Table of Contents
See also:
Dates of creation of all indexed pages
Begin with the answers.
Then one day, perhaps, you'll find the final question.
"The Chinese Maze Murders"
by
Robert Hans van Gulik
(1910-1967) It is
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
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Saint Petersburg Paradox:
What would you pay to play the Petersburg game?
- You can only prove a
negative (a lack of counterexamples).
- Stochastic proofs
leave only a vanishing uncertainty.
- Heuristic arguments
establish the likelihood of a conjecture.
- Too few tools
are not enough to prove a given statement.
- Center of an arc
determined with straightedge and compass.
- Surface areas:
Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
- Special points in a triangle.
Euler's line and Euler's circle.
- Elliptic arc:
Length of the arc of an ellipse between two points.
- Perimeter of an ellipse.
Exact formulas and simple ones.
- Surface area of a spheroid
(oblate or prolate ellipsoid of revolution).
- Surface of an ellipse.
- Quadratic equations in the plane
describe ellipses, parabolas, or hyperbolas.
- Volume of an ellipsoid [spheroid].
- Centroid of a circular segment.
Find it with Guldin's (Pappus) theorem.
- 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.
- Complete metric space:
All Cauchy sequences are convergent.
- Locally compactness set
contains 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.
- The product topology
makes projections continuous on a cartesian product.
- Connected sets
can't be split by open sets; they needn't be
path-connected.
- Homeomorphic sets.
An homeomorphism is a bicontinuous function.
- 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.
- Planar angles
(from one direction to another) are signed quantities.
- Solid angles
are to spherical patches what planar angles are to circular arcs.
- Circular measures:
Angles and solid angles aren't quite dimensionless...
- Formulas for solid angles
subtended by patches with simple shapes.
- Curvature of a planar curve:
Variation of inclination with distance dj/ds.
- Curvature and torsion
of a three-dimensional curve.
- Geodesic and nornal curvatures and
torsions of a curve drawn on a surface.
- Lines of curvature and geodesic lines.
Extremal curvature and least length.
- 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.
- 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.
- Planar curves
rolling without slipping while rotating about two fixed points.
- Congruent ellipses
roll against each other while revolving around their foci.
- Elliptical gears:
A family of gears which include ellipses and sine curves.
- Cycloidal gears :
Traditional profiles used by watchmakers.
- Epicycloidal gears :
Philippe de la Hire (1640-1718).
- Involute tooth profile
provides a constant rotational speed ratio.
- Harmonic Drive:
The flexspline has 2 fewer teeth than the circular spline.
- Hexahedra.
The cube is not the only polyhedron with 6 faces.
- Fat tetragonal antiwedge:
Chiral hexahedron of unit volume and least area.
- Enumeration of polyhedra:
Convex polyhedra with n faces and k edges.
- The 5 Platonic solids:
Cartesian coordinates of the vertices.
- The 13 Archimedean solids
and their duals (Catalan solids).
- Some special polyhedra
may have a traditional (mnemonic) name.
- Polyhedra in certain families
are named after one prominent polygon.
- 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.
- 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.
- 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 raised 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.
- Symmetric polynomials of 3 variables:
Obtain the value of one from 3 others.
- Geometric progression of 6 terms. Sum is 14,
sum of squares is 133.
- Quartic equation involved in the classic
"Ladders in an Alley" problem.
- 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.
- Adjugate of a matrix:
The 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.
- 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.
- 0 to 60 mph in 4.59 s,
may not always mean 201.96 feet.
- 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).
- 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.
- 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.
- Analytic continuation:
Power series that converge on overlapping disks.
- Decimated power series
are equal to finite sums involving roots of unity.
- 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 (¥).
- 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.
- The existence of nonmeasurable sets
is guaranteed by the Axiom of Choice.
- 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.
- 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.
- 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.
- 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).
- 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.
- 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.
- Group homomorphism:
The image of a product is the product of the images.
- The symmetric group on
a set 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.
- 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 PÎF[x]. The 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.
- Modules are vectorial structures over
a ring of scalars (instead of a field).
- Banach spaces are
complete normed vector spaces.
- Dual space:
The set of all [continuous] linear functions with scalar values.
- 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.
- 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).
- 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...
- The p-adic metric
can be used to define p-adic numbers analytically.
- The reciprocal of a p-adic number
computed by successive approximations.
- Hasse's local-global principle.
Established for the quadratic case in 1920.
- Rotating digit patterns (in base g)
may double the corresponding values.
- 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 numbers:
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.
- For almost all numbers,
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:
Mental magic in a classroom.
- The 5-card trick of Fitch Cheney:
4 cards tell the fifth one.
- Generalizing the 5-card trick
and Devil's Poker...
- Kruskal's Count.
- Paths to God.
- Stacked Deck.
- Magic Age Cards.
- Ternary Cards.
- Magical 21 (or 27).
- Boolean Magic.
- Perfect Faro Shuffles.
- 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 must happen with fewer holes than pigeons.
- 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 Euphorbe...
- 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.
- 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.
- 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.
- 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:
The concept attributed to Sir Francis Bacon (1590).
- History of the Scientific Method.
- Distinguishing between Science and
Pseudoscience.
- 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 "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.
- Ball in a Bowl:
Pure rolling increases the period of oscillation by 18.3%.
- Hooke's Law:
Motion of a mass suspended to a spring.
- Speed of an electron
estimated with the Bohr model of the atom.
- Thermal expansion coefficients:
Cubical coefficients and linear ones.
- Waves in a solid:
P-waves (fastest), S-waves, E-waves (thin rod), SAW...
- Rayleigh Wave:
The quintessential surface acoustic wave (SAW).
- 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.
- Moments about a point or a plane are
convenient mathematical fictions.
- Moment of inertia of a spherical distribution
or an homogeneous ellipsoid.
- Perpendicular Axis Theorem:
Axis of rotation perpendicular to a thin plate.
- 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.
- Momenta of homogeneous bodies.
List of common examples.
- Rigid pendulum
moving under its own weight about a fixed horizontal axis.
- Reversible pendulum
swings with the same period around two distinct axes.
- All physical theories
have a limited range of validity.
- Gravity vs. Electrostatics:
Straight comparisons.
- Airy weighs the Earth
by timing a pendulum at the bottom of a mine.
- Rigid equilateral triangle
formed by three gravitating bodies.
- The five Lagrange points
of two gravitating bodies in circular orbit.
- 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.
- 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.
- 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 varying magnetic flux.
- 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 1900.
- 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.
- The gauge of retarded potentials:
is it really the Lorenz gauge?
- 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.
- Power radiated by an accelerated charge:
The Larmor formula (1897).
- Lorentz-Dirac equation
for the motion of a point charge is of third order.
- 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.
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 horizontally.
- Tesla turbine.
Stack of spinning disks with outer intake and inner outflow.
- Observers in motion:
A simple-minded derivation of the Lorentz Transform.
- Adding up velocities:
The combined speed can never be more 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.
- 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.
- 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.
- 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.
- 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.
- 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).
- 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.
- Latent heat (L)
is the heat transferred in a change of phase.
- Calorimetric coefficients, adiabatic
coefficient (g) heat capacities, etc.
- 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.
- 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.
- 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.
- 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.
- Gibbs Function (free energy):
Its sign tells the direction of spontaneity.
- Labile is
not quite the same as unstable.
- Inks:
India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, 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.
- Redox Reactions:
Oxidizers are reduced by accepting electrons...
- 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 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.
- 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).
- 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.
- 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.
- Nuclear fusion is what powers the stars.
- Brown dwarves glow from gravitational
contraction, without igniting fusion.
- The Jeans mass
above which gases at temperature T collapse by gravitation.
- Main sequence: The evolution of a typical star.
- 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 X-ray source:
A small accretor in tight orbit around a donor star.
- Astronomical unit:
The precise definition of a standard unit of length.
- 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: This is home.
- The asteroid belt:
Planetoids and bolids between Mars and Jupiter.
- The 4 giant gaseous planets:
Jupiter, Saturn, Uranus, Neptune.
- The 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?
Anything besides the 6 ancient planets, Uranus & Neptune?
- 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.
- 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.
- Alcohol solutions
are rated by volume not by mass.
- Mixing solutions
to obtain a predetermined intermediate rating.
- 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 :
- Distance to the horizon
is proportional 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.
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.
- 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 exponent
to fine-tune the approximation.
- Zafary's formula.
- 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.
- 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 would 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
was proven by Grisha Perelman in 2002.
- 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.
- 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.
- 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.
- 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.
- 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.
- Larger Numbers, like 18034...
- Extending the Roman system.
- 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.
- 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.
- 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).
- The equality symbol ( = ).
The "equal sign" dates back to the 16th century.
- The double-harpoon equilibrium symbol
denotes chemical equilibrium.
- "Lines" among symbols:
Vinculum, bar, solidus, virgule, slash, macron, etc.
- The infinity symbol
( ¥ ) introduced in 1655 by John Wallis (1616-1703).
- Transfinite numbers:
Mathematical symbols for the many faces of infinity.
- Chrevron symbols:
Intersection (highest below) or union (lowest above).
- Disjoint union. Square "U" or
inverted p symbol.
- Blackboard bold: Doublestruck
symbols are often used for 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.
Unabridged Answers (monographs and complements):
- Sagan's number: The number of stars,
compared to earthly grains of sand.
- The Sand Reckoner: Archimedes fills the cosmos
with grains of sand.
- About Zero.
- Wilson's Theorem.
- Counting Polyhedra:
A tally of polyhedra with n faces and k edges.
Hall of Fame:
- Numericana's list
of distinguished Web authors in Science...
- Giants of Science:
Towering characters in the history of Science.
- 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).
- Escutcheons of Science (Armorial):
Coats-of-arms of illustrious scientists.
|
Note:
The above numbering may change, don't use it for reference purposes.
|
Noted
Numericana fans
(and/or contributors) in alphabetical order:
- Max Alekseyev, Silent Circles.
- 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:
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