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
 |
|
- Only a
negative deserves a proof (no counterexamples).
- Proving by induction
the truth of infinitely many things.
- Stochastic proofs
leave only a vanishing uncertainty.
- Heuristic arguments
state the likelihood of a conjecture.
- Computer-aided proofs:
The 4-color theorem was so proved in 1976.
- Lack of a good approach
doesn't invalidate a statement.
- All metric prefixes:
Current SI prefixes, obsolete prefixes, bogus prefixes...
- Prefixes for units of information (multiples
of the bit). Brontobyte hoax.
- Density one.
Relative and absolute density precisely defined.
- Acids
yielding a mole of H+ per liter are
normal (1N) solutions.
- Calories:
Thermochemical calorie, gram-calorie, IST calorie (and Btu).
- Horsepowers:
hp, metric horsepower and boiler horsepower.
- The standard acceleration of gravity
has been 9.80665 m/s2 since 1901.
Time:
- Tiny durations;
zeptosecond (zs, 10-21s) &
yoctosecond (ys, 10-24s).
- A jiffy is either a light-cm
or 10 ms (tempons and chronons are shorter).
- The length of a second.
Solar time, ephemeris time, atomic time.
- The length of a day.
Solar day, atomic day, sidereal or Galilean day.
- Scientific year = 31557600
SI seconds (»
Julian year of 365.25 solar days).
Length:
- The International inch
(1959) is 999998/1000000 of a US Survey inch.
- The typographer's point
is exactly 0.013837" = 0.3514598 mm.
- How far is a league?
Land league, nautical league.
- Radius of the Earth
and circumference at the Equator.
- Extreme units of length.
The very large and the very small.
Surface Area:
- Acres, furlongs, chains and square inches...
Volume, Capacity:
- Capitalization of units.
You only have a choice for the liter (or litre ).
- Drops or minims:
Winchester, Imperial or metric. Teaspoons and ounces.
- Fluid ounces:
American ounces (fl oz) are about 4% larger than British ones.
- Gallons galore:
Winchester (US) vs. Imperial gallon (UK), dry gallon, etc.
- US bushel
and Winchester units of capacity (dry = bushel, fluid = gallon).
- Kegs and barrels: A keg of beer is half a barrel,
but not just any "barrel".
Mass, "Weight":
- Tiny units of mass.
A hydrogen atom is about 1.66 yg.
- Solar mass:
The unit of mass in the astronomical system of units.
- Technical units of mass.
The slug and the hyl.
- Customary units of mass
which survive in the electronic age.
- The poids de marc system:
18827.15 French grains to the kilogram.
- A talent was the mass of a cubic foot of water.
- Tons:
short ton, long ton (displacement ton), metric ton (tonne), assay ton...
- Other tons: Energy
(kiloton, toe, tce), cooling power, thrust, speed.
- Scientific notation:
Nonzero numbers given as multiples of powers of ten.
- So many "significant" digits
imply a result of limited precision.
- Standard deviation
specifies the uncertainty in the precision of a result.
- Engineering notation
reduces a number to a multiple of a power of 1000.
- The quadratic formula
is numerically inadequate in common cases.
- Devising robust formulas
which feature a stable floating-point precision.
- The Beaufort scale
is now defined in terms of wind speed.
- The Saffir / Simpson scale for hurricanes.
- The Fujita scale for tornadoes.
- Decibels: A general-purpose logarithmic
scale for relative power ratios.
- Sound intensity level (SIL)
is well-defined; SPL is an approximation.
- Apparent and absolute magnitudes of stars.
- Acidity. The pH scale was
invented by Søren Sørensen in 1909.
- The Richter scale for earthquakes and
other sudden energy releases.
- The scale of animals
according to Galileo Galilei.
- Jumping fleas...
compared to jumping athletes...
- Drag coefficient
of a sphere as a function of the Reynolds number R.
Physical Constants:
- For the utmost in precision,
physical constants are derived in a certain order.
- Primary conversion factors
between customary systems of units.
6+1 Basic Dimensionful Physical Constants
(Proleptic SI)
- Speed of Light in a Vacuum
(Einstein's Constant): c = 299792458 m/s.
- Magnetic Permeability of the Vacuum:
An exact value defining the ampere.
- Planck's constant:
The ratio of a photon's energy to its frequency.
- Boltzmann's constant:
Relating temperature to energy.
- Avogadro's number:
The number of things per mole of stuff.
- Mechanical Equivalent of Light
(683 lm/W at 540 THz) defines the candela.
- Newton's constant of gravitation
and a futuristic definition of the second.
Fundamental Mathematical Constants:
- 0: Zero is the most fundamental
and most misunderstood of all numbers.
- 1 and -1: The unit numbers.
- p ("Pi"):
The ratio of the circumference of a circle to its diameter.
- Ö2:
The diagonal of a square of unit side. Pythagoras' Constant.
- f:
The diagonal of a regular pentagon of unit side. The Golden Number.
- Euler's e:
Base of the exponential function which equals its own derivative.
- ln(2): The alternating sum of the reciprocals
of the integers.
- An engineering favorite: The decimal
logarithm of 2.
- Euler-Mascheroni Constant
g :
Limit of [1 + 1/2 + 1/3 +...+ 1/n] - ln(n).
- Catalan's Constant G :
The alternating sum of the reciprocal odd squares.
- Apéry's Constant
z(3) : The sum of the reciprocals of the perfect cubes.
- Imaginary i:
If "+1" is a step forward, "+ i" is a step sideways to the left.
Exotic Mathematical Constants:
- Delian constant:
21/3 is the solution to the classical
duplication of the cube.
- Gauss's constant:
Reciprocal of the arithmetic-geometric mean of 1
and Ö2.
- Mertens constant:
The limit of [1/2 + 1/3 + 1/5 +...+ 1/p] - ln(ln p)
- Artins's constant is the proportion of
long primes in decimal or binary.
- Ramanujan-Soldner constant
(m): Positive root of the logarithmic integral.
- The Omega constant:
W(1) is the solution of the equation x exp(x) = 1.
- Feigenbaum constant
(d) and the related reduction parameter
(a).
Some Third-Tier Mathematical Constants:
- Brun's Constant:
A standard uncertainty (s)
means a 99% level of ±3s
- Prévost's Constant:
The sum of the reciprocals of the Fibonacci numbers.
- Grossman's Constant: One
recurrence converges for only one initial point.
- Ramanujan's Number:
exp(p Ö163) is almost an integer.
- Viswanath's Constant: Mean growth
in random additions and subtractions.
- Copeland-Erdös Number: Almost
all numbers are normal, it's one of them!
- Always change your first guess
if you're always told another choice is bad.
- The Three Prisoner Problem
predated Monty Hall and Marilyn by decades.
- Seating N children at a round table
in (N-1)! different ways.
- How many Bachet squares?
A 1624 puzzle using the 16 court cards.
- Choice Numbers:
C(n,p) is the number of ways to choose p items among n.
- C(n+2,3) three-scoop sundaes.
Several ways to count them (n flavors).
- C(n+p-1,p) choices
of p items among n different types, allowing duplicates.
- How many new intersections
of straight lines defined by n random points.
- Face cards.
The probability of getting a pair of face cards is less than 5%.
- Homework Central:
Aces in 4 piles, bad ICs, airline overbooking.
- Binomial distribution.
Defective units in a sample of 200.
- Siblings with the same birthday.
What are the odds in a family of 5?
- Covariance:
A generic example helps illustrate the concept.
- Variance of a binomial distribution,
derived from general principles.
- Standard deviation.
Two standard formulas to estimate it.
- Markov's inequality
is used to prove the Bienaymé-Chebyshev inequality.
- Bienaymé-Chebyshev inequality:
Valid for any probability distribution.
- Inclusion-Exclusion:
One approach to the probability of a union of 3 events.
- The "odds in favor" of poker hands:
A popular way to express probabilities.
- Probabilities of a straight flush in 7-card stud.
Generalizing to "q-card stud".
- Probabilities of a straight flush
among 26 cards (or any other number).
- The exact probabilities
in 5-card, 6-card, 7-card, 8-card and 9-card stud.
- Rearrangements of CONSTANTINOPLE
so no two vowels are adjacent...
- Four-letter words from
POSSESSES: Counting with generating functions.
- How many positive integers below 1000000
have their digits add up to 19?
- Polynacci Numbers:
Flipping a coin n times without p tails in a row.
- Winning in finitely many flips or losing endlessly...
- 252 decreasing sequences
of 5 digits (2002 nonincreasing ones).
- How many ways are there
to make change for a dollar? Closed formulas.
- Partitioniong an amount
into the parts minted in a certain currency.
- The number of rectangles
in an N by N chessboard-type grid.
- The number of squares
in an N by N grid: 0, 1, 5, 14, 30, 55, 91, 140, 204...
- Screaming Circles:
How many tries until there's no eye contact?
- Average distance
between two random points on a segment, a disk, a cube...
- Average distance
between two points on the surface of a sphere.
- Sizes of playing cards:
French, Bridge, Poker, French tarot, etc.
- How playing cards are made:
Either 2 layers of paper or "100% plastic".
- Suits:
Spades, hearts, diamonds & clubs (swords, cups, coins & wands).
- The four court cards:
Ace, king, queen, jack (king, queen, cavalier, page).
- The Mameluke 52-card standard deck
with 3 figure cards per suit.
- 78-card tarot deck:
21 trumps, 1 fool, 4 suits of 14 (incl. 4 court cards).
- The Major Arcana:
Trumps and fool of the tarot deck, in occult parlance.
- Names of the court cards
in the French tradition. Hundred Years' War.
- 48-card Aluette deck:
Latin suits, mimicks and names of special cards.
- Jokers from Euchre (1857) found their
way into Poker in the 1870's.
- The 40-card Spanish baraja deck lacks 8, 9 & 10.
- The 32-card piquet deck lacks 2-6.
French Belote and German Skat.
- Skat: The most popular German card game (32-card deck).
- 24-card deck for Euchre (single deck)
and Pinochle (double deck).
- Happy Families:
44-card British deck of 11 families of 4 (1851).
- Jeu des 7 familles:
42=card French deck of 7 families of 6 (1876).
- 1000 Bornes:
106 cards for a boardless car-racing game (1954).
- Set® cards:
Combinatorics of a modern 81-card ternary deck (1974).
- Poisson Processes:
Random arrivals happening at a constant rate (in Bq).
- Simulating a poisson process
with a uniform random number generator.
- Markov Processes:
When only the present influences the future...
- The Erlang B Formula
assumes callers don't try again after a busy signal.
- Markov-Modulated Poisson Processes
may look like Poisson processes.
- The Utility Function:
A dollar earned is usually worth less than a dollar lost.
- St. Petersburg's Paradox:
What would you pay to play the Petersburg game?
- Condorcet's Paradox:
A group of rational voters need not behave rationally.
- Center of an arc
determined with straightedge and compass.
- Surface areas:
Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
- Power of a point with respect to a circle.
- Euler's line goes through
the orthocenter, the centroid and the circumcenter.
- Euler's circle
is tangent to the incircle and the excircles (Feuerbach, 1822).
- Barycentric coordinates & trilinears
examplify homogeneous coordinates.
- Elliptic arc:
Length of the arc of an ellipse between two points.
- Perimeter of an ellipse.
Exact formulas and simple ones.
- Surface of an ellipse.
- Volume of an ellipsoid
(either a spheroid or a scalene ellipsoid).
- Surface area of a spheroid
(oblate or prolate ellipsoid of revolution).
- Quadratic equations in the plane
describe ellipses, parabolas, or hyperbolas.
- Centroid of a circular segment.
Find it with Guldin's (Pappus) theorem.
- Parabolic arc of given extremities
with a prescribed apex between them.
- Focal point of a parabola.
y = x 2 / 4f (where f is the focal distance).
- Parabolic telescope:
The path from infinity to focus is constant.
- Make a cube go through a hole in a smaller cube.
- Octagon: The relation between side and diameter.
- Constructible regular polygons
and constructible angles (Gauss).
- Areas of regular polygons of unit side:
General formula & special cases.
- For a regular polygon of given perimeter,
the more sides the larger the area.
- Curves of constant width:
Reuleaux Triangle and generalizations.
- Irregular curves of constant width.
With or without any circular arcs.
- Solids of constant width.
The three-dimensional case.
- Constant width in higher dimensions.
- Fourth dimension.
Difficult to visualize, but easy to consider.
- Volume of a hypersphere
and hyper-surface area, in any dimensionality.
- Hexahedra. The cube is not the only
polyhedron with 6 faces.
- Descartes-Euler Formula:
F-E+V=2 but restrictions apply.
- Metric spaces:
The motivation behind more general topological spaces.
- Abstract topological spaces are defined
by calling some subsets open.
- Closed sets are sets (of a
topological space) whose complements are open.
- Subspace F of E:
Its open sets are the intersections with F of open sets of E.
- Separation axioms:
Flavors of topological spaces, according to Trennung.
- Compactness of a topological space:
Any open cover has a finite subcover.
- Real-valued continuous functions on compact sets
attain their extremes.
- Borel sets.
Tribes form the topological foundation for measure theory.
- Locally compact sets
contain a compact neighborhood of every point.
- General properties of sequences
characterize topological properties.
- Continuous functions let
the inverse image of any open set be open.
- Restrictions remain continuous.
Continuous extensions may be impossible.
- The product topology
makes projections continuous on a cartesian product.
- Connected sets
can't be split by open sets. The empty set is connected.
- Path-connected sets
are a special case of connected sets.
- Homeomorphic sets.
An homeomorphism is a bicontinuous function.
- Arc-connected spaces are path-connected.
The converse need not be true.
- Homotopy:
A progessive transformation of a function into another.
- The Fundamental Group:
The homotopy classes of all loops through a point.
- Homology and Cohomology.
Poincaré duality.
- Descartes-Euler Formula:
F-E+V = 2, but restrictions apply.
- Euler Characteristic:
c (chi) extended beyond its traditional definition...
- Winding number of a continuous
planar curve about an outside point.
- Fixed-point theorems
by Brouwer, Shauder and Tychonoff.
- Turning number of a planar curve with a
well-defined oriented tangent.
- Real projective plane and Boy's surface.
- Hadwiger's
additive continuous functions of d-dimensional rigid bodies.
- Eversion of the sphere.
An homotopy can turn a sphere inside out.
- Classification of surfaces:
"Zero Irrelevancy Proof" (ZIP) by J.H. Conway.
- Braid groups: strands, braids and pure braids.
- Complete metric space:
A space in which all Cauchy sequences converge.
- Flawed alternatives to completeness.
- Banach spaces
are complete normed vector spaces.
- Fréchet spaces
are generalized Banach spaces.
- Fractional exponents were first conceived by
Nicole d'Oresme (c. 1360).
- The von Koch curve (and snowflake):
Dimension of self-similar objects.
- Hausdorff dimension is revealed by a covering with
balls of radius < e.
- The Julia set and the Fatou set of an analytic function
are complementary.
- The Mandelbrot set was so named by
Adrien Douady & John H. Hubbard.
- Planar angles
(from one direction to another) are signed quantities.
- Bearing:
Unless otherwise specified, this is the angle west of north.
- Solid angles
are to spherical patches what planar angles are to circular arcs.
- Circular measures:
Angles and solid angles aren't quite dimensionless...
- Solid angle formed by a trihedron :
Van Oosterom & Strackee (1983).
- Solid angle subtended by a rhombus.
Apex of a right rhombic pyramid.
- Formulas for solid angles
subtended by patches with simple shapes.
- Right ascension and declination.
Precession of celestial coordinates (a,d).
- Curvature of a planar curve:
Variation of inclination with distance dj/ds.
- Curvature and torsion
of a three-dimensional curve.
- Distinct curvatures and geodesic
torsion of a curve drawn on a surface.
- The two fundamental quadratic forms
at a point of a parametrized surface.
- Lines of curvature:
Their normal curvature is extremal at every point.
- Geodesic lines.
Least length is achieved with zero geodesic curvature.
- Meusnier's theorem:
Tangent lines have the same normal curvature.
- Gaussian curvature of a surface.
The Gauss-Bonnet theorem.
- Parallel-transport of a vector
around a loop. Holonomic angle of a loop.
- Total curvature of a curve.
The Fary-Milnor theorem for knotted curves.
- Linearly independent components
of the Riemann curvature tensor.
- Cartesian equation of a straight line:
passing through two given points.
- Confocal Conics:
Ellipses and hyperbolae sharing the same pair of foci.
- Spiral of Archimedes:
Paper on a roll, or groove on a vinyl record.
- Hyperbolic spiral: The inverse of
the Archimedean spiral.
- Catenary:
The shape of a thin chain under its own weight.
- Witch of Agnesi.
How the versiera (Agnesi's cubic) got a weird name.
- Folium of Descartes.
- Lemniscate of Bernoulli:
A quartic curve shaped like the infinity symbol.
- Along a Cassini oval,
the product of the distances to the two foci is constant.
- Limaçons of Pascal:
The cardioid (unit epicycloid) is a special case.
- On a Cartesian oval,
the weighted average distance to two poles is constant.
- The envelope of a family of curves
is everywhere tangent to one of them.
- The evolute of a curve
is the locus of its centers of curvature.
- Involute of a curve:
Trajectory of a point of a line rolling on that curve.
- Parallel curves share their
normals, along which their distance is constant.
- The nephroid (or
two-cusped epicycloid )
is a catacaustic of a circle.
- Freeth's nephroid: A special
strophoid of a circle.
- Bézier curves are
algebraic splines. The cubic type is the most popular.
- Piecewise circular curves:
The traditional way to specify curved forms.
- Intrinsic equation
[curvature as a function of arc length] may have spikes.
- The quadratrix (trisectrix)
of Hippias squares the circle and trisects angles.
- The parabola
is constructible with straightedge and compass.
- Mohr-Mascheroni constructions
use the compass alone (no straightedge).
- Glossary
of terms related to gears.
- Gear ratio:
Ratio of the input rotation to the output rotation (signed).
- Planar curves
rolling without slipping while rotating about two fixed points.
- Congruent ellipses
roll against each other while revolving around their foci.
- Elliptic gears:
A family of gears which include ellipses and sine curves.
- Watchmaker gears:
Ogival surfaces for pinions & radial planes for wheels.
- La Hire's theorem :
An hypocycloid of ratio 2 is a straight line.
- Cycloidal gear:
Epicycloidal addendum curve & hypocycloidal dedendum.
- The law of conjugate action
was formulated by Leonhard Euler (c. 1754).
- Involute tooth profiles
provide a uniform rotational speed ratio.
- Harmonic Drive: A
flexspline with 2 fewer teeth than the circular spline.
- Circular arc helical gears:
E. Wildhaber (1923) & M.L. Novikov (1956).
- Double circular arc helical gears
were standardized by the Chinese in 1981.
- Hexahedra.
The cube is not the only polyhedron with 6 faces.
- Fat tetragonal antiwedge:
Chiral hexahedron of unit volume and least area.
- Duality:
To a face of a polyhedron corresponds a vertex of its dual.
- Enumeration of polyhedra:
Convex polyhedra with n faces and k edges.
- The 5 Platonic solids:
Cartesian coordinates of the vertices.
- Symmetries
may equate all commensurate components of a polyhedron.
- Equimetric polyhedra
feature constant measures for all elements of a kind.
- The 13 Archimedean solids
and their duals (Catalan solids).
- Polyhedra in certain families
are named after one prominent polygon.
- Some special polyhedra
may have a traditional (mnemonic) name.
- Deltahedra
have equilateral triangular faces. Only 8 deltahedra are convex.
- Johnson Polyhedra and the associated nomenclature.
- Polytopes
are the n-dimensional counterparts of 3-D polyhedra.
- A simplex of touching unit spheres
may allow a center sphere to bulge out.
- Regular Antiprism:
Height and volume of a regular n-gonal antiprism.
- The Szilassi polyhedron
features 7 pairwise adjacent hexagonal faces.
- Wooden buckyball:
Cutting 32 blocks to make a truncated icosahedron.
- Zonogons, zonohedra, zonotopes.
Zones and zonoids.
- Space-filling polyhedra:
Cuboctahedron, truncated octahedron, etc.
- 16 possible standard dice
(opposite faces add up to 7). Two handedness.
- 30 labelings of a die:
For 16 of them, opposite faces never add up to 7.
- 3-sided spindle: A 9-hedron with 6 unstable faces.
- Nontransitive dice:
Every die is dominated by another die from the set.
- Sicherman dice
yield any total with the same probability as a regular pair.
- Percentile dice
and other ways several dice can have equiprobable sums.
- Polyhedral dice
were popularized by rôle playing games.
- Commonly available dice sizes:
Small, medium, large, jumbo and giant.
- Convex isohedra
are fair dice, by reason of symmetry between faces.
- Round dice: Outer isohedral marks
& steel ball in an isogonal cavity.
- Scalene isogonal polyhedra.
Their duals are scalene isohedra.
- Juryeonggu:
Korean die with 8 hexagonal faces and 6 square ones.
- Fairness of a non-isohedral die
may depend on the way it's tossed.
- Are there any intrinsically fair dice
which aren't isohedral?
- Necessary conditions
an absolutely fair die must satisfy.
- Quasistatic
probability is proportional to the solid angle a face subtends.
- Thermal tossing
puts a face of minimal height at the bottom.
- Balanced mesohedral dice
are fair for both quasistatic and thermal tossing.
- Mesodecahedron: Mesohedron with 10 faces.
- Mesopentahedron: The mesohedral proportions of a rhombic pyramid.
- Mesoheptahedron: Mesohedron with 7 faces.
- Statistical bias of unfair dice.
Isohedra. The symmetry of fair dice.
- Classification of all convex isohedra.
Intrinsic fair dice.
- Disphenoids are tetrahedra where
opposing edges have equal length.
- The hexakis icosahedra (120 faces)
include the disdyakis triacontahedra.
- Two chiral fair dice: No mirror symmetry,
24 or 60 pentagonal faces.
- A pseudo-isohedral die. Its faces are congruent, but is it fair?
- The bridges of Königsberg:
Eulerian graphs and the birth of graph theory.
- Undirected graphs are digraphs
with symmetrical adjacency matrix.
- Adjacency matrix
of a directed graph (digraph) or of a bipartite graph.
- The 3-utilities problem:
Providing 3 cottages with water, gas & electricity.
- Silent Circles:
An enumeration based on adjacency matrices (Alekseyev).
- Silent Prisms:
Modifying the screaming game for short-sighted people.
- Tallying
markings of one edge per node where no edge is marked twice.
- Line graph: Nodes of L(G) are edges of G
(connected iff adjacent in G).
- Transitivity:
Vertex-transitive and/or edge-transitive graphs.
- Factorial zero is 1, so is an empty product;
an empty sum is 0.
- Anything to the power of 0
is equal to 1, including 0 to the power of 0.
- Idiot's Guide to Complex Numbers.
- Using the Golden Ratio (f)
to express the 5 [complex] fifth roots of unity.
- "Multivalued" functions are functions defined over
a Riemann surface.
- Square roots are inherently ambiguous for
negative or complex numbers.
- The difference of two numbers,
given their sum and their product.
- All symmetric polynomials of 3 variables
are determined by the first three.
- Geometric progression of 6 terms. Sum is 14,
sum of squares is 133.
- Quartic equation involved in the classic
"Ladders in an Alley" problem.
- Chebyshev polynomials
give cos(nq)
as a function of cos q
- Legendre polynomials and
zonal harmonics.
- Laguerre polynomials. Hypergeometric
confluent function.
- Hermite polynomials. Eigenstates
of the quantum harmonic oscillator.
- Permutation matrices
include the identity matrix and the exchange matrix.
- Operations on matrices
are conveniently defined using Dirac's notation.
- The determinant is proportional
to any completely antisymmetrical form.
- Minors. First minors
are obtained by deleting one row and one column.
- Adjugate of a matrix:
Tranpose matrix of its cofactors A adj(A) = det(A) I
- Eigenvectors and eigenvalues
of an operator or a matrix.
- The characteristic polynomial
of an operator doesn't depend on the basis.
- Cayley-Hamilton theorem:
A matrix vanishes its characteristic polynomial.
- Vandermonde matrix:
The successive powers of elements in its second row.
- Generalized Vandermonde matrix
involving fractional powers...
- Cholesky decomposition L L*
of a positive-definite Hermitian matrix.
- Toeplitz matrix: Constant diagonals.
- Circulant matrix:
Cyclic permutations of the first row.
- Wendt's Determinant:
The circulant of the binomial coefficients.
- Hankel matrix: Constant skew-diagonals.
- Catberg matrix: Hankel matrix of the
reciprocal of Catalan numbers.
- Hadamard matrix:
Unit elements and orthogonal columns.
- Sylvester matrix of two polynomials
has their resultant for determinant.
- The discriminant of a polynomial
is the resultant of itself and its derivative.
- Numerical functions:
Polynomial, rational, algebraic, transcendental, special.
- Trigonometric functions:
Memorize a simple picture for 3 basic definitions.
- Solving triangles with the
law of sines, law of cosines, and law of tangents.
- Spherical trigonometry:
Triangles drawn on the surface of a sphere.
- Sum of tangents of two half angles,
in terms of sums of sines and cosines.
- The absolute value of the sine of a complex number.
- Exact solutions to transcendental equations.
- All positive rationals
(& square roots) as trigonometric functions of zero!
- The sine function:
How to compute it numerically.
- Chebyshev economization
saves billions of steps in common computations.
- The Gamma function:
Its definitions, properties and special values.
- Lambert's W function
is used to solve practical transcendental equations.
- Pochhammer's symbol:
Upper factorial of k increasing factors: x(x+1)...
- Gauss's hypergeometric function:
2+1 parameters (and one variable).
- Kummer's transformations relate
different hypergeometric expressions.
- Sum of the reciprocal of Catalan numbers,
in closed hypergeometric form.
- Derivative:
The slope of a function and/or something more abstract.
- Integration: The Fundamental Theorem of Calculus.
- Integration by parts:
Reducing an integral to another one.
- Length of a parabolic arc.
- Top height of a curved bridge
with a 5280 ft span and a 5281 ft length.
- Sagging:
A cable which spans 28 m and sags 30 cm is 28.00857 m long.
- The length of the arch of a cycloid
is 4 times the diameter of the wheel.
- Integrating the cube root of the tangent function.
- Changing inclination
to a particle moving along a parabola.
- Algebraic area of a "figure 8"
may be the sum or the difference of its lobes.
- Area surrounded by an oriented planar loop
which may intersect itself.
- Linear differential equations of higher
order and/or in several variables.
- Theory of Distributions:
Convolution products and their usage.
- Laplace Transforms:
The Operational Calculus of Oliver Heaviside.
- Integrability
of a function and of its absolute value.
- Analytic functions of a linear operator;
defining f (D) when D is d/dx...
- Ordinary differential equations. Several examples.
- A singular change of variable may not be valid
over a maximal domain.
- Vertical fall against fluid resistance
(valid for viscous and quadratic drag).
- Jet propulsion:
Expelling stuff at speed u makes (u-v) m remain constant.
- Riccati equation:
When y' is a quadratic function of y...
- Generalizing the
fundamental theorem of calculus.
- Vectorial surface
dotted into an observing direction gives apparent area.
- Practical identities of vector calculus.
Optimization:
Operations Research, Calculus of Variations
- Stationary points (saddlepoints)
are where all partial derivatives vanish.
- Single-variable optimization:
Derivative vanishes at any interior extremum.
- Extrema of a function of two variables
obey a second-order inequality.
- Saddlepoints of a multivariate function.
One equation for each variable.
- Lagrange multipliers:
Constrained optimization of an objective function.
- Minimizing the lateral surface area of a cone
of given base and volume.
- Euler-Lagrange equations
hold along the path of a stationary integral.
- Noether's theorem:
One conserved quantity for each Lagrangian symmetry.
- The brachistochrone curve is a cycloid
(in a uniform gravitational field).
- Isoperimetric Inequality:
The largest area enclosed by a loop of unit length.
- Plateau
extended the calculus of variations from paths to membranes.
- Embedded minimal surfaces:
Plane, catenoid, helicoid, Costa's surface, etc.
- Connecting blue dots to red dots
in the plane, without any crossings...
- The shortest way to connect 3 dots
can be to join them to a fourth point.
- The Honeycomb Theorem:
A conjecture of old, proved by Thomas Hales.
- Counterexamples to Kelvin's conjecture.
Unit spatial tiles of least area.
- Cauchy sequences
help define real numbers rigorously.
- Permuting the terms of a series
may change its sum arbitrarily.
- Two decreasing divergent series
may have a convergent minimum!
- Uniform convergence
of continuous functions makes the limit continuous.
- Defining integrals:
Cauchy, Riemann, Darboux, Lebesgue.
- Cauchy principal value of an integral.
- Fourier series.
A simple example.
- Infinite sums evaluated with Fourier series.
- A double sum is often the product of two sums,
which may be Fourier series.
- At a jump,
a Fourier series is the half-sum of its left and right limits.
- Gibbs phenomenon;
9% overshoot of partial Fourier series near a jump.
- Method of Froebenius
about a regular singularity of a differential equation.
- Laurent series
of a function about one of its poles.
- Cauchy's Residue Theorem
is helpful to compute difficult definite integrals.
- Tame complex functions:
Holomorphic and meromorphic functions.
- Taylor's expansion
of a differentiable function as a power series.
- Radius of convergence.
The convergence disk of a complex power series.
- The exponential series.
Proving that exp (x) exp (y) = exp (x+y)
- Analytic continuation:
Power series that converge on overlapping disks.
- Decimated power series
are equal to finite sums involving roots of unity.
- Summing geometric series:
Equating things that match over some domain.
- Desirable properties of summation methods yield rules for handling them.
- Formal series are kets.
Linear summation methods are bras.
- Euler summation (1746).
- All Nörlund summations
are linear, stable, regular and consistent (1919).
- Abel summation .
- Lindelöf summation (1903).
- Borel summation (1899).
- Mittag-Leffler summation method (1908).
- Weierstrass summation:
Summation by analytic continuation (1842).
- Zeta-function regularization (1916).
Dubbed heat-kernel regularization.
- The Mercator series is the integral of
the geometric series (1668).
- Ramanujan's irregular summation (1913).
"Sum" of the harmonic series.
- Summations of p-adic integers
for a special radix or for all of them.
- Moments, Stieltjes functions and Stieltjes series.
- Shanks' transformation.
- Richardson extrapolation.
- Asymptotic analysis and asymptotic series.
- Stirling's approximation and Stirling's series.
- Convolution
as an inner operation among numerical functions.
- Duality:
The product of a bra by a ket
is a (complex) scalar.
- A distribution
associates a scalar to every test function.
- Schwartz functions
are suitable rapidly decreasing test functions.
- Tempered distributions
are functionals over Schwartz functions.
- The Fourier Transform
associates a tempered distribution to another.
- Parseval's theorem (1799).
The Fourier transform is unitary.
- Noteworthy distributions and their
Fourier transforms:
- Dirac's d and the
uniform distribution ( f (x) = 1).
- The signum function
sign(x) and its transform:
i / ps
- The Heaviside step function H(x) = ½ (1+sign(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.
- Sampling formula:
The unit comb
(
 )
is its own Fourier transform.
- Far image of a picture on translucent film
is its Fourier transform.
- The Radon transform
(used in lateral tomography) is easily inverted.
- Competing definitions of the Fourier
transform. For the record.
- Discrete Fourier Transform,
defined as a unitary involution.
- The Barber's Dilemma is
not a paradox, if analyzed properly.
- What is infinity?
There's more to it than a pretty symbol (¥).
- Peano's axioms
provide a rigorous definition of the set of natural integers.
- There are more real than rational numbers.
Cantor's diagonal argument.
- Cantor's ternary set.
A vanishing set of reals equipollent to the whole line.
- The axioms of set theory:
Fundamental axioms and the Axiom of Choice.
- Equivalents and alternatives
to the axiom of choice.
- The existence of nonmeasurable sets
is guaranteed by the Axiom of Choice.
- Functions and applications
are special types of binary relations.
- A set is smaller than its powerset:
A simple proof applies to all sets.
- Transfinite cardinals
describe the various sizes of infinite sets.
- The continuum hypothesis:
Is the continuum the smallest uncountable set?
- Transfinite ordinals:
Counting to infinity... and beyond.
- Surreal numbers
include reals, transfinite ordinals, infinitesimals & more.
- Numbers:
From integers to surreals. From reals to quaternions and beyond.
- A set belongs to a class
in NBG (a conservative extension of ZFC).
- The number 1 is not prime.
Good definitions allow simple theorems.
- Composite numbers are not prime,
but the converse need not be true...
- Two prime numbers whose sum is equal to their product.
- Gaussian integers:
Factoring into primes on a two-dimensional grid.
- The least common multiple
may be obtained without factoring into primes.
- Standard Factorizations: n4 + 4
is never prime for
n > 1 because...
- Euclid's algorithm
gives the GCD and the related Bézout coefficients.
- Bézout's Theorem:
The GCD of p and q is of the form u p + v q.
- Greatest Common Divisor (GCD)
defined for all commensurable numbers.
- Linear equation in integers
can be solved using Bézout's theorem.
- Pythagorean Triples:
Right triangles whose sides are coprime integers.
- The number of divisors of an integer.
- Perfect squares are the only integers
with an odd number of divisors.
- The product of all divisors
is often a perfect square.
- Perfect numbers and Mersenne primes.
Do odd perfect numbers exist?
- Multiperfect & hemiperfect numbers.
Whole or half-integral abundancies.
- Fast exponentiation by repeated squaring.
- Partition function.
How many collections of positive integers add up to 15?
- A Lucas sequence
whose oscillations never carry it back to -1.
- A bit sequence with intriguing
statistics. Counting squares between cubes.
- Binet's formulas:
N-th term of a sequence obeying a linear recurrence.
- The square of a Fibonacci number
is almost the product of its neighbors.
- D'Ocagne's identity
relates conjugates products of Fibonacci numbers.
- Catalans's identity
generalizes Cassini's Identity.
- Faulhaber's formula
gives the sum of the p-th powers of the first n integers.
- Multiplicative functions:
If a and b are coprime, then
f(ab) = f(a) f(b).
- Moebius function:
Getting N values with O(N Log(Log N)) additions.
- Dirichlet convolution
is especially interesting for multiplicative functions.
- Dirichlet powers of arithmetic functions
(especially, the Möbius function).
- Dirichlet powers of multiplicative functions
are given by a superb formula.
- Totally multiplicative functions are
the simplest multiplicative functions.
- Dirichlet characters are
important totally multiplicative functions.
- Euler products
and generalized zeta functions.
- Modular arithmetic may be used to find the last digits
of very large numbers.
- Powers of ten
expressed as products of two factors without zero digits.
- Divisibility by 7, 13, and 91
(or by B2-B+1 in base B).
- Lucky 7's. Any integer divides a number composed
of only 7's and 0's.
- The decimal representation of rational numbers
is ultimately periodic.
- Midy's theorem: Properties of periods in radix-B numeration
- Numbers with two decimal expansions.
E.g., 1 and 0.99999999999999...
- Binary and/or hexadecimal numeration
for floating-point numbers as well.
- Extract a square root the old-fashioned way.
- Ternary system:
Is base 3 really the best radix for positional numeration?
- A prime number
is a positive integer with 2 distinct divisors (1 and itself).
- Euclid's proof:
There are infinitely many primes.
- Dirichlet's theorem:
There are infinitely many primes of the form kN+a.
- Green-Tao theorem:
Arbitrarily long arithmetic progressions of primes.
- Von Mangoldt's function
is Log p for a power of a prime p, 0 otherwise.
- Prime Number Theorem:
The probability that N is prime is roughly 1/ln(N).
- The average number of factors
of a large number N is Log N.
- The average number of distinct prime factors
of N is Log Log N.
- The largest known prime:
Historical records, old and new.
- The Lucas-Lehmer Test
checks the primality of a Mersenne number fast.
- Formulas giving only primes
may not help with new primes.
- Ulam's Lucky Numbers
and other sequences generated by sieves.
- Chinese remainder theorem:
Remainders define an integer, within limits.
- Modular arithmetic:
The formal algebra of congruences, due to Gauss.
- Fermat's little theorem:
For any prime p not dividing a, ap-1 is 1 modulo p.
- Euler's totient function:
f(n) counts the integers coprime to n, from 1 to n.
- Fermat-Euler theorem:
If a is coprime to n,
a to the f(n) is 1 modulo n.
- Carmichael's reduced totient function
(l) : A special divisor of the totient.
- 91 is a pseudoprime
to half of the bases coprime to itself.
- Carmichael Numbers:
An absolute pseudoprime n divides
an-a for any a.
- Chernik's Carmichael numbers:
3 prime factors (6k+1)(12k+1)(18k+1).
- Other products
that yield a Carmichael number iff every factor is prime.
- Large Carmichael numbers
may be obtained in various ways.
- Conjecture:
Any odd integer coprime to its totient has Carmichael multiples.
- Monoids feature
an associative operation and a neutral element.
- The inverse of an element
comes in 2 flavors that coincide when both exist.
- Free monoid:
All the finite strings (words) in a given alphabet.
- Raising something to the power of an integer.
- Groups
are monoids in which every element is invertible.
- A subgroup is a group
contained in another group.
- Generators of a group
are not contained in any proper subgroup.
- Lagrange's theorem:
The order of a subgroup divides the order of the group.
- Normal subgroups
and their quotients in a group.
- Homomorphism:
The image of a product is the product of the images.
- The symmetric group on
E consists of all the bijections of E onto itself.
- Inner automorphisms:
Inn(G) is isomorphic to G modulo its center.
- The conjugacy class formula
uses conjugacy to tally elements of a group.
- Simple groups
are groups without nontrivial normal subgroups.
- The derived subgroup of a group
is generated by its commutators.
- Direct product of two groups
(called a direct sum for additive groups).
- Groups of small orders.
Basic families: Cyclic groups, dihedral groups, etc.
- Enumeration
of "small" groups. How many groups of order n?
- Classification of finite simple groups,
by Gorenstein and many others.
- Sporadic groups:
Tits Group, 20 relatives of Fischer's Monster, 6 pariahs.
- Classical groups:
Their elements depend on parameters from a field.
- The Möbius group
consists of homographic transformations of
 È{¥}.
- Lorentz transformations
may change spatial orientation or time direction.
- Symmetries of the laws of nature:
A short primer.
- Rings
are sets endowed with addition, subtraction and multiplication.
- Divisors of zero,
idempotent elements and nilpotent elements.
- Nonzero characteristic:
Least p for which all sums of p like terms vanish.
- Ideals
within a ring are multiplicatively absorbent additive subgroups.
- Quotient ring, modulo an ideal:
The residue classes modulo that ideal.
- Cauchy multiplication
is well-defined for "formal power series" over a ring.
- Ring of polynomials
whose coefficients are in a given ring.
- Galois rings.
Residues of modular polynomials, modulo one of them.
- Vocabulary: We consider
skew fields to be noncommutative. Some don't.
- Fields are commutative rings where
every nonzero element has a reciprocal.
- Wedderburn's Theorem:
Finite division rings are necessarily commutative.
- Every finite integral domain is a
field.
- Galois fields are the
finite fields. Their orders are powers of primes.
- The trivial field is a singleton.
It's the only field where 0 is invertible.
- Splitting field of P in F[x] :
Smallest extension of F where P fully factors.
- Conway's Nim-Field is algebraically complete.
It contains infinite ordinals.
- Ternary multiplication
compatible with ternary addition (without "carry").
Vector Spaces (over a field)
and Modules (over a ring)
- Vectors were originally just
differences between points in ordinary space...
- Abstract vector spaces:
Vectors can be added, subtracted and scaled.
- Dimension of a vector space:
The number of its independent generators.
- Subspaces. Intersection. Sum.
Direct sums of supplementary subspaces.
- Linear maps between vector spaces respect
addition and scaling.
- Quotient of two vector spaces.
Hyperplanes have codimension 1.
- Fundamental theorem of linear algebra
and rank theorem.
- Modules are vectorial structures over
a ring of scalars (instead of a field).
- Normed vector spaces.
The fundamental properties of a norm.
- Dual space:
The set of all [continuous] linear functions with scalar values.
- Lebesgue spaces.
Sequence spaces exemplify more general types.
- Tensors:
Multilinear functions of vectors and covectors with scalar values.
- Algebra: A vector space
with a scalable and distributive internal product.
- Clifford algebra: Unital
associative algebra endowed with a quadratic form.
- Things that are not vectorial
because they're not defined intrinsically.
- David Hestenes proposed
geometric calculus as a denotational unification.
- Convex sets in a real vector space.
- Aspect ratio:
Smallest ro largest width (i.e., height to diameter ratio).
- A norm is characterized
by a closed convex body, symmetric about 0.
- Convex hull: Conv (S) is
the smallest convex set containing the set S.
- Closed halfspaces
generate all closed convex sets by intersection.
- Polar of a closed convex set.
Dot-product duality among convex bodies.
- Separating hyperplane
(in the loose sense) between disjoint convex sets.
- A compact convex
can be strictly separated from a disjoint closed convex.
- Two disjoint open convexes
are separated by a nonintersecting hyperplane.
- Functionals assign scalar values to
some functions over an infinite set I.
- Eduard Helluy (1912):
The space C[a,b] (continuous functions over [a,b]).
- Hahn-Banach extension theorem.
Extending a dominated functional.
- Hahn-Banach separation theorem.
A different view of the same result.
- The ring of p-adic integers.
Objects with infinitely many radix-p digits.
- Polyadic integers: Greek naming of p-adic integers.
- What if p isn't prime?
Dealing with divisors of zero.
- Decadic Integers:
The strange realm of 10-adic integers (composite radix).
- Decadic puzzle:
A tribute to the late columnist J.A.H. Hunter (1902-1986).
- The field of p-adic numbers.
Quotient field of the ring of p-adic integers.
- Dividing two p-adic numbers
looks like "long division", only backwards...
- Overbar notation,
for p-adic and rational numbers alike.
- The p-adic metric
can be used to define p-adic numbers analytically.
- The reciprocal of a p-adic number
computed by successive approximations.
- Ratios of rational integers have two representations:
g-adic and radix-g.
- Solving algebraic equations in p-adic integers.
 -linear maps between
and Qp are discontinuous at every point.
- Hasse's local-global principle.
Established for the quadratic case in 1920.
- Rotating digit patterns (in base g)
may double the corresponding values.
- Rotating digits one place to the left divides some integers by k.
- Pseudoprimes to base a.
Poulet numbers are pseudoprimes to base 2.
- Weak pseudoprimes to base a :
Composite integers n which divide
(an-a).
- Counting the bases to which
a composite number is a pseudoprime.
- Strong pseudoprimes to base a
are less common than Euler pseudoprimes.
- The witnesses of a composite number:
At least 75% of nontrivial bases.
- Rabin-Miller Test:
An efficient and trustworthy stochastic primality test.
- The product of 3 primes
is a pseudoprime when all pairwise products are.
- Wieferich primes
are scarce but there ought to be infinitely many of them.
- Super-pseudoprimes:
All their composite divisors are pseudoprimes.
- Maximal super-pseudoprimes
have no super-pseudoprime multiples.
- Jevons Number. Factoring
8616460799 is now an easy task.
- Challenges help tell
special-purpose and general-purpose methods apart.
- Special cases of a priori
factorizations are helpful to number theorists.
- Trial division may be used
to weed out the small prime factors of a number.
- Ruling out factors can speed up trial
divison in special cases.
- Recursively defined sequences (over a
finite set) are ultimately periodic.
- Pollard's rho factoring
method is based on ultimately periodic sequences.
- Pollard's p-1 Method finds
prime factors p for which p-1 is smooth.
- Williams' p+1 Method is based on the
properties of Lucas sequences.
- Lenstra's Elliptic Curve Method generalizes
Pollard's p-1 approach.
- Dixon's method: Combine small square residues into
a solution of x 2
º y 2
- Motivation:
On the prime factors of some quadratic forms...
- Quadratic residues:
Half of the nonzero residues modulo an odd prime p.
- Euler's criterion:
A quadratic residue raised to the power of (p-1)/2 is 1.
- The Legendre symbol (a|p)
extends to values of p besides odd primes.
- The law of quadratic reciprocity
states a simple but surprising fact.
- Gauss' Lemma expresses a
Legendre symbol as a product of many signs.
- Eisenstein's Lemma: A variation
of Gauss's lemma allows a simpler proof.
- One of many proofs of the
law of quadratic reciprocity.
- Artin's Reciprocity.
- What is a continued fraction?
Example: The expansion of p.
- The convergents of a number
are its best rational approximations.
- Large partial quotients
allow very precise approximations.
- Regular patterns
in the continued fractions of some irrational numbers.
- In almost all cases,
partial quotients are ≥ k with probability lg(1+1/k).
- Elementary operations on continued fractions.
- Expanding functions as continued fractions.
- Engel expansions of positive numbers
are nondecreasing integer sequences.
- Pierce expansions of numbers from 0 to 1.
Strictly increasing sequences.
- Counterfeit Coin:
In 3 weighings, find an odd object among 12, 13 or 14.
- Counterfeit Penny Problem:
Find an odd object in the fewest weighings.
- Seven-Eleven: Four prices
with a sum and product both equal to 7.11.
- Equating a right angle and an obtuse angle,
with a clever false proof.
- Choosing a raise:
Trust common sense, beware of fallacious accounting.
- 3 men pay $30 for a $25 hotel room,
the bellhop keeps $2... Is $1 missing?
- Chameleons:
A situation is unreachable because of an invariant quantity.
- Sam Loyd's 14-15 puzzle
also involves an invariant quantity (and 2 orbits).
- Einstein's riddle:
5 distinct colors, nationalities, drinks, smokes and pets.
- Numbering n pages
of a book takes this many digits (formula).
- The Ferry Boat Problem (by Sam Loyd):
To be or not to be ingenious?
- Hat overboard !
What's the speed of the river?
- All digits once and only once:
48 possible sums (or 22 products).
- 2-people bridge crossed by 4 people (U2).
Four paces, one flashlight!
- Managing supplies
to travel 6 days while carrying enough for only 4 days.
- Go south, east, north and you're back...
not necessarily to the North Pole!
- Icosapolis:
Numbering a 5 by 4 grid so adjacent numbers differ by at least 4.
- Unusual mathematical boast for people born
in 1806, 1892, or 1980.
- Puzzles for extra credit:
From Chinese remainders to the Bookworm Classic.
- Simple geometrical dissection:
A proof of the Pythagorean theorem.
- Early bird saves time by walking to
meet incoming chauffeur.
- Sharing a meal:
A man has 2 loaves, the other has 3, a stranger has 5 coins.
- Fork in the road:
Find the way to Heaven by asking only one question.
- Proverbial Numbers:
Words commonly associated with some numbers.
- Riddles:
The Riddle of the Sphinx and other classics, old and new.
- Martin Gardner (1914-2010)
described himself as "strictly a journalist".
- Flexagons:
Hexaflexagons were popularized by Martin Gardner in 1956.
- Polyominoes:
The 12 pentominoes and other tiles invented by Sol Golomb.
- Soma: 7 nonconvex solids
consisting of 3 or 4 cubes make a larger cube.
- Tessellations by convex pentagons.
The contributions of Marjorie Rice.
- Kites and Darts.
The aperiodic tilings of Roger Penrose.
- Ambigrams:
Calligraphic spellings which change when rotated or flipped.
- The Game of Life.
John Conway's endearing cellular automaton (1970).
- Rubik's Cube:
Ernõ Rubik (1974) Singmaster (1979) Gardner (1981).
- It's impossible to tie a knot
without letting go of the ends of the string.
- On the limited knowledge of Man.
An Indian legend...
- 1089: Subtract a 3-digit number and its reverse,
then...
- Multiples of Nine: A secret symbol is revealed.
- Casting Out Nines: A missing digit is revealed.
- Mass media mentalism
by David Copperfield (1992).
- Grey Elephants in Denmark:
Classroom mental magic.
- Fitch Cheney's 5-card trick:
4 cards tell the fifth one.
- Generalizing the 5-card trick
and Devil's Poker...
- Kruskal's Count.
- Paths to God.
- Stacked Deck.
- Enigma Card Trick.
- Magic Age Cards.
- Ternary Cards.
- Magical 21 (or 27).
- Boolean Magic.
- Perfect Faro Shuffles.
- Equal Numbers of Heads !
- Deceit and lying.
- Misdirection.
- Find the Lady.
- Cups and balls.
One of the most ancient tricks.
- Chop cup.
Invented by "Chop-Chop" Wheatley in 1954.
- Invisible Thread Reel (ITR) by
James George (1992).
- Force and Reveal: A whole class of magic tricks.
- Dots and Boxes: The "Boxer's Puzzle" position of Sam Loyd.
- The Game of Nim:
Remove items from one of several rows. Don't play last.
- Sprague-Grundy numbers
are defined for all positions in impartial games.
- Moore's Nim:
Remove something from at most (b-1) rows. Play last.
- Normal Kayles:
Knocking down a pin or two adjacent pins may split a row.
- Grundy's Game:
Split a row into two unequal rows, if at all possible.
- Wythoff's Game:
Take either from one heap or equally from both heaps.
- Nalimov Tables:
Perfect analysis of endgame situations.
- Evaluation function:
Estimating a quiescent position statically.
- Minimax search tree:
The basic paradigm for analyzing two-player games.
- Alpha-beta pruning.
Iin a minimax search, some alternatives can be ignored.
- Hash tables.
How to avoid analyzing the same position more than once.
- Short chess games.
Checkmates occuring during the opening moves.
- Classic traps:
Fishing pole, etc.
- The pigeonhole principle:
What's entailed by fewer holes than pigeons.
- Among 70 distinct integers between 1 and 200,
two must differ by 4, 5 or 9.
- n+1 of the first 2n integers
always include two which are coprime.
- Largest sets of small numbers with at
most k pairwise coprime integers.
- Ramsey's Theorem:
Monochromatic complete subgraphs of a large graph.
- Infinite alignment among infinitely many lattice points
in the plane? Nope.
- Infinite alignment in a lattice sequence with bounded
gaps? Almost...
- Large alignments in a lattice sequence with bounded
gaps. Yeah!
- Van der Waerden's theorem: Long
monochromatic arithmetic progressions.
- Ford circles: Nonintersecting circles
touching the real line at rational points.
- Farey series:
The rationals from 0 to 1, with a bounded denominator.
- The Stern-Brocot tree
features every positive rational once and only once.
- Any positive rational
is a unique ratio of two consecutive Stern numbers.
- Pick's formula gives the area of a lattice polygon
by counting lattice points.
History :
- Earliest mathematics on record.
Before Thales was Euphorbus...
- Indian numeration
became a positional system with the introduction of zero.
- Roman numerals are awkward for larger numbers.
- The invention of logarithms:
Napier, Bürgi, Briggs, St-Vincent, Euler.
- The earliest mechanical calculators.
W. Shickard (1623) & Pascal (1642).
- The Fahrenheit Scale:
100°F was meant to be the normal body temperature.
- The revolutionary innovations
which brought about new civilizations.
Nomenclature & Etymology :
- The origin of the word "algebra",
and also that of "algorithm".
- The name of the avoirdupois system
is from a pristine form of French.
- Long Division:
Cultural differences in long division layouts.
- Is a parallelogram a trapezoid?
In a mathematical context [only?], yes it is...
- Naming polygons.
Greek only please; use hendecagon not "undecagon".
- Chemical nomenclature:
Sequential names are systematic or traditional.
- Fractional prefixes:
hemi (1/2)
sesqui (3/2)
hemipenta (5/2)
hemisesqui (3/4).
- Matches, phosphorus, and
phosphorus sesquisulphide.
- Zillion. Naming large numbers.
- Zillionplex. Naming huge numbers.
- Abbreviations:
Abbreviations of scholarly Latin expressions.
- "Resp." is a
mathematical symbol with its own syntax.
- Typography of long numbers.
- Intervals
denoted with square brackets (outward for an excluded extremity).
- Dates in the simplest ISO 8601
form (with customary time stamps or not).
- The names of operands
in common numerical operations.
- Spoken numbers.
- Pronouncing mathematical expressions,
like native English speakers do.
- PEMDAS:
A mnemonic for a rule that should not be taught.
- Physical units:
Their products and their ratios.
- The word respectively
doesn't have the same syntax as "resp."
- The heliocentric system
was known two millenia before Copernicus.
- The assistants of Galileo
and the mythical experiment at the Tower of Pisa.
- Switching calendars:
Newton was not born the year Galileo died.
- The Lorenz Gauge is due to
Ludwig Lorenz (1829-1891) not H.A. Lorentz.
- Special Relativity was first formulated
by Henri Poincaré.
- The Fletcher-Millikan "oil-drop" experiment
isn't entirely due to Millikan.
- Collected errata about customary physical units.
- Portrait of Legendre:
The mathematician was confused with a politician.
- The iconography used for Apollonius of Perga
was meant for another man.
- Dubious quotations:
Who really said that?
- Classical geometry describes an
homogeneous space indifferent to scale.
- Obliquity of the ecliptic
in the time of Eratosthenes (276-194 BC).
- Vertical wells at Syene are
completely sunlit only once a year, aren't they?
- Eratosthenes sizes up the Earth:
700 stadia per degree of latitude.
- The distance to the Moon
was computed by Aristarchus and Hipparchus.
- Latitude and longitude:
The spherical grid of meridians and parallels.
- Itinerary units:
The land league and the nautical league.
- Amber, compass and lightning:
Glimpses of electricity and magnetism.
- On the nature of physical laws:
The example of gravitation.
- Controlled Experiment:
A concept attributed to Sir Francis Bacon (1590).
- History of the Scientific Method.
- Distinguishing between Science and
Pseudoscience.
- Faster-than-light neutrinos?
How the media butcher the scientific method.
- What is time? Why don't we remember the future?
- The beginning of time. Was there anything before that?
- Time machines: Unavoidable microscopically,
impossible macroscopically.
- Determinism precludes the arrow of time.
- The notion of force.
Statics, mechanical advantage and virtual work.
- Speed.
Allowing the division of unlike quantities (distance and time).
- Mean-speed theorem.
The distance traveled at constant acceleration.
- The timing experiments of Galileo:
From the pendulum to falling bodies.
- The true period of a pendulum
is proportional to 1 / agm ( 1 , cos A/2 ).
- The parabola of a cannonball,
compared to Aristotle's triangular path.
- Conservation of momentum
is key to Newton's three laws of motion.
- The work done to a point-mass
equals the change in its kinetic energy.
- Relativistic work done
and the corresponding change in relativistic energy.
- Relativistic thermodynamics:
A point-mass endowed with internal heat.
- Spacecraft speeds up upon reentry
into the upper atmosphere.
- Lewis Carroll's monkey
climbs a rope over a pulley, with a counterweight.
- Two-ball drop
can make one ball bounce up to 9 times the dropping height.
- Normal acceleration =
Square of speed divided by the radius of curvature.
- Roller-coasters must rise
more than half a radius above any loop-the-loop.
- Conical pendulum:
A hanging bob whose trajectory is an horizontal circle.
- Conical pendulum constrained by a hemisphere:
The string tension.
- Ball in a Bowl:
Pure rolling increases the period of oscillation by 18.3%.
- Hooke's Law:
Simple harmonic motion of a mass suspended to a spring.
- Speed of an electron
estimated with the Bohr model of the atom.
- Hardest Stuff:
Diamond is no longer the hardest material known to science.
- Hardness is an elusive
nonelastic property, distinct from stiffness.
- Hot summers, hot equator!
The distance to the Sun is not the explanation.
- Kelvin's Thunderstorm:
Using falling water drops to generate high voltages.
- The Coriolis effect:
A dropped object falls to the east of the plumb line.
- Terminal velocity
of an object falling in the air.
- Angular momentum and torque.
Spin and orbital angular momentum.
- Rotation vector
of a moving rigid body (and/or "frame of reference").
- Angular momentum equals
moment of inertia times angular velocity.
- Kinetic energy of a solid:
Sum of its translational and rotational energies.
- Moments about a point or a plane are
convenient mathematical fictions.
- Moment of inertia of a spherical distribution
or an homogeneous ellipsoid.
- Moment of inertia of the Earth is equal
to 0.330695 M a 2.
- Perpendicular Axis Theorem:
Axis of rotation perpendicular to a lamina.
- The Parallel Axis Theorem: Moment
of inertia about an off-center axis.
- Moment of inertia of a thick plate,
derived from the parallel axis theorem.
- Moment of inertia of a right cone
or conical frustum.
- Momenta of homogeneous bodies.
List of common examples.
- Rigid pendulum
moving under its own weight about a fixed horizontal axis.
- Reversible pendulum.
The same period around two distinct axes.
- All physical theories
have a limited range of validity.
- Gravity vs. Electrostatics:
Straight comparisons.
- Binet's formulas:
Deriving Kepler's laws for two orbiting bodies.
- Airy weighs the Earth
by timing a pendulum deep in a mine.
- Rigid equilateral triangle
formed by three gravitating bodies.
- The five Lagrange points
of two gravitating bodies in circular orbit.
- Geosynchronous Orbit:
Semimajor radius of 36000 km around the Earth.
- The gravitational self-energy
of a ball (mass M, radius R) is -0.6 GM2/R
- Tides on Earth:
Dominant rôle of the Moon. Lesser rôle of the Sun.
- Asteroid 99942 Apophis:
Near-Earth objects and gravitational keyholes.
- Mass distributions of galaxies.
Evidence for the existence of dark matter.
- Coefficients of friction: ds the kinetic one.
- Example
involving a nontrivial choice between static and kinetic regimes.
- Minimum inclination of a ladder
leaning against a frictionless wall.
- Spinning cylinder on an horizontal plane:
The skidding before pure roll.Drawing a dotted line on a blackboard.
- Coefficient of restitution (e)
Ratio of initial to final closing speed.
- Billiard and pool tables:
Sizes, slate bed, cloth, rails and cushions.
- Billiard balls: Phenolic resin
binding a dense powder has replaced ivory.
- Cue sticks:
Butts and shafts. Basic construction. Anti-squirt technology.
- The contents of a cue case
reflect the player's basic choices.
- Cue tips.
Leather and phenolic tips.
- Two types of billiard chalk to
reduce hand friction or increase tip friction.
- Normal trajectory of a billiard ball:
A parabola followed by a straight line.
- Making the cue ball stop
after hitting the object ball.
- The impossible 90° cut-shot
made possible with extreme english.
- Squirt
between cue and cue ball with extreme English (vertical spin axis).
- Jump shots.
Legal and illegal ways to send the cue ball up in the air.
- Matrix methods: Transformations of a
ray's inclination and radial distance.
- A crystal ball (index n and radius R)
has focal length f = R / (2n-2).
- Lensmaker's formula Focal lens as a
function of signed curvatures.
- Concave mirrors create enlarged
virtual images of objects in front of them.
- Opposition effect increases albedo
by eliminating micro-shadows.
- Huygens' Principle. A convenient fiction to
describe wave propagation.
- Diffraction occurs when when a wave
emanates from a bounded source.
- Young's
double-slit experiment demonstrates the wavelike nature of light.
- Celerity is the speed with which
phase propagates.
- Standing waves feature stationary nodes
and antinodes.
- Chladni patterns: The lines formed by nodes
in an oscillating plate.
- Snell's Law (1621) gives the angle of refraction
(if anything is refracted).
- Birefringence. Discovery of
polarization (Erasmus Bartholinus, 1669).
- Brewster's angle is the incidence which yields
a 100% polarized reflection.
- Fresnel equations:
Reflected or refracted intensities of polarized light.
- Stokes parameters:
A standard description of the state of polarization.
- Transverse wave on a rope:
(celerity) 2 = (tension) / (linear mass density).
- Dispersion relation: Pulsatance
vs. wave number; frequency vs. wavelength.
- Group velocity
is the traveling speed of a beat phenomenon.
- Rayleigh scattering
makes the sky blue and sunsets red.
- Index of refraction of water
for light of different colors.
- A spherical drop
reflects light back (red up to 42.34° & violet up to 40.58°).
- The length
of a rainbow: Mathematical digression.
Lasers : From masers to laser beams
- Stimulated emission is crucial to
blackbody equilibrium (Einstein, 1916).
- Bose-Einstein Statistics is what explains
stimulated emission of bosons.
- Population inversion : When energetic
states are abnormally abundant.
- LASER Cavity "Light Amplification by
Stimulated Emission of Radiation".
- Gaussian beam. The shape of an ideal laser beam.
- Fermat's principle
(least time) for light (c.1655) predates Newton.
- Maupertuis principle of
least action (1744).
- Virtual Work: A substitute
for Newton's laws that cancels constraint forces.
- Phase Space: A phase
describes completely the state of a classical system.
- Either velocities or momenta are added
to configuration to specify a phase.
- Relativistic point-mass:
Lagrangian, Hamiltonian and free momentum.
- Charge in a magnetic field:
The canonical momentum isn't the linear one.
- The Lagrangian
is a function of positions and velocities.
- The Hamiltonian
depends on positions and momenta.
- Poisson brackets: An
abstract synthetic view of analytical mechanics.
- Liouville's theorem: The
Hamiltonian phase volume doesn't change.
- Noether's theorem:
Conservation laws express the symmetries of physics.
- Field theory:
Lagrangian function of a continuum of values and velocities.
- Clarifications:
Vector calculus (Heaviside) & microscopic view (Lorentz).
- The vexing problem of units is a thing of the past
if you stick to SI units.
- The Lorentz force on a test particle
defines the local electromagnetic fields.
- Electrostatics (1785):
The study of the electric field due to static charges.
- Electric capacity
is an electrostatic concept (adequate at low frequencies).
- Electrostatic multipoles:
The multipole expansion of an electrostatic field.
- Birth of electromagnetism (1820):
Electric currents create magnetic fields.
- Biot-Savart Law:
The static magnetic induction due to steady currents.
- Magnetic scalar potential:
A multivalued static scalar field.
- Magnetic monopoles do not exist :
A law stating a fact not yet disproved.
- Ampère's law (1825):
The law of static electromagnetism.
- Faraday's law (1831):
Electric circulation induced by magnetic flux change.
- Self-induction
received by a circuit from the magnetic field it produces.
- Ampère-Maxwell law:
Dynamic generalization (1861) of Ampère's law.
- Putting it all together:
Historical paths to Maxwell's electromagnetism.
- Maxwell's equations
unify electricity and magnetism dynamically (1864).
- Continuity equation:
Maxwell's equations imply conservation of charge.
- Waves
anticipated by Faraday, Maxwell & FitzGerald. Observed by Hertz.
- Electromagnetic energy density and
the flux of the Poynting vector.
- Planar electromagnetic waves:
The simplest type of electromagnetic waves.
- Maxwell-Bartoli radiation pressure.
First detected by P. Lebedev in 1899.
- Electromagnetic potentials
are postulated to obey the Lorenz gauge.
- Solutions to Maxwell's equations,
as retarded or advanced potentials.
- Electrodynamic fields
corresponding to retarded potentials.
- Electrodynamic fields
corresponding to advanced potentials.
- The gauge of retarded potentials:
is it really the Lorenz gauge?
- Power radiated by an accelerated charge:
The Larmor formula (1897).
- Lorentz-Dirac equation
for the motion of a point charge is of third order.
- Molecular electric dipole moments.
First studied by Peter Debye in 1912.
- Force exerted on a dipole
by a nonuniform field.
- Torque on a dipole is proportional
to its cross-product into the field.
- Electric and magnetic dipoles:
Dipolar solutions of Maxwell's equations.
- Static distributions of magnetic dipoles
can be emulated by steady currents.
- Static distributions of electric dipoles
are equivalent to charge distributions.
- Field at the center of a uniformly magnetized or polarized
sphere of any size.
- Sign reversal in
magnetic and electric fields from matching dipoles.
- Relativistic dipoles: A moving magnet
develops an electric moment.
Magnetism,
Electromagnetic Properties of Matter
- Magnetization and polarization
describe densities of bound dipoles.
- Distinct magnetization and polarization
gauges may yield the same field.
- Maxwell's equations in matter:
Electric displacement & magnetic strength.
- Electric susceptibility
is the propensity to be polarized by an electric field.
- Electric permittivity and magnetic permeability.
Related to susceptibilities.
- Paramagnetism:
Weak positive susceptibility.
- Diamagnetism:
Lorentz force
turns orbital moments against an external B.
- Magnetic levitation:
How to skirt the theorem of Samuel Earnshaw (1842).
- Pyrolytic carbon:
The most diamagnetic substance, at room temperature.
- Bohr-van Leeuwen Theorem:
Diamagnetism and paramagnetism cancel ?!
- Thermodynamics of dielectric matter:
dU = E.dD + ...
- Ferromagnetism:
Permanent magnetization without an external field.
- Antiferromagnetism:
When adjacent dipoles tend to oppose each other...
- Ferrimagnetism:
With two kinds of dipoles, partial cancellation may occur.
- Magneto-optical effect
discovered by Faraday on September 13, 1845.
- Ohm's Law: Current density is proportional
to electric field: j = s E.
- Homopolar motor: The first electric
motor, by Michael Faraday (1831).
- Faraday's disk can generate huge currents
at a low voltage.
- Magic wheels:
Two repelling ring magnets mounted on the same axle.
- Beakman's motor.
Current switches on and off as the coil spins.
- Tesla turbine.
Stack of spinning disks with outer intake and inner outflow.
- Aristotle's plenism.
Downfall of the Horror Vacui doctrine (17th century).
- Vacuum tubes.
Heated filaments, grids and electrons moving in a vacuum.
- Dirac's equation
predicted positrons as holes in a bizarre vacuum.
- The Quantum Vacuum.
The vacuum isn't empty. Structure of the vacuum.
- Observers in motion:
A simple-minded derivation of the Lorentz Transform.
- Adding up parallel velocities:
The combined speed can never exceed c.
- Combining velocities
using an angle measured in a moving frame.
- The headlight effect:
An isotropic source will radiate forward if it moves.
- Closing speed: between objects may decrease faster than c.
- Fizeau's empirical relation
between refractive index (n) and Fresnel drag.
- The Harress-Sagnac effect
used to measure rotation with fiber optic cable.
- Combining relativistic speeds:
Using rapidity, the rule is transparent.
- Relative velocity of two photons:
Undefined if they have the same direction
- Minkowski spacetime:
Lorentz transform applies to 4-vector coodinates.
- The Lorentz transform expressed vectorially
for a boost of speed V.
- Wave vector:
The 4-dimensional gradient of the phase describes a wave.
- Doppler shift:
The relativistic effect is not purely radial.
- Relativistic momentum
and Einstein's relation between mass and energy.
- Kinetic energy: At low speed, the relativistic
energy varies like ½ mv 2.
- Photons and other massless particles:
Finite energy at speed c.
- The de Broglie celerity (u) is
inversely proportional to a particle's speed.
- Compton diffusion:
The result of collisions between photons and electrons.
- The Klein-Nishina formula:
gives the cross-section in Compton scattering.
- Compton effect is suppressed
for visible light and bound electrons.
- Elastic shock:
Energy transfer is v.dp.
(None is seen from the barycenter.)
- Photon-photon scattering
is like an elastic collision of two photons.
- Cherenkov effect:
When an electron exceeds the celerity of light...
- Constant acceleration
over an entire lifetime will take you pretty far.
- Photons
are quanta of light that are both wavelike and corpuscular.
- The photoelectric effect
was explained by Albert Einstein in 1905.
- Henri Becquerel
and the discovery of natural radioactivity (1896).
- Pierre & Marie Curie:
The discovery of new radioactive elements (1898).
- Geiger-Marsden experiment:
There's a tiny dense nucleus inside the atom!
- Alpha-decay:
Polonium (Po-210, Z=84) decays into Lead (Pb-206, Z=82).
- Mass Defect:
In a nuclear reaction, the Q-value balances the mass change.
- The standard decay
modes: a, b-, 2b-, b+, e (electron capture) or IT.
- The 4 radioactive series:
Thorium, Neptunium, Uranium and Actinium.
- Other decay modes:
Proton or neutron emission, fission and spallation.
- The Geiger counter
measures the activity flux of ionizing radiation.
- Scintillation
allows quantitive measurements of a gamma spectrum.
- Cross-section:
The target's size depends on the projectile's speed.
- Artificial radioactivity:
Neutron bombardment creates unstable nuclides.
- Chain reactions:
When neutron-induced decays produce more neutrons...
- Critical mass:
The smallest mass that will allow runaway chain reactions.
- Thermonuclear bombs.
Nuclear fusion ignited by fission devices.
- Carbon-dating:
Radiocarbon ratio starts decaying when an organism dies.
- Fusion of deuterons:
Helium is formed with liberation of energy.
- The Proton-Proton chain fusion
powers all stars less than 1.5 solar masses.
- Tokamak reactors:
Deuterium-Tritium fusion (DT) is the easiest to ignite.
- Farnsworth-Hirsch fusor:
Controlled fusion on a desktop. Neutron source.
- Polywell reactor:
The design advocated by the late Robert Bussard.
- Amateur nuclear physics:
Demystifying nuclear energy and radioactivity.
- The Radioactive Boyscout
and other misguided experimenters.
- The Harress-Sagnac effect
seen by an observer rotating with an optical loop.
- Relativistic rigid motion
is an equilibrium modified at the speed of sound.
- In the Euclidean plane:
Contravariance and covariance.
- In the Lorentzian plane:
Contravariance and covariance revisited.
- Tensors of rank n+1 are linear maps
that send a vector to a tensor of rank n.
- Signature of the quadratic form
defined by a given metric tensor.
- Covariant and contravariant coordinates
of rank-n tensors, in 4 dimensions.
- The metric tensor and its inverse.
Lowering and raising indices.
- Partial derivatives along
contravariant or covariant coordinates.
- Christoffel symbols:
Coordinates of the partial derivatives of basis vectors.
- Covariant derivatives.
Absolute differentiation. The nabla operator
Ñ.
- Contravariant derivatives: The lesser-known
flavor of absolute derivatives.
- The antisymmetric part of Christoffels symbols
form a fundamental tensor.
- Totally antisymmetric spacetime torsion
is described by a vector field.
- Levi-Civita symbols:
Antisymmetric with respect to any pair of indices.
- Einstein's equivalence principle
implies vanishing spacetime torsion.
- Ricci's theorem: The covariant derivative
of the metric tensor vanishes.
- Curvature: The Ricci tensor
is a contraction of the Riemann tensor.
- The Bianchi identity shows that the
Einstein tensor is divergence free.
- Stress tensor:
Flow of energy density is density of [conserved] momentum.
- Einstein's Field Equations:
16 equations in covariant form (Einstein, 1915).
- Free-falling bodies: Their trajectories
are geodesics in curved spacetime.
- The "anomalous" precession of Mercury's perihelion
is entirely relativistic.
- Schwarzschild metric:
The earliest exact solution to Einstein's equations.
- What is mass?
- Unruh temperature experienced by an accelerating observer.
- Electromagnetism: Covariant expressions, using tensors.
- Kaluza-Klein theory of electromagnetism
involves a fifth dimension.
- Harvard Tower Experiment:
The slow clock at the bottom of the tower.
- Shapiro time delay:
The effect on radar signals of gravitational time dilation.
String Theory and other "Theories of Everything"
- Unification:
Consistency is required. Actual high-energy unification is not.
- Kaluza-Klein Theory:
Postulating an extra dimension for electromagnetism.
- 1960's hadron physics:
Regge trajectories begat constant-tension strings.
- Gabriele Veneziano:
The magic of Euler's beta and gamma functions.
- Leonard Susskind (1940-):
The basic idea of a fundamental string.
- Joël Scherk (1946-1979) & John Schwarz:
Rediscovering gravity.
- Michael Green & John Schwarz:
Hoping for a Theory of Everything.
- String Quintet:
Five different consistent string theories!
- M-Theory:
Ed Witten's 11-dimensional brainchild, unveiled at String '95.
- The brane world scenarios
of Lisa Randall and Burt Ovrut.
- The Magdeburg hemispheres are held together
by more than a ton of force.
- The ideal gas laws of
Boyle, Mariotte, Charles, Gay-Lussac, and Avogadro.
- Joule's law: Internal energy of an ideal
gas depends only on temperature.
- Deflating a tire:
Releasing pressurized gas into the atmosphere.
- The Van der Waals equation and other interesting
equations of state.
- Virial equation of state.
Virial expansion coefficients. Boyle's temperature.
- Viscosity
is the ratio of a shear stress to the shear strain rate it induces.
- Permeability and permeance:
Vapor barriers and porous materials.
- Resonant frequencies of air in a box.
- The Earth's atmosphere.
Pressure at sea-level and total mass above.
- The first hot-air balloon
(Montgolfière) was demonstrated on June 4, 1783.
- Sulfur hexafluoride
is a very heavy gas and a good electrical insulator.
- Viscosity:
The transport of microscopic momentum.
- Brownian motion and
Einstein's estimate of molecular sizes.
- Thermal Conductivity:
The transport of microscopic energy.
- Diffusivity:
The transport of chemical concentration.
- Speed of Sound:
Reversible transport of a pressure disturbance in a fluid.
- Complex pulsatance:
s = s+iw
(damping constant + imaginary pulsatance)
- Complex impedance:
Resistance and reactance.
- Quality Factor (Q).
Ratio of maximal stored energy to dissipated power.
- Nullators and norators: Strange dipoles
for analog electronic design.
- Corner frequency
of a simple first-order low-pass filter. -3 dB bandwidth.
- Second-order passive low-pass filter,
with inductor and capacitor.
- Sallen-Key filters: Active filters
do not require inductors.
- Lowpass Butterworth filter of order n :
The flattest low-frequency response.
- Linkwitz-Riley crossover filters
are used in modern active audio crossovers.
- Chebyshev filters:
Ripples in either the passband or the stopband.
- Elliptic (Cauer) filters
encompass all Butterworth and Chebyshev types.
- Legendre filters
maximal roll-off rate for monotonous frequency response.
- Gegenbauer filters:
From Butterworth to Chebyshev, via Legendre.
- Phase response of a filter.
- Bessel-Thomson filters:
Phase linearity and group delay.
- Gaussian filters: Focusing on
time-domain communication pulses.
- Linear phase equiripple: Ripples
in group delay to go beyond Bessel filters.
- DSL filters allow POTS
below 3400 Hz & block digital data above 25 kHz.
- Raising the Titanic, with (a lot of) hydrogen.
- Gravitational Subway:
From here to anywhere on Earth, in 42 minutes.
- In a vacuum tube, a drop to the center of the
Earth would take 21 minutes.
- The aeolipile: This
ancient steam engine demonstrates jet propulsion.
- Edward Somerset of Worcester (1601-1667):
Steam fountain blueprint.
- Denis Papin (1647-1714):
Pressure cooking and the first piston engine.
- Thomas Savery (c.1650-1715):
Two pistons and an independent boiler.
- Thomas Newcomen (1663-1729)
& John Calley: Atmospheric engine.
- Nicolas-Joseph Cugnot (1725-1804):
The first automobile (October 1769).
- James Watt (1736-1819):
Steam condenser and Watt governor.
- Richard Trevithick (1771-1833)
and the first railroad locomotives.
- Sadi Carnot (1796-1832):
Carnot's cycle. The theoretical efficiency limit.
- Sir Charles Parsons (1854-1931):
The modern steam turbine (1884).
- Drinking Bird:
Room-temperature engine based on evaporative cooling.
- Elementary concept of temperature.
The zeroth law of thermodynamics.
- Conservation of energy:
The first law of thermodynamics.
- Increase of Entropy:
The second law of thermodynamics.
- State variables:
Extensive and intensive quantities.
- Entropy
is missing information, a measure of disorder.
- Nernst Principle
(third law): Entropy is zero at zero temperature.
- Thermodynamic potentials
are convenient alternatives to internal energy.
- Calorimetric coefficients, adiabatic
coefficient (g) heat capacities, etc.
- Relations between isothermal
and isentropic coefficients.
- The thermal Grüneisen parameter.
- Entropy of a Van der Waals fluid
as derived from its equation of state.
- Dulong-Petit Law (1819).
The molar heat capacity of a metal is about 3 R.
- Thermal effects of molecular vibrations
at moderate temperatures.
- Latent heat (L)
is the heat transferred in a change of phase.
- Cryogenic coefficients:
Lower temperature with an isenthalpic expansion.
- Relativistic Thermodynamics:
A moving body appears cooler.
- Inertia of energy
for an object at nonzero temperature.
- Stefan's Law:
A black body radiates as the fourth power of its temperature.
- The "Fourth Law":
Is there really an upper bound to temperature?
- Hawking radiation:
On the entropy and temperature of a black hole.
- Partition function:
The cornerstone of the statistical approach.
- Elastic properties
Reversible deformations in perfectly resilient materials.
- Hysteresis and resilience.
Stored elastic energy is never fully recovered.
- Elastomers.
Unsaturated rubbers are cured by sulfur vulcanization.
- Thermal expansion coefficients:
Cubical scalar and linear tensor.
- Invar anomaly:
The low thermal expansion of 36% Ni / 86% Fe alloy.
- Waves in a solid:
P-waves (fastest), S-waves, E-waves (thin rod), SAW...
- Thermodynamics of acoustics:
Dynamic coefficients and isothermal ones.
- Rayleigh Wave:
The quintessential surface acoustic wave (SAW).
- Laplace's Demon:
Deducing past and future from a detailed snapshot.
- Maxwell's Demon:
Trading information for entropy.
- Shockley's Ideal Diode Equation:
Diodes don't violate the Second Law.
- Szilard's engine & Landauer's Principle:
Thermodynamic cost of forgetting.
- Lagrange multipliers.
One multiplier for each constraint of an optimization.
- Microcanonical equilibrium.
Isolated system: All states are equiprobable.
- Equipartition of energy.
Every degree of freedom gets an equal share.
- Canonical equilibrium:
Boltzmann factor in a heat bath.
- Grand-canonical equilibrium
when chemical exchanges are possible.
- Bose-Einstein statistics:
One state may be occupied by many particles.
- Fermi-Dirac statistics:
One state is occupied by at most one particle.
- Boltzmann statistics:
The low-occupancy limit (most states unoccupied).
- Maxwell-Boltzmann distribution
of molecular speeds in an ideal gas.
- Partition function:
The cornerstone of the statistical approach.
- Quantum Logic:
The surprising way quantum probabilities are obtained.
- Swapping particles
either negates the quantum state or leaves it unchanged.
- The Measurement Dilemma:
What makes Schrödinger's cat so special?
- Matrix Mechanics:
Like measurements, matrices don't commute.
- Schrödinger's Equation:
Nonrelativistic quantum particle in a classical field.
- Noether's Theorem:
Conservation laws express the symmetries of physics.
- Kets are Hilbert vectors
(duals of bras) on which observables operate.
- Observables are operators
explicitely associated with physical quantities.
- Commutators
are the quantities which determine uncertainty relations.
- Uncertainty relations
hold whenever the commutator does not vanish.
- Spin is a form of angular momentum
without a classical equivalent.
- Pauli matrices: Three 2 by 2 matrices
with eigenvalues +1 and -1.
- Quantum Entanglement:
The singlet and triplet states of two electrons.
- Bell's inequality
is violated for the singlet state of two electron spins.
- Generalizations of Pauli matrices
beyond spin ½.
- Density operators
are quantum representations of imperfectly known states.
- Hamilton's analogy equates
the principles of Fermat and Maupertuis.
- Box confinement by a finite potential
in one dimension and 3 dimensions.
- Rotator:
Quantization of the angular momentum.
- Harmonic oscillator.
- Coulomb potential:
Classification of chemical orbitals.
- Elementary particles:
Quarks and leptons. Electroweak bosons. Graviton?
- Second Quantization:
Particles are modes of a quantized field.
- Bethe-Salpeter Equation:
A relativistic equation for bound-state problems.
- Measuring chemical stuff in
moles (mol) makes stoichiometry obvious.
- Modern distillation
(alembic = still-head) is due to Mary the Jewess.
- The retort
was a prominent tool of alchemists and chemists for centuries.
- Production and distillation of alcohol.
Its origins and limitations.
- Black powder:
An ancient explosive, still used as a propellant (gunpowder).
- Predicting explosive reactions:
A useful but oversimplified rule of thumb.
- Thermite
generates temperatures hot enough to weld iron.
- Enthalpy of Formation:
The tabulated data which gives energy balances.
- Exothermic crystallization of
sodium acetate trihydrate ("hot ice").
- Gibbs Function (free energy):
Its sign tells the direction of spontaneity.
- Berthollet's Law of Mass Action
governs every chemical equilibrium.
- Labile is
not quite the same as unstable.
- Inks:
India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.
- Traditional pigments:
Carbon black, vermillion, brazilin, malachite, etc.
- Beeswax is dominated by a long-chain
ester (a "wax") called mycerin.
- Pine pitch & cedar pitch:
Two similar products with different properties.
- Gum Arabic:
The magic bullet of ancient chemistry.
- Ancient acids:
From vinegar and lemon juice to vitriolic acid and more.
- Gold Chemistry:
Aqua regia ("Royal Water") dissolves gold and platinum.
- Who was the "father" of modern chemistry?
- Birth of organic chemistry:
Urea was first made chemically in 1828.
- Aliphatic saturated hydrocarbons are
called alkanes.
- Unsaturated hydrocarbons feature
some carbons tied by multiple bonds.
- Functional groups determine
the basic reactions of organic chemistry.
- The oxidation number
increases by oxidation and decreases by reduction.
- Salt bridges put
solutions in electrical contact but prevent transfers of ions.
- Nernst equation: The
voltage induced by different concentrations.
- Redox Reactions:
Oxidizers are reduced by accepting electrons...
- Basic glassware:
Flasks, funnels, tubes, bulbs, condensers, etc.
- PTFE = Polytetrafluoroethylene = Teflon®.
- Ground-glass joints:
Standard glass-to-glass conical joints have a 1:10 taper.
- Titration.
Measuring the volume of a reactant of known concentration.
- Chemistry set from a bygone era (if memory serves).
- Waterlock: 1 g of sodium polyacetate
can hold 825 mL of water.
- Negative-X: Water ignites a mixture of zinc and
ammonium nitrate.
- Nitrogen triiodide Is an extremely unstable explosive when dry.
- The normal body temperature is
37°C (98.6°C) or is it?
- Normal blood pressure.
Systolic (max.) and diastolic (min.) pressures.
- Normal pulse. 1 Hz (one hertz)
is 60 beats per minute.
- Blood circulation
(1628). Discovered by William Harvey (1578-1657).
- Respiration is a form of combustion
(Lavoisier and Laplace, 1780).
- Normal caloric intake.
100 W of power is about 2065 kcal/day.
- International Unit
(IU) is an arbitrarily-defined rating of biological activity.
- Concentration
is an amount (either mass or moles) per volume.
- Glycosylated hemoglobin
(HbA1c) relates to average blood glucose (bG).
- Medical abbreviations
commonly used in prescriptions and elsewhere.
- Kant's Island Universes:
The Universe is filled with separate galaxies.
- The Cosmological Principle:
The Universe is homogeneous and isotropic.
- The Big Bang:
An idea of Georges Lemaître mocked by Fred Hoyle.
- The Cosmic Microwave Background (CMB):
Its spectrum and density.
- Cosmic redshift (z) of light from a Universe
(1+z) times smaller than now.
- Multiple choices and misguided explanations
for cosmic redshifts.
- Hubble Law relates redshift
and distance for comoving points.
- Omega (W):
The ratio of the density of the Universe to the critical density.
- Look-Back Time: The time ellapsed since
observed light was emitted.
- Distance: In a cosmological context,
there are several flavors of distances.
- Comoving points follow
the expansion of the Universe.
- The Anthropic Principle:
An unsatisfactory type of absolute constraint.
- Dark matter & dark energy:
Gravity betrays the existence of dark stuff.
- The Pioneer Effect:
The anomalous escape of the Pioneer spaceprobe.
Galaxies and large-scale structure of the Universe
- The local group is dominated
by the Milky Way & Andromeda galaxies.
- The virgo cluster dominates our corner of the Universe.
- Superclusters are the largest objects in the Universe.
- Nuclear fusion is what powers the stars.
- Brown dwarves
glow from gravitational contraction. Fusion isn't ignited.
- Red dwarves can burn hydrogen for trillions of years.
- The Jeans mass
above which gases at temperature T collapse by gravitation.
- Main sequence: The evolution of a typical star.
- Metallicity (Z) measures the abundance of
all elements beyond helium.
- Eta Carinae and hypergiants.
The most massive stars possible.
- Betelgeuse and red supergiants.
- Rigel and blue supergiants.
- Planetary nebulae: Aftermaths of stellar explosions.
- White dwarfs: The ultimate fate of our Sun
and other small stars.
- Neutron stars: Remnants from the
supernova collapse of medium stars.
- Stellar black holes form when
supermassive stars run out of nuclear fuel.
- Binary stars:
Pairs of unlike stars often gravitate around each other.
- Binary X-ray source:
A small accretor in tight orbit around a donor star.
- Astronomical unit (au).
Successive definitions of a standard unit of length.
- Mean distance between the Sun and the Earth,
A tad above 1 au.
- Parsec:
Triangulating interstellar distances, using the motion of the Earth.
- The solar corona
is a very hot region of rarefied gas.
- Solar radiation:
The Sun has radiated away about 0.03% of its mass.
- The Titius-Bode Law:
A numerical pattern in solar orbits?
- The 4 inner rocky planets:
Mercury, Venus, Earth, Mars.
- Earth and Moon: This is home.
- The asteroid belt:
Planetoids and bolids between Mars and Jupiter.
- The 4 outer giant gaseous planets:
Jupiter, Saturn, Uranus, Neptune.
- Discovery of Neptune:
Urbain Le Verrier scooped John Couch Adams.
- Pluto
and other Kuiper Belt Objects (KBO).
- Sedna
and other planetoids beyond the Kuiper Belt.
- What's a planet?
The latest definition excludes Pluto.
- Heliosphere and Heliopause:
The region affected by solar wind.
- Oort's Cloud
is a cometary reservoir at the fringe of the Solar System.
- Easy conversion between
Fahrenheit and Celsius scales: F+40 = 1.8 (C+40)
Automotive :
- Car speed
is proportional to tire size & engine rpm, divided by gear ratio.
- 0 to 60 mph in 4.59 s,
may not always mean 201.96 feet.
- Car acceleration. Guessing the curve from standard data.
- "0 to 60 mph" time,
obtained from vehicle mass and actual average power.
- Thrust is the ratio of power to speed
[measured along direction of thrust].
- Power as a function of chamber size
for internal combustion engines.
- Optimal gear ratio
to maximize top speed on a flat road (no wind).
Surface Areas :
- Heron's formula (for the area of a triangle)
is related to the Law of Cosines.
- Brahmagupta's formula
gives the area of a quadrilateral (cyclic or not).
- Bretschneider's formula
for a quadrilateral of given sides and diagonals.
- Vectorial area of a quadrilateral:
Half the cross-product of its diagonals.
- Parabolic segment:
2/3 the area of circumscribed parallelogram or triangle.
Volumes :
- Content of a cylindrical tank (horizontal axis),
given the height of the liquid.
- Volume of a spherical cap, or content of an
elliptical vessel, at given level.
- Content of a cistern
(cylindrical with elliptical ends), at given fluid level.
- Volume of a cylinder or prism,
possibly with tilted [nonparallel] bases.
- Volume of a conical frustum:
Formerly a staple of elementary education...
- Volume of a sphere...
obtained by subtracting a cone from a cylinder !
- The volume of a tetrahedron
is the determinant of three edges, divided by 6.
- Volume of a wedge of a cone.
Averages :
- Splitting a job evenly between two unlike workers.
- Splitting a job unevenly between two unlike workers.
- Mixing solutions
to obtain a predetermined intermediate rating.
- Alcohol solutions
are rated by volume not by mass.
- Mixing alcohol solutions
to obtain an exact rating by volume (ABV).
- Special averages:
harmonic (for speeds), geometric (for rates), etc.
- Mean Gregorian month:
either 30.436875 days, or 30.4587294742534...
- The arithmetic-geometric mean is
related to a complete elliptic integral.
Geodesy and Astronomy :roportional to the square root of your altitude.
- Distance between two points
on a great circle at the surface of the Earth.
- Euclidean distance between two cities,
along a line through the Earth.
- Geodetic coordinates:
Point of elevation h at latititude
j and longitude q.
- The figure of the Earth.
Geodetic and geocentric latitudes.
- Kepler's Third Law:
The relation between orbital period and orbit size.
- Creation and Discovery in Science.
- Search for Extraterrestrial Intelligence.
If we listen, we must talk.
- The Anthropic Principle:
The laws of physics must allow human life.
- Science and Politics:
Political support for Science makes a society worthy.
- What's Mathematics anyway?
The groundwork of scientific knowledge.
Below are topics not yet integrated with the rest of this site's navigation.
- Circumference of an ellipse:
Exact series and approximate formulas.
- Ramanujan I and Lindner formulas:
The journey begins...
- Ramanujan II:
An awesome approximation from a mathematical genius.
- Hudson's Formula
and other Padé approximations.
- Peano's Formula:
The sum of two approximations with cancelling errors.
- The YNOT formula
(Maertens, 2000. Tasdelen, 1959).
- Euler's formula is the first step in
an exact expansion.
- Naive formula:
p ( a + b )
features a -21.5% error for elongated ellipses.
- Cantrell's Formula:
A modern attempt with an overall accuracy of 83 ppm.
- From Kepler to Muir.
Lower bounds and other approximations.
- Relative error cancellations in
symmetrical approximative formulas.
- Complementary convergences of two
series. A nice foolproof algorithm.
- Elliptic integrals & elliptic functions.
Traditional symbols vs. computerese.
- Padé approximants
are used in a whole family of approximations...
- Improving Ramanujan II
over the whole range of eccentricities.
- The Arctangent Function
as a component of several approximate formulas.
- Abed's formula uses a parametric exponentA>.
Improved looks for a brainchild of Shahram Zafary.
- Rivera's formula gives the
perimeter of an ellipse with 104 ppm accuracy.
- Better accuracy from
Cantrell, building on his own previous formula
- Rediscovering
a well-known exact expansion due to Euler (1773).
- Exact expressions for the
circumference of an ellipse: A summary.
- Surface Area of a Scalene Ellipsoid:
The general formula isn't elementary.
- Thomsen's Formula:
A simple symmetrical approximation.
- Approximate formulas
for the surface area of a scalene ellipsoid.
- Nautical mile:
"Average" minute of latitude on an oblate spheroid.
- Great ellipses
have the same center as the ellipsoid they are drawn on.
- Area enclosed by a curve
drawn on the surface of an oblate spheroid.
- Pseudo-straight boundaries
of areas varying quadratically with longitude.
- The Magnetic Field of the Earth.
- Life (1): The mysteries of evolution.
- Life (2): The origins of life on Earth.
- Life (3):
Does extraterrestrial life exist?
Is there intelligence out there?
- Nemesis:
A distant companion of the Sun could cause periodical extinctions.
- Current Challenges to established dogma.
- Unexplained artifacts and sightings.
- The Riemann Hypothesis:
{Re(z) > 0 &
z(z) = 0} Þ
{Re(z) = ½}.
- P = NP ? Can we find
in polynomial time what we can check that fast?
- Collatz sequences
go from n to n/2
or (3n+1)/2. Do they all lead to 1?
- The Poincaré Conjecture (1904).
Proven by Grisha Perelman in 2002.
- Fermat's Last Theorem (1637).
Proven by Andrew Wiles in 1995.
- The ABC conjecture.
- The only magic hexagon.
- The law of small numbers applied to conversion factors.
- Quadratic formulas yielding long sequences of prime numbers.
- The area under a Gaussian curve
involves the square root of p
- Exceptional simple Lie groups.
- Monstrous Moonshine in Number Theory.
- Oldest open
mathematical problem: Are there any odd perfect numbers?
- Magnetic field of the Earth:
South side is near the geographic north pole.
- From the north side,
a counterclockwise angle is positive by definition.
- What initiates the wind?
Well, primitive answers were not so wrong...
- Why "m"
for the slope of a linear function y = m x + b ? [in US textbooks]
- The diamond mark on US tape measures
corresponds to 8/5 of a foot.
- Naming the largest possible number,
in n keystrokes or less (Excel syntax).
- The "odds in favor" of poker hands:
A popular way to express probabilities.
- Reverse number sequence(s)
on the verso of a book's title page.
- Living species:
About 1400 000 have been named, but there are many more.
- Dimes and pennies:
The masses of all current US coins.
- Pound of pennies:
The dollar equivalent of a pound of pennies is increasing!
- Nickels per gallon:
Packing more than 5252 coins per gallon of space.
- Geodetic coordinates,
based on the Reference Ellipsoid defined in 1980.
- Geocentric coordinates
are almost never used in geography or astronomy.
- Distance from the center of the Earth
to points located at the surface.
- The volume of the Grand Canyon:
2 cm (3/4") over the entire Earth.
- The Oldest City in the World:
Damascus or Jericho?
- USA (States & Territories):
Postal and area codes, capitals, statehoods, etc.
- Inventing Money: Brass in China, electrum
in Lydia, gold and silver staters.
- Prices of Precious Metals:
Current market values (Gold, Silver. Pt, Pd, Rh).
- Medieval sysyem:
12 deniers to a sol.
- Ancien Régime French monetary system.
- British coinage before decimalization.
- Exchange rates
when the euro was born.
- Worldwide circulation of currencies.
- Counterfeit Coin:
In 3 weighings, find an odd object among 12, 13 or 14.
- Counterfeit Penny Problem:
Find an odd object in the fewest weighings.
- Explicit tables for detecting
one odd marble among 41, in 4 weighings.
- Find-a-birthday:
Detect an odd marble among 365, in 6 weighings.
- Error-correcting codes for ternary numeration.
- If the counterfeit is known to be heavier,
fewer weighings may be sufficient.
- Fossil calendars:
420 million years ago, a month was only 9 short days.
- Julian Day Number (JDN)
Counting days in the simplest of all calendars.
- The Week has not always been a period of seven days.
- Egyptian year of 365 days:
Back to the same season after over 1500 years.
- Heliacal rising of Sirius: Sothic dating.
- Coptic Calendar:
Reformed Egyptian calendar based on the Julian year.
- The Julian Calendar: Year starts March 25.
Every fourth year is a leap year.
- Anno Domini:
Counting roughly from the birth of Jesus Christ.
- Gregorian Calendar:
Multiples of 100 not divisible by 400 aren't leap years.
- Counting the days between dates,
with a simple formula for month numbers.
- Age of the Moon,
based on a mean synodic month of 29.530588853 days.
- Easter Sunday
is defined as the first Sunday after the Paschal full moon.
- The Muslim Calendar:
The Islamic (Hijri) Calendar (AH = Anno Hegirae).
- The Jewish Calendar:
An accurate lunisolar calendar, set down by Hillel II.
- Zoroastrian Calendar.
- The Zodiac:
Zodiacal signs and constellations. Precession of equinoxes.
- The Iranian Calendar.
Solar Hejri [SH] or Anno Persarum [AP].
- The Chinese Calendar.
- The Japanese Calendar.
- Mayan System(s):
Haab (365), Tzolkin (260), Round (18980), Long Count.
- Indian Calendar:
The Sun goes through a zodiacal sign in a solar month.
- Post-Gregorian Calendars:
Painless improvements to the secular calendar.
- Geologic Time Scale:
Beyond all calendars.
Roman Numerals
(Archaic, Classic and Medieval)
- Roman Numeration:
Ancient rules and medieval ones.
- An easy conversion table
for numbers up to 9999.
- Larger Numbers, like 18034...
- Extending the Roman system.
- The longest year so far, in terms of Roman numerals.
- IIS (or HS) is for sesterce
(originally, 2½ asses, "unus et unus et semis").
- Standard jokes.
- Limericks.
- Proper credit may not always be possible.
- Trick questions can be legitimate ones.
- Ignorance is bliss:
Why not read all that mathematical stuff faster ?
- Silly answers to funny questions.
- Why did the chicken cross the road?
Scientific and other explanations.
- Humorous or inspirational quotations
by famous scientists and others.
- One great quote to be translated
into as many languages as possible.
- Famous Last Words:
Proofs that the guesses of experts are just guesses.
- Famous anecdotes.
- Parodies, hoaxes, and practical jokes.
- Omnia vulnerant,
ultima necat: The day of reckoning.
- Funny Units:
A millihelen is the amount of beauty that launches one ship.
- Funny Prefixes:
A lottagram is many grams; an electron is 0.91 lottogram.
- The Lamppost Theory:
Only look where there's enough light.
- Is it insanity
or just a viable alternative to orthodoxy?
- Anagrams:
Rearranging letters may reveal hidden meanings ;-)
- Mnemonics:
Remembering things and/or making fun of them.
- Acronyms:
Funny ones and/or alternate interpretations of serious ones.
- Usenet Acronyms:
If you can't beat them, join them (and HF, LOL).
- Adobe's Symbol font:
Endangered standard HTML mathematical symbols.
- The equality symbol ( = ).
The "equal sign" dates back to the 16th century.
- The double-harpoon symbol
denotes chemical equilibrium.
- Line components:
Vinculum, bar, solidus, virgule, slash, macron, etc.
- The infinity symbol
( ¥ ) introduced in 1655 by John Wallis (1616-1703).
- Transfinite numbers
and the many faces of mathematical infinity.
- Chrevron symbols:
Intersection (highest below) or union (lowest above).
- Disjoint union. Square "U" or
inverted p symbol.
- Blackboard bold: Doublestruck
characters denote sets of numbers.
- The integration sign
( ò ) introduced by Leibniz at the dawn of Calculus.
- The end-of-proof box (or tombstone)
is called a halmos symbol (QED).
- Two "del" symbols:
¶ for partial derivatives, and
Ñ for Hamilton's nabla.
- The rod of Asclepius:
Medicine and the 13th zodiacal constellation.
- The Caduceus:
Scepter of Hermes, symbol of commerce (not medicine).
- Tetractys: Mystical Pythagorean symbol,
"source of everflowing Nature".
- The Borromean Rings: Three interwoven rings which are
pairwise separate.
- The Tai-Chi Mandala: The taiji
(Yin-Yang) symbol was Bohr's coat-of-arms.
- Dangerous-bend symbol:
Introduced by Bourbaki, popularized by Knuth.
Monographs and Complements
- About Zero.
- Wilson's Theorem.
- Counting Polyhedra:
A tally of polyhedra with n faces and k edges.
- Sagan's number: The number of stars,
compared to earthly grains of sand.
- The Sand Reckoner: Archimedes fills the cosmos
with grains of sand.
- Numericana's list
of distinguished Web authors in Science...
- Giants of Science:
Towering characters in Science history.
- Two Solvay conferences
helped define modern physics, in 1911 and 1927.
- Physical Units:
A tribute to the late physicist Richard P. Feynman.
- The many faces of Nicolas Bourbaki
(b. January 14, 1935).
- Lucien Refleu
(1920-2005). "Papa" of 600 mathematicians. [ In French ]
- Roger Apéry (1916-1994)
and the irrationality of z(3).
- Hergé (1907-1983):
Tintin and the Science of Jules Verne (1828-1905).
- Other biographies: Dulong, Galois, Tannery, Vessiot,
Drach, Glénisson...
- Escutcheons of Science (Armorial):
Coats-of-arms of illustrious scientists.
In-Depth Reviews of Great Products
- Printer :
The HP 82240B thermal printer has been standard since 1989.
- Modifier keys.
Lesser-used functions require several keystrokes.
- Infinity:
Unsigned algebraic infinity and signed topological infinities.
- Physical units:
A built-in feature inherited from the HP-28 (1986).
- Bug reports: Severe problems and minor ones.
- Complex functions:
Complex values & arguments. Complex variables.
- RPL programming
("Reverse Polish LISP") originated with the HP-28.
- Easter eggs: Unofficial features, just for fun.
HP-35s: Released on the 35th birthday of the HP-35
- Modifier keys.
Lesser-used functions require several keystrokes.
- Unit conversions: °F/°C | HMS |
°/rad | lb/kg | mi/km | in/cm | gal/L.
- 40 physical constants (and one mathematical constant) in a single menu.
- Bug reports: Severe problems and minor ones.
- Complex functions:
Complex values & arguments. Complex variables.
- Programming:
Recorded keystroke sequences. Tests, loops & subroutines.
- Modifier keys.
Lesser-used functions require several keystrokes.
- Physical units:
A very nice afterthought, with a few rough edges.
- Analytical functions
may present discontinuity cliffs in the complex realm.
- Wrong!
0 to the 0th power should be 1. ¥
and +¥ shouldn't be equal.
- 68000 Assembly Programming:
A primer without the help of an assembler.
- The clock frequency of your calculator
measured with 0.1% accuracy.
- TI's BASIC.
A built-in interpreted language not designed for speed.
- Pretty 2D algebraic displays
can only be edited in their 1D version.
- The keypad:
One shift-key suffices with the introduction of multi-tap.
- Integer arithmetic.
Numbers with more than 6 digits cannot be factorized.
- 20 pairs of unit conversions...
and not a single inaccuracy. (That's rare!)
- 20 physical constants
listed by name, with their units. 9 are numbered.
- Bug reports: From minor gripes to more severe flaws.
- History of the "natural" Casio scientific calculator series.
- Mode 4: Hexadecimal or octal arithmetic on 32-bit integers.
- Mode 7: Tabulate a function (or a pair of functions with "plus" version).
- Scientific constants: Consistent values recommended by CODATA (2010).
- Conversion factors between units: A few inaccuracies & one typo.
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Note:
The above numbering may change, don't use it for reference purposes.
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Noted
Numericana fans
(and/or contributors) in alphabetical order:
- Stuart Errol Anderson, Enumeration of Polyhedra.
- Max Alekseyev, Silent Circles.
- Gottfried Barthel, Generic Carmichael Numbers.
- David Cantrell, Perimeter of an Ellipse
(2001, 2004, etc.)
- Scott Cram, Magician
(Grey Matters).
- Joe Crump, Carmichael Divisors.
- Paul Godfrey, Lanczos Formula.
- Michel Marcus, Hemiperfect Numbers
(11/2,
13/2,
15/2,
17/2).
- Ed Pegg, Jr. Recreational Mathematician
(MathPuzzle).
- Belisha Price, Polyhedra.
- Knud Thomsen, Surface Area of an Ellipsoid.
- Robin Whitty, Theorems.
- Jochen Wilke, Escutcheons of Science.
Guest Authors:
Public-Domain Texts:
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