. Rivera's formula gives the perimeter of an ellipse with 104 ppm accuracy.
  1453. Better accuracy from Cantrell, building on his own previous formula
  1454. Rediscovering  a well-known exact expansion due to Euler (1773).
  1455. Exact expressions for the circumference of an ellipse:  A summary.

    Surface of an Ellipsoid

  1456. Surface Area of a Scalene Ellipsoid:  The general formula isn't elementary.
  1457. Thomsen's Formula:  A simple symmetrical approximation.
  1458. Approximate formulas  for the surface area of a scalene ellipsoid.
  1459. Nautical mile:  "Average"  minute of latitude  on an oblate spheroid.
  1460. Great ellipses  have the same center as the ellipsoid they are drawn on.
  1461. Area enclosed by a curve  drawn on the surface of an oblate spheroid.
  1462. Pseudo-straight boundaries  of areas varying quadratically with longitude.

    The Unexplained

  1463. The Magnetic Field of the Earth.
  1464. Life (1):  The mysteries of evolution.
  1465. Life (2):  The origins of life on Earth.
  1466. Life (3):  Does extraterrestrial life exist?  Is there intelligence out there?
  1467. Nemesis: A distant companion of the Sun could cause periodical extinctions.
  1468. Current Challenges to established dogma.
  1469. Unexplained artifacts and sightings.

    Open Questions (or tough answers)

  1470. The Riemann Hypothesis:   {Re(z) > 0   &   z(z) = 0}   Þ   {Re(z) = ½}.
  1471. Twin primes conjecture:  One of the oldest open mathematical questions.
  1472. P = NP ?   Can we  find  in polynomial time what we can  check  that fast?
  1473. Collatz sequences  go from  n  to  n/2  or  (3n+1)/2.  Do they all lead to  1?
  1474. The Poincaré Conjecture (1904).  Proven by  Grisha Perelman  in 2002.
  1475. Fermat's Last Theorem (1637).  Proven by  Andrew Wiles  in 1995.
  1476. The ABC conjecture.

    Mathematical Miracles

  1477. The only magic hexagon.
  1478. The law of small numbers applied to conversion factors.
  1479. Quadratic formulas yielding long sequences of prime numbers.
  1480. The area under a Gaussian curve  involves the square root of  p
  1481. Exceptional simple Lie groups.
  1482. Monstrous Moonshine in Number Theory.


  1483. Oldest  open  mathematical problem:  Are there any  odd  perfect numbers?
  1484. Magnetic field of the EarthSouth side is near the geographic north pole.
  1485. From the north side,  a counterclockwise angle is positive  by definition.
  1486. What initiates the wind?  Well, primitive answers were not so wrong...
  1487. Why "m" for the slope of a linear function  y = m x + b ?  [in US textbooks]
  1488. The diamond mark on US tape measures corresponds to 8/5 of a foot.
  1489. Naming the largest possible number, in n keystrokes or less (Excel syntax).
  1490. The "odds in favor" of poker hands: A popular way to express probabilities.
  1491. Reverse number sequence(s) on the verso of a book's title page.
  1492. Living species: About 1400 000 have been named, but there are many more.
  1493. Dimes and pennies:  The masses of all current US coins.
  1494. Pound of pennies: The dollar equivalent of a pound of pennies is increasing!
  1495. Nickels per gallon:  Packing more than 5252 coins per gallon of space.

    Geography, Geographical Trivia

  1496. Geodetic coordinates,  based on the  Reference Ellipsoid  defined in 1980.
  1497. Geocentric coordinates  are almost never used in geography or astronomy.
  1498. Distance from the center of the Earth  to points located at the surface.
  1499. The volume of the Grand Canyon:  2 cm (3/4") over the entire Earth.
  1500. The Oldest City in the World:  Damascus or Jericho?
  1501. USA (States & Territories):  Postal and area codes, capitals, statehoods, etc.

    Money, Currency, Precious Metals

  1502. Inventing Money: Brass in China, electrum in Lydia, gold and silver staters.
  1503. Prices of Precious Metals:  Current market values (Gold, Silver. Pt, Pd, Rh).
  1504. Medieval system:  12  deniers  to a  sol.
  1505. Ancien Régime  French monetary system.
  1506. British coinage  before decimalization.
  1507. Exchange rates  when the  euro  was born.
  1508. Worldwide circulation  of currencies.

    The Counterfeit Penny Problem

  1509. Counterfeit Coin:  In 3 weighings, find an odd object among 12, 13 or 14.
  1510. Counterfeit Penny Problem: Find an odd object in the fewest weighings.
  1511. Explicit tables  for detecting  one  odd marble among  41,  in  4  weighings.
  1512. Find-a-birthday:  Detect an odd marble among 365, in 6 weighings.
  1513. Error-correcting codes for ternary numeration.
  1514. If the counterfeit is known to be heavier, fewer weighings may be sufficient.

    Calendars & Chronology

  1515. Fossil calendars: 420 million years ago, a  month  was only 9  short  days.
  1516. Julian Day Number (JDN) Counting days in the simplest of all calendars.
  1517. The Week has not always been a period of seven days.
  1518. Egyptian year of 365 days: Back to the same season after over 1500 years.
  1519. Heliacal rising of Sirius: Sothic dating.
  1520. Coptic Calendar: Reformed Egyptian calendar based on the Julian year.
  1521. The Julian Calendar: Year starts March 25. Every fourth year is a leap year.
  1522. Anno Domini: Counting roughly from the birth of Jesus Christ.
  1523. Gregorian Calendar: Multiples of 100 not divisible by 400 aren't leap years.
  1524. Counting the days between dates, with a simple formula for month numbers.
  1525. Age of the Moon, based on a mean synodic month of  29.530588853 days.
  1526. Easter Sunday is defined as the first Sunday after the Paschal full moon.
  1527. The Muslim Calendar:  The Islamic (Hijri) Calendar (AH = Anno Hegirae).
  1528. The Jewish Calendar:  An accurate lunisolar calendar, set down by Hillel II.
  1529. Zoroastrian Calendar.
  1530. The Zodiac:  Zodiacal signs and constellations.  Precession of equinoxes.
  1531. The Iranian Calendar.  Solar Hejri [SH]  or  Anno Persarum  [AP].
  1532. The Chinese Calendar.
  1533. The Japanese Calendar.
  1534. Mayan System(s)Haab (365), Tzolkin (260), Round (18980), Long Count.
  1535. Indian Calendar:  The Sun goes through a zodiacal sign in a solar month.
  1536. Post-Gregorian CalendarsPainless  improvements to the secular calendar.
  1537. Geologic Time Scale:  Beyond all calendars.

    Roman Numerals  (Archaic, Classic and Medieval)

  1538. Roman Numeration:  Ancient rules and medieval ones.
  1539. An easy conversion table  for numbers up to  9999.
  1540. Larger Numbers, like 18034...
  1541. Extending the Roman system.
  1542. The longest year so far,  in terms of Roman numerals.
  1543. IIS (or HS) is for sesterce (originally, 2½ asses, "unus et unus et semis").
  1544. Roman fractions.  A rudimentary  duodecimal  system.


  1545. Standard jokes.
  1546. Limericks.
  1547. Proper credit may not always be possible.
  1548. Trick questions can be legitimate ones.
  1549. Ignorance is bliss:  Why not read all that mathematical stuff  faster ?
  1550. Silly answers to funny questions.
  1551. Why did the chicken cross the road?  Scientific and other explanations.
  1552. Humorous or inspirational quotations by famous scientists and others.
  1553. One great quote  to be translated into as many languages as possible.
  1554. Famous Last Words:  Proofs that the guesses of experts are just guesses.
  1555. Famous anecdotes.
  1556. Parodies, hoaxes, and practical jokes.
  1557. Omnia vulnerant, ultima necat:  The day of reckoning.
  1558. Funny Units: A millihelen is the amount of beauty that launches one ship.
  1559. Funny Prefixes: A lottagram is many grams; an electron is 0.91 lottogram.
  1560. The Lamppost Theory:  Only look where there's enough light.
  1561. Is it  insanity  or just a viable alternative to orthodoxy?
  1562. Anagrams: Rearranging letters may reveal hidden meanings ;-)
  1563. Mnemonics: Remembering things and/or making fun of them.
  1564. Acronyms: Funny ones and/or alternate interpretations of serious ones.
  1565. Usenet Acronyms: If you can't beat them, join them (and HF, LOL).

    Scientific Symbols and Icons

  1566. Adobe's Symbol font:  Endangered standard HTML mathematical symbols.
  1567. The equality symbol ( = ).  The "equal sign" dates back to the 16th century.
  1568. The double-harpoon symbol  denotes  chemical equilibrium.
  1569. Line components: Vinculum, bar, solidus, virgule, slash, macron, etc.
  1570. The infinity symbol ( ¥ ) introduced in 1655 by John Wallis (1616-1703).
  1571. Transfinite numbers  and the many faces of mathematical infinity.
  1572. Chrevron symbols:  Intersection (highest below)  or  union (lowest above).
  1573. Disjoint union.  Square "U" or  inverted  p  symbol.
  1574. Blackboard boldDoublestruck  characters denote  sets of numbers.
  1575. The integration sign ( ò ) introduced by Leibniz at the dawn of Calculus.
  1576. The end-of-proof box (or tombstone) is called a halmos symbol  (QED).
  1577. Two "del" symbols  for partial derivatives, and  Ñ  for Hamilton's nabla.
  1578. The rod of Asclepius:  Medicine and the 13th zodiacal constellation.
  1579. The Caduceus:  Scepter of Hermes, symbol of  commerce  (not medicine).
  1580. Tetractys:  Mystical Pythagorean symbol, "source of everflowing Nature".
  1581. The Borromean Rings: Three interwoven rings which are pairwise separate.
  1582. The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.
  1583. Dangerous-bend symbol:  Introduced by  Bourbaki,  popularized by  Knuth.  
     Gerard Michon

    Monographs and Complements  

  1584. About Zero.
  1585. Wilson's Theorem.
  1586. Counting Polyhedra:  A tally of polyhedra with n faces and k edges.
  1587. Sagan's number:  The number of stars, compared to earthly grains of sand.
  1588. The Sand Reckoner:  Archimedes fills the cosmos with grains of sand.  
     Gerard Michon

    Numericana Hall of Fame  

  1589. Numericana's list of distinguished Web authors in Science...
  1590. Giants of Science:  Towering characters in Science history.
  1591. Two Solvay conferences  helped define modern physics, in 1911 and 1927.
  1592. Physical Units: A tribute to the late physicist Richard P. Feynman.
  1593. The many faces of Nicolas Bourbaki  (b. January 14, 1935).
  1594. Lucien Refleu  (1920-2005).  "Papa" of 600 mathematicians.  [ In French ]
  1595. Taupe Laplace.  [ In French ].
  1596. Roger Apéry  (1916-1994)  and the irrationality of  z(3).
  1597. Hergé (1907-1983):  Tintin and the Science of Jules Verne (1828-1905).
  1598. Other biographies:  Dulong, Galois, Tannery, Vessiot, Drach, Glénisson...
  1599. Escutcheons of Science (Armorial):  Coats-of-arms of illustrious scientists.  
     Gerard Michon

    In-Depth Reviews of Great Products  

    Top HP Calculators:  HP-49g+ and HP-50g

  1600. Printer :  The  HP 82240B thermal printer  has been standard since 1989.
  1601. Modifier keys.  Lesser-used functions require several keystrokes.
  1602. Infinity:  Unsigned algebraic infinity and signed topological infinities.
  1603. Physical units:  A built-in feature inherited from the HP-28  (1986).
  1604. Bug reports:  Severe problems and minor ones.
  1605. Complex functions:  Complex values & arguments.  Complex variables.
  1606. RPL programming  ("Reverse Polish LISP")  originated with the HP-28.
  1607. Easter eggs:  Unofficial features, just for  fun.

    HP-35s:  Released on the 35th birthday of the  HP-35

  1608. Modifier keys.  Lesser-used functions require several keystrokes.
  1609. Unit conversions: °F/°C  |  HMS  |  °/rad  |  lb/kg  |  mi/km  |  in/cm  |  gal/L.
  1610. 40 physical constants  (and one mathematical constant)  in a single menu.
  1611. Bug reports:  Severe problems and minor ones.
  1612. Complex functions:  Complex values & arguments.  Complex variables.
  1613. Programming:  Recorded keystroke sequences.  Tests, loops & subroutines.

    Top TI Calculators:  TI-92, TI-92+, TI-89, Voyage 200

  1614. Modifier keys.  Lesser-used functions require several keystrokes.
  1615. Physical units:  A very nice afterthought, with a few rough edges.
  1616. Analytical functions  may present discontinuity  cliffs  in the complex realm.
  1617. Wrong!  0 to the 0th power  should  be 1.  ¥ and   shouldn't  be equal.
  1618. 68000 Assembly Programming:  A primer without the help of an assembler.
  1619. The clock frequency of your calculator  measured with 0.1% accuracy.
  1620. TI's BASIC.  A built-in interpreted language not designed for speed.
  1621. Pretty 2D algebraic displays  can only be edited in their 1D version.

    Texas-Instruments Scientific Calculator:  TI-36X Pro

  1622. The keypad:  One shift-key suffices with the introduction of  multi-tap.
  1623. Integer arithmetic.  Numbers with more than 6 digits cannot be factorized.
  1624. 20 pairs of unit conversions...  and not a single inaccuracy.  (That's rare!)
  1625. 20 physical constants  listed by name, with their units.  9 are numbered.
  1626. Bug reports:  From minor gripes to more severe flaws.

    Sharp EL-W516 Calculator   (EL-W516x, EL-W516xBSL...)

  1627. The line of Sharp scientific calculators.  What the model numbers mean.
  1628. The basics.  Color-coded multiple functions.
  1629. Entering and exiting special modes.
  1630. Solving cubic equations.

    Affordable Casio Calculators:  fx-115es,  fx-991es,  fx570es

  1631. History  of the "natural" Casio scientific calculator series.
  1632. Mode 4:  Hexadecimal or octal arithmetic on 32-bit integers.
  1633. Mode 7:  Tabulate a function  (or a pair of functions with "plus" version).
  1634. Scientific constants:  Consistent values recommended by  CODATA (2010).
  1635. Conversion factors between units:  A few inaccuracies  &  one typo.

    Canon's  F-792SGA  Calculator  (2013) :

  1636. The basics.
  1637. The good and the bad.

Note: The above numbering may change, don't use it for reference purposes.

Noted   Numericana fans  (and/or contributors)  in alphabetical order:

Guest Authors:

Public-Domain Texts:

 Gerard Michon  Gerard Michon
 (UCLA Bruins)

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