Nov 16, 2009

Final Answers - Science - NUMERICANA

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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)

Measurements and Units

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

    The Art of Rounding Numbers

  31. Scientific notation:  Nonzero numbers given as multiples of  powers of ten.
  32. So many "significant" digits  imply a result of limited precision.
  33. Standard deviation  gives the  precision  of a result as a form of  uncertainty.
  34. Engineering notation  reduces a number to a multiple of a power of 1000.
  35. The quadratic formula  is numerically inadequate in common cases.
  36. Devising robust formulas  which feature a stable floating-point precision.

    Scales and Ratings: Measuring without Units

  37. The Beaufort scale  is now defined in terms of wind speed.
  38. The Saffir / Simpson scale  for hurricanes.
  39. The Fujita scale  for tornadoes.
  40. The Richter scale  for earthquakes and other sudden energy releases.
  41. Decibels:  A general-purpose logarithmic scale for relative power ratios.
  42. Apparent and absolute magnitudes  of stars.

    Scaling and Scale Invariance

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

    Numerical Constants: Mathematical & Physical Constants

    Physical Constants:
  46. For the utmost in precision,  physical constants are derived in a certain order.
  47. Primary conversion factors  between customary systems of units.
    6+1 Basic Dimensionful Physical Constants  (Proleptic SI)
  48. Speed of Light in a Vacuum (Einstein's Constant):   c = 299792458 m/s.
  49. Magnetic Permeability of the Vacuum: An exact value defining the ampere.
  50. Planck's constant:  The ratio of a photon's energy to its frequency.
  51. Boltzmann's constant:  Relating temperature to energy.
  52. Avogadro's number:  The number of things per mole of stuff.
  53. Mechanical Equivalent of Light (683 lm/W at 540 THz) defines the candela.
  54. Newton's constant of gravitation  and a futuristic definition of the second.
    Fundamental Mathematical Constants: 
  55. 0:  Zero is the most fundamental and most misunderstood of all numbers.
  56. 1 and -1:  The unit numbers.
  57. p ("Pi"): The ratio of the circumference of a circle to its diameter.
  58. Ö2: The diagonal of a square of unit side.  Pythagoras' Constant.
  59. f: The diagonal of a regular pentagon of unit side.  The Golden Number.
  60. Euler's  e:  The base of the exponential function which equals its own derivative.
  61. ln(2):  The alternating sum of the reciprocals of the integers.
  62. Euler-Mascheroni Constant  g :  Limit of   [1 + 1/2 + 1/3 +...+ 1/n] - ln(n).
  63. Catalan's Constant  G :  The alternating sum of the reciprocal odd squares.
  64. Apéry's Constant  z(3) :  The sum of the reciprocals of the perfect cubes.
  65. Imaginary  i:  If "+1" is a step forward, "+ i" is a step sideways to the left.
    Exotic Mathematical Constants: 
  66. Delian constant:  21/3  is the solution to the classical  duplication of the cube.
  67. Gauss's constant:  The reciprocal of the arithmetic-geometric mean of 1 and Ö2.
  68. Mertens constant:  The limit of   [1/2 + 1/3 + 1/5 +...+ 1/p] - ln(ln p)
  69. Artins's constant  is the proportion of long primes in decimal or binary.
  70. Ramanujan-Soldner constant (m):  Positive root of the logarithmic integral.
  71. The Omega constant:  W(1) is the solution of the equation   x exp(x) = 1.
  72. Feigenbaum constant (d) and the related reduction parameter (a).
    Some Third-Tier Mathematical Constants: 
  73. Brun's Constant:  A standard uncertainty  (s)  means a 99% level of  ±3s
  74. Prévost's Constant:  The sum of the reciprocals of the Fibonacci numbers.
  75. Grossman's Constant:  One recurrence converges for only one initial point.
  76. Ramanujan's Number:   exp(p Ö163)   is almost an integer.
  77. Viswanath's constant:  Mean growth in random additions and subtractions.

    Counting, Combinatorics, Probability

  78. Always change your first guess if you're always told another choice is bad.
  79. The Three Prisoner Problem predated Monty Hall and Marilyn by decades.
  80. Seating N children at a round table in (N-1)! different ways.
  81. How many Bachet squares?  A 1624 puzzle using the 16 court cards.
  82. Choice Numbers: C(n,p) is the number of ways to choose p items among n.
  83. C(n+2,3) three-scoop sundaes. Several ways to count them (n flavors).
  84. C(n+p-1,p) choices of p items among n different types, allowing duplicates.
  85. How many new intersections of straight lines defined by n random points.
  86. Face cards. The probability of getting a pair of face cards is less than 5%.
  87. Homework Central: Aces in 4 piles, bad ICs, airline overbooking.
  88. Binomial distribution. Defective units in a sample of 200.
  89. Siblings with the same birthday. What are the odds in a family of 5?
  90. Covariance:  A generic example helps illustrate the concept.
  91. Variance of a binomial distribution, derived from general principles.
  92. Standard deviation. Two standard formulas to estimate it.
  93. The Markov inequality  is used to prove the  Bienaymé-Chebyshev inequality.
  94. The Bienaymé-Chebyshev inequality  is valid for  any  probability distribution.
  95. Inclusion-Exclusion: One approach to the probability of a union of 3 events.
  96. The "odds in favor" of poker hands: A popular way to express probabilities.
  97. Probabilities of a straight flush in 7-card stud. Generalizing to "q-card stud".
  98. Probabilities of a straight flush among 26 cards  (or any other number).
  99. The exact probabilities in 5-card, 6-card, 7-card, 8-card and 9-card stud.
  100. Rearrangements of  CONSTANTINOPLE  so no two vowels are adjacent...
  101. Four-letter words from  POSSESSES:  Counting with generating functions.
  102. How many positive integers below 1000000  have their digits add up to 19?
  103. Polynacci Numbers:  Flipping a coin n times without  p  tails in a row.
  104. 252 decreasing sequences of 5 digits  (2002 nonincreasing ones).
  105. How many ways are there to make change for a dollar?  Closed formulas.
  106. Partitioniong an amount into the parts minted in a certain currency.
  107. The number of rectangles in an N by N chessboard-type grid.
  108. The number of squares in an N by N grid:  0, 1, 5, 14, 30, 55, 91, 140, 204...
  109. Screaming Circles:  How many tries until there's no eye contact?
  110. Average distance between two random points on a segment, a disk, a cube...
  111. Average distance between two points on the surface of a sphere.

    Stochastic Processes & Stochastic Models

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

    "Utility" and Decision Analysis

  117. The Utility Function: A dollar earned is usually worth less than a dollar lost.
  118. Saint Petersburg Paradox: What would you pay to play the Petersburg game?

    Mathematical Proofs

  119. You can only prove a negative (a lack of counterexamples).
  120. Stochastic proofs  leave only a  vanishing  uncertainty.
  121. Heuristic arguments  establish the likelihood of a conjecture.
  122. Too few tools  are not enough to prove a given statement.

    Geometry and Topology   (for Polyhedra page, see below)

  123. Center of an arc determined with straightedge and compass.
  124. Surface areas: Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
  125. Special points in a triangle. Euler's line and Euler's circle.
  126. Elliptic arc: Length of the arc of an ellipse between two points.
  127. Perimeter of an ellipse. Exact formulas and simple ones.
  128. Surface area of a spheroid  (oblate or prolate ellipsoid of revolution).
  129. Surface of an ellipse.
  130. Quadratic equations in the plane describe ellipses, parabolas, or hyperbolas.
  131. Volume of an ellipsoid [spheroid].
  132. Centroid of a circular segment. Find it with Guldin's (Pappus) theorem.
  133. Focal point of a parabola. y = x 2 / 4f (where f is the focal distance).
  134. Parabolic telescope: The path from infinity to focus is constant.
  135. Make a cube go through a hole in a smaller cube.
  136. Octagon: The relation between side and diameter.
  137. Constructible regular polygons  and constructible angles (Gauss).
  138. Areas of regular polygons of unit side: General formula & special cases.
  139. For a regular polygon of given perimeter, the more sides the larger the area.
  140. Curves of constant width: Reuleaux Triangle and generalizations.
  141. Irregular curves of constant width. With or without any circular arcs.
  142. Solids of constant width. The three-dimensional case.
  143. Constant width in higher dimensions.
  144. Fourth dimension. Difficult to visualize, but easy to consider.
  145. Volume of a hypersphere and hper-surface area, in any number of dimensions.
  146. Hexahedra. The cube is not the only polyhedron with 6 faces.
  147. Descartes-Euler Formula: F-E+V=2 but restrictions apply.


  148. Metric spaces:  The motivation behind more general  topological  spaces.
  149. Topological spaces:  Singling out abstract "open" subsets defines a  topology.
  150. Closed sets  are sets (of a topological space) whose complements are open.
  151. Subspace F of E:  Its open sets are the intersections with F of open sets of E.
  152. Separation axioms:  Flavors of topological spaces, according to  Trennung.
  153. Compactness of a topological space:  Every open cover has a  finite  subcover.
  154. Complete metric space:  All Cauchy sequences are convergent.
  155. Local compactness:  Every point of the set has a  compact neighborhood  in it.
  156. General properties of sequences  are indicative of topological characteristics.
  157. Continuous functions  are such that the  inverse image  of any open set is open.
  158. The product topology  makes projections continuous on a cartesian product.
  159. Connected sets  cannot be split by open sets; they need not be path-connected.
  160. Homeomorphic sets  are related by a bicontinuous function  (homeomorphism).
  161. Homotopy:  A progessive transformation of a  function  into another.
  162. The Fundamental Group:  The homotopy classes of all loops through a point.
  163. Homology and Cohomology.  Poincaré duality.
  164. Descartes-Euler Formula:  F-E+V = 2, but restrictions apply.
  165. Euler Characteristic:   c   (chi)  extended beyond its traditional definition...
  166. Winding number  about point O of a continuous planar curve not containing O.
  167. Fixed-point theorems  by  BrouwerShauder  and  Tychonoff.
  168. Turning number  of a planar curve with a well-defined oriented tangent.
  169. Real projective plane  and Boy's surface.
  170. Hadwiger's  additive continuous functions of d-dimensional rigid bodies.
  171. Eversion of the sphere.  An homotopy  can  turn a sphere inside out.
  172. Classification of surfaces:  "Zero Irrelevancy Proof" (ZIP) by J.H. Conway.
  173. Braid groups:  strands, braids and pure braids.

    Angles and Solid Angles:

  174. Planar angles  separate two directions.  In an oriented plane, they are  signed.
  175. Solid angles  are to spherical patches what planar angles are to circular arcs.
  176. Circular measures:  Angles and solid angles aren't quite dimensionless quantities.
  177. Formulas for solid angles  subtended by patches with simple shapes.

    Curvature and Torsion:

  178. Curvature of a planar curve:  The variation of inclination with distance  dj/ds.
  179. Curvature and torsion  of a three-dimensional curve.
  180. Geodesic and nornal curvatures and torsions  of a curve drawn on a surface.
  181. Lines of curvature and geodesic lines.  Lines of extremal curvature or least length.
  182. Meusnier's theorem:  Tangent lines have the same  normal curvature.
  183. Gaussian curvature of a surface and its integral:  The  Gauss-Bonnet theorem.
  184. Parallel-transport of a vector around a closed curve.  Holonomic angle of a loop.
  185. Total curvature of a curve.  The Fary-Milnor theorem for knotted curves.
  186. Linearly independent components  of the  Riemann curvature tensor.

    Planar Curves:

  187. Cartesian equation of a straight line:  passing through two given points.
  188. Confocal Conics:  Ellipses and hyperbolae sharing the same pair of  foci.
  189. Spiral of Archimedes:  Paper on a roll, or groove on a vinyl record.
  190. Catenary:  The shape of a thin chain under its own weight.
  191. Witch of Agnesi.  How the versiera (Agnesi's cubic) got a weird name.
  192. Folium of Descartes.
  193. Lemniscate of Bernoulli:  The shape of the infinity symbol is a quartic curve.
  194. Along a Cassini oval, the product of the distances to the two foci is constant.
  195. Limaçons of Pascal:  The cardioid  (unit epicycloid) is a special case.
  196. On a Cartesian oval, the weighted average distance to two poles is constant.
  197. Parallel curves  share the same normal, along which their distance is constant.
  198. Bézier curves  are algebraic splines.  The cubic type is the most popular.
  199. Piecewise circular curves:  The traditional way to specify curved forms.
  200. Intrinsic equation  [curvature as a function of arc length]  may include  spikes.
  201. The quadratrix (or trisectrix) of Hippias can square the circle and trisect angles.
  202. The parabola  is a curve that's  constructible  with straightedge and compass.
  203. Mohr-Mascheroni constructions  use the compass alone  (no straightedge).


  204. Glossary  of terms related to gears.
  205. Planar curves  rolling without slipping while rotating about two fixed points.
  206. Congruent ellipses  roll against each other while revolving around their foci.
  207. Elliptical gears:  A family of gears which include ellipses and sine curves.
  208. Cycloidal gears :  Traditional profiles used by watchmakers.
  209. Epicycloidal gears :  Philippe de la Hire (1640-1718).
  210. Involute tooth profile  provides a constant rotational speed ratio.
  211. Harmonic Drive:  The  flexspline  has 2 fewer teeth than the  circular spline.

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

  212. Hexahedra.  The cube is not the only polyhedron with 6 faces.
  213. Fat tetragonal antiwedge:  Chiral hexahedron of  least area  for a given volume.
  214. Enumeration of polyhedra: Tally of polyhedra with n faces and k edges.
  215. The 5 Platonic solids: Cartesian coordinates of the vertices.
  216. The 13 Archimedean solids  and their  duals  (Catalan solids).
  217. Some special polyhedra may have a traditional (mnemonic) name.
  218. Polyhedra in certain families are named after one of their prominent polygons.
  219. Deltahedra have equilateral triangular faces. Only 8 deltahedra are convex.
  220. Johnson Polyhedra and the associated nomenclature.
  221. Polytopes are the n-dimensional counterparts of 3-D polyhedra.
  222. A simplex of touching unit spheres may allow a center sphere to bulge out.
  223. Regular Antiprism:  Height and volume of a regular n-gonal antiprism.
  224. The Szilassi polyhedron  features 7 pairwise adjacent hexagonal faces.
  225. Wooden buckyball:  Cutting 32 blocks to make a truncated icosahedron.

    Graph Theory

  226. The bridges of KönigsbergEulerian graphs  and the birth of  graph theory.
  227. Undirected graphs are digraphs with  symmetrical  adjacency matrix.
  228. Adjacency matrix of a directed graph  (digraph)  or of a  bipartite graph.
  229. Silent Circles:  An enumeration based on adjacency matrices  (Max Alekseyev).
  230. Silent Prisms:  Another version of the screaming game, for short-sighted people.
  231. Tallying  all markings of one edge per node in which no edge is marked twice.
  232. Line graph:  The nodes of L(G) are edges of G  (connected  iff  adjacent in G).
  233. Transitivity:  Vertex-transitive and/or edge-transitive graphs.


  234. Factorial zero is 1, so is an empty product; an empty sum is 0.
  235. Anything raised to the power of 0 is equal to 1, including 0 to the power of 0.
  236. Idiot's Guide to Complex Numbers.
  237. Using the Golden Ratio (f) to express the 5 [complex] fifth roots of unity.
  238. "Multivalued" functions are functions defined over a Riemann surface.
  239. Square roots are inherently ambiguous for negative or complex numbers.
  240. The difference of two numbers, given their sum and their product.
  241. Symmetric polynomials of 3 variables: Obtain the value of one from 3 others.
  242. Geometric progression of 6 terms. Sum is 14, sum of squares is 133.
  243. Quartic equation involved in the classic "Ladders in an Alley" problem.

    Matrices and Determinants

  244. Permutation matrices  include the identity matrix and the exchange matrix.
  245. Operations on matrices  are conveniently defined using  Dirac's notation.
  246. Vandermonde matrix:  The successive powers of elements in its second row.
  247. Toeplitz matrix:  Constant diagonals.
  248. Circulant matrix:  Cyclic permutations of the first row.
  249. Wendt's Determinant:  The circulant of the binomial coefficients.
  250. Hankel matrix:  Constant skew-diagonals.
  251. Catberg matrix:  Hankel matrix of the reciprocal of Catalan numbers.
  252. Hadamard matrix:  Unit elements and orthogonal columns.
  253. Sylvester matrix  of two polynomials has their resultant for determinant.
  254. The discriminant of a polynomial is the resultant of itself and its derivative.

    Trigonometry, Elementary Functions, Special Functions

  255. Numerical functions: Polynomial, rational, algebraic, transcendental, special...
  256. Trigonometric functions:  Memorize a simple picture for 3 basic definitions.
  257. Solving triangles with the law of sines, law of cosines, and law of tangents.
  258. Spherical trigonometry:  Triangles drawn on the surface of a sphere.
  259. Sum of tangents of two half angles, in terms of sums of sines and cosines.
  260. The absolute value of the sine of a complex number.
  261. Exact solutions to transcendental equations.
  262. All positive rationals (and their square roots) as trigonometric functions of zero!
  263. The sine function: How to compute it numerically.
  264. Chebyshev economization saves billions of operations on routine computations.
  265. The Gamma function: Its definition(s) properties and values.
  266. Lambert's W function is used to solve practical transcendental equations.

    Hypergeometric Functions

  267. Pochhammer's symbolUpper factorial of k increasing factors, starting with x.
  268. Gauss's hypergeometric function:  2+1 parameters (and one variable).
  269. Kummer's transformations relate different values of the hypergeometric function.
  270. Sum of the reciprocal of Catalan numbers, in closed hypergeometric form.


  271. Derivative: Usually, the slope of a function, but there's a more abstract approach.
  272. Integration: The Fundamental Theorem of Calculus.
  273. 0 to 60 mph in 4.59 s, may not always mean 201.96 feet.
  274. Integration by parts:  Reducing an integral to another one.
  275. Length of a parabolic arc.
  276. Top height of a curved bridge spanning a mile, if its length is just a foot longer.
  277. Sagging:  A cable which spans 28 m and sags 30 cm is 28.00857 m long.
  278. The length of the arch of a cycloid is 4 times the diameter of the wheel.
  279. Integrating the cube root of the tangent function.
  280. Changing inclination to a particle moving along a parabola.
  281. Algebraic area of a "figure 8" may be the sum or the difference of its lobes.
  282. Area surrounded by an oriented planar loop  which  may  intersect itself.
  283. Linear differential equations of higher order and/or in several variables.
  284. Theory of Distributions:  Convolution products and their usage.
  285. Laplace Transforms: The Operational Calculus of Oliver Heaviside.
  286. Integrability of a function and of its absolute value.
  287. Analytic functions of a linear operator; defining  f (D) when D is d/dx...

    Differential Equations

  288. Ordinary differential equations. Several examples.
  289. A singular change of variable  is valid over a domain which may not be maximal.
  290. Vertical fall  against fluid resistance  (including both viscous and quadratic drag).

    Differential Forms  &  Vector Calculus

  291. Generalizing the  fundamental theorem of calculus.
  292. The surface of a loop  is a vector determining its apparent area in any direction.
  293. Practical identities  of vector calculus

    Optimization:  Operations Research, Calculus of Variations

  294. Stationary points  (or  saddlepoints )  are where  all  partial derivatives vanish.
  295. Single-variable optimization:  Derivative vanishes unless the variable is extreme.
  296. Extrema of a function of two variables  must satisfy a  second-order  condition.
  297. Saddlepoints of a multivariate function.  One equation to satisfy per variable.
  298. Lagrange multipliers:  Optimizing an objective function under various constraints.
  299. Minimizing the lateral surface area of a cone  of given base and volume.
  300. Euler-Lagrange equations  hold along the path of a  stationary  integral.
  301. Noether's theorem:  A symmetry of the integrand yields a conserved quantity.
  302. Isoperimetric Inequality:  The largest area enclosed by a loop of unit perimeter.
  303. Plateau's problem  extends the calculus of variations from paths to membranes.
  304. Embedded minimal surfaces: Plane, catenoid, helicoid, Costa's surface, etc.
  305. Connecting blue dots to red dots  in the plane, without any crossings...
  306. The shortest way to connect 3 dots  can be to join them to a  fourth  point.
  307. The Honeycomb Theorem:  A conjecture of old, proved by  Thomas Hales.
  308. Counterexamples to Kelvin's conjecture.  Tiling space with unit cells of least area.

    Analysis, Convergence, Series, Complex Analysis

  309. Cauchy sequences help define real numbers rigorously.
  310. Permuting the terms of a series may change its sum arbitrarily.
  311. Uniform convergence implies properties for the limit of a sequence of functions.  Augustin Cauchy   (1789-1857)  Joseph Fourier   (1768-1830)
  312. Defining integrals: Cauchy, Riemann, Darboux, Lebesgue.
  313. Cauchy principal value of an integral.
  314. Fourier series. A simple example.
  315. Infinite sums may sometimes be evaluated with Fourier Series.
  316. A double sum is often the product of two sums, which may be Fourier series.
  317. At a jump, the sum of a Fourier series is the half-sum of its left and right limits.
  318. Gibbs phenomenon; 9% overshoot of partial Fourier series near a jump.
  319. Method of Froebenius about a regular singularity of a differential equation.
  320. Laurent series of a function about one of its poles.
  321. Cauchy's Residue Theorem is helpful to compute difficult definite integrals.
  322. Tame complex functions: Holomorphic and meromorphic functions.

    Complex Power Series  &  Analytic Continuations

  323. Radius of convergence.  The convergence disk of a complex power series.
  324. Analytic continuation:  Power series that coincide whenever their disks overlap.
  325. Decimated power series are equal to finite sums involving  roots of unity.

    Fourier Transform  &  Tempered Distributions  Joseph Fourier   (1768-1830)

  326. Convolution  as an inner operation among numerical functions.
  327. Duality:  The product of a  bra  by a  ket  is a (complex) scalar.
  328. distribution  associates a scalar to every  test function.
  329. Schwartz functions  are suitable  rapidly decreasing  test functions.
  330. Tempered distributions  are functionals over Schwartz functions.
  331. The Fourier Transform  associates a  tempered distribution  to another.  Antoine Parseval   (1755-1836)
  332. Parseval's theorem  (1799).  The Fourier transform is unitary.
  333. 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.
  334. Poisson summation formula:  The unit comb  (Shah function)  is its own Fourier transform.
  335. Far image of a translucent film  is the Fourier transform of its optical density.
  336. The Radon transform, corresponding to lateral tomography, is easily inverted.
  337. Competing definitions of the Fourier transform.  For the record.

    Discrete Fourier Transforms  &  Fast Fourier Transform

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

    Set Theory and Logic

  339. The Barber's Dilemma. Not a paradox if analyzed properly.
  340. What is infinity? More than a pretty symbol (¥).
  341. There are more real than rational numbers. Cantor's argument.
  342. Cantor's ternary set.  A vanishing set or reals  equipollent  to the whole line.
  343. The axioms of set theory: Fundamental axioms and the Axiom of Choice.
  344. A set is smaller than its powerset:  A simple proof applies to all sets.
  345. Transfinite cardinals, transfinite ordinals: Two different kinds of infinite numbers.
  346. The continuum hypothesis:  What's between the countable and the continuum?
  347. Surreal Numbers:  These include reals, transfinite ordinals, infinitesimals & more.
  348. Numbers:  From integers to surreals.  From reals to quaternions and  beyond.

    Integer Arithmetic, Number Theory

  349. The number 1 is not prime, as definitions are chosen to make theorems simple.
  350. Composite numbers are not prime, but the converse need not be true...
  351. Two prime numbers whose sum is equal to their product.
  352. Gaussian integers:  Factoring into primes on a two-dimensional grid.
  353. The least common multiple may be obtained without factoring into primes.
  354. Standard Factorizations:   n4 + 4   is never prime for   n > 1   because...
  355. Euclid's algorithm gives the greatest common divisor and Bézout coefficients.
  356. Bézout's Theorem:  The GCD of p and q is of the form  u p + v q.
  357. Greatest Common Divisor  (GCD)  defined for all commensurable numbers.
  358. Linear equation in integers  can be solved using  Bézout's theorem.
  359. Pythagorean Triples:  Right triangles whose sides are coprime integers.
  360. The number of divisors of an integer.
  361. Perfect numbers and Mersenne primes.
  362. Multiperfect and hemiperfect numbers  divide  twice the sum of their divisors.
  363. Fast exponentiation by repeated squaring.
  364. Partition function. How many collections of positive integers add up to 15?
  365. A Lucas sequence whose oscillations never carry it back to -1.
  366. A bit sequence  with intriguing statistics.  Counting squares between cubes.
  367. Binet's formulas: N-th term of a sequence obeying a second-order recurrence.
  368. The square of a Fibonacci number  is  almost  the product of its neighbors.
  369. D'Ocagne's identity  relates conjugates products of Fibonacci numbers.
  370. Catalans's identity  generalizes  Cassini's Identity  (about Fibonacci squares).
  371. Faulhaber's formula gives the sum of the p-th powers of the first n integers.
  372. Multiplicative functions:  If a and b are  coprime, then  f(ab) =  f(a) f(b).
  373. Moebius function:  Getting  N  values with only  O(N Log(Log N))  additions.
  374. Dirichlet convolution is especially interesting for  multiplicative  functions.
  375. Dirichlet powers of arithmetic functions  (especially, of the Moebius function).
  376. Dirichlet powers of multiplicative functions  are given by a  superb formula.
  377. Totally multiplicative functions are the simplest type of multiplicative functions.
  378. Dirichlet characters are important  totally multiplicative functions.
  379. Euler products  and generalized zeta functions.

    Positional Numeration  &  Number Systems

  380. Modular Arithmetic may be used to find the last digit(s) of very large numbers.
  381. Powers of ten expressed as products of two factors  without zero digits.
  382. Divisibility by 7, 13, and 91 (or by B2-B+1 in base B).
  383. Lucky 7's.  Any integer divides a number composed of only 7's and 0's.
  384. Binary and/or hexadecimal numeration for floating-point numbers as well.
  385. Extract a square root the old-fashioned way.
  386. Ternary system:  Is base 3  really  the best radix for positional numeration?

    Prime Numbers

  387. A prime number  is a positive integer with just two distinct divisors (1 and itself).
  388. Euclid's proof:  There are infinitely many primes.
  389. Dirichlet's theorem:  There are infinitely many primes of the form  kN+a.
  390. Green-Tao theorem:  There are arbitrarily long arithmetic progressions of primes.
  391. The von Mangoldt function  is  Log p  for a power of a prime p,  0 otherwise.
  392. The Prime Number Theorem:  The probability that N is prime is roughly 1/ln(N).
  393. The average number of factors  of a large number  N  is  Log N.
  394. The average number of distinct prime factors  of  N  is  Log Log N.
  395. The largest known prime:  Historical records, old and new.
  396. The Lucas-Lehmer Test  checks the primality of a Mersenne number  very fast.
  397. Formulas giving only primes  may not help with new primes.
  398. Ulam's Lucky Numbers  and other sequences generated by sieves.

    Modular Arithmetic

  399. Chinese Remainder Theorem:  How remainders define an integer (within limits).
  400. Modular arithmetic: The algebra of congruences, formally introduced by Gauss.
  401. Fermat's little theorem:   For any prime p, ap-1 is 1 modulo p, unless p divides a.
  402. Euler's totient functionf(n) is the number of integers coprime to n, from 1 to n.
  403. Fermat-Euler theorem:  If  a is coprime to n,  then a to the  f(n)  is 1 modulo n.
  404. Carmichael's reduced totient function (l) : A very special divisor of the totient.
  405. 91 is a pseudoprime to half of the bases coprime to itself.
  406. Carmichael Numbers:  An absolute pseudoprime  n divides  (an - a)  for any a.
  407. Chernik's Carmichael numbers:  3 prime factors   (6k+1)(12k+1)(18k+1).
  408. Large Carmichael numbers may be obtained in various ways.
  409. Conjecture: Any odd number coprime to its totient has a Carmichael multiple.

    Group Theory and Symmetries

  410. Monoids  are endowed with an  associative  operation and a  neutral element.
  411. The inverse of an element  comes in two flavors which coincide when both exist.
  412. Free monoid:  All the finite strings (words) in a given  alphabet.
  413. Groups  are monoids in which  every element  is invertible.
  414. A subgroup is a group  contained in another group.
  415. Generators  of a group are not contained in any  proper  subgroup.
  416. Lagrange's Theorem:  The order of a subgroup divides the order of the group.
  417. Normal subgroups  and their quotients in a group.
  418. Group homomorphism:  The image of a product is the product of the images.
  419. The symmetric group  on a set E consists of all the bijections of E onto itself.
  420. Inner automorphisms:  Inn(G)  is isomorphic to the quotient of  G  by its center.
  421. The conjugacy class formula  uses conjugacy to tally elements of a group.
  422. Simple groups  are groups without  nontrivial  normal subgroups.
  423. The derived subgroup  of a group is  generated  by its  commutators.
  424. Direct product of two groups  (also called a  direct sum  for additive groups).
  425. Groups of small orders  and their families:  Cyclic groups, dihedral groups, etc.
  426. Enumeration  of "small" groups.  How many groups of order n?
  427. Classification of finite simple groups,  by Gorenstein and many others (1982).
  428. Sporadic groupsTits Group, 20 relatives of Fischer's Monster, 6 pariahs.
  429. Classical groups:  Their elements depend on parameters from a  field.
  430. The Möbius group  consists of homographic transformations of  CÈ{¥}.
  431. Lorentz transformations  may  change spatial orientation or time direction.
  432. Symmetries of the laws of nature:  A short primer.

    Ring Theory

  433. Rings  are sets endowed with addition, subtraction and multiplication.
  434. Nonzero characteristic:  The least  p  for which all sums of  p  like terms vanish.
  435. Ideals  within a ring are  multiplicatively absorbent  additive subgroups.
  436. Quotient ring, modulo an ideal:  The residue classes modulo that ideal.
  437. Cauchy multiplication  is well-defined for "formal power series" over a ring.
  438. Ring of polynomials  whose coefficients are in a given ring.
  439. Galois rings.  Residues of modular polynomials,  modulo  one of them.

    Fields, Galois Fields and Skew Fields

  440. Vocabulary:  We consider  skew fields  to be  noncommutative.  Some don't.
  441. Fields  are commutative rings where every nonzero element has a reciprocal.
  442. Wedderburn's TheoremFinite  division rings are commutative  (they're fields).
  443. Every  finite  integral domain is a field.  A corollary of  Wedderburn's theorem.
  444. Galois fields  are the  finite fields.  Their orders are powers of prime numbers.
  445. The trivial field  has a single element.  It's the only field where 0 has a reciprocal.
  446. The splitting field  of PÎF[x]  is the smallest extension of F where P fully factors.
  447. The Nim-Field  is algebraically complete.  It contains [surreal] infinite ordinals.
  448. Ternary multiplication  compatible with  ternary addition  (without "carry").

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

  449. Vectors  were originally just differences between points in ordinary space...
  450. Abstract vector spacesVectors can be added, subtracted and  scaled.
  451. Modules  are vectorial structures over a  ring of scalars  (instead of a  field).
  452. Banach spaces  are  complete  normed vector spaces.
  453. Dual space:  The set of all [continuous] linear functions with scalar values.
  454. Tensors:  Multilinear functions of vectors and covectors with scalar values.
  455. An algebra  is a vector space with a scalable and distributive internal product.
  456. Clifford algebras are unital associative algebras endowed with a quadratic form.
  457. Physical things that are not vectorial  because they're not defined  intrinsically.
  458. David Hestenes has proposed  geometric algebra  as a denotational unification.

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

  459. The ring of p-adic integers  includes objects with infinitely many radix-p digits.
  460. Polyadic integers:  Greek naming of p-adic integers.
  461. What if p isn't prime?  Dealing with  divisors of zero.
  462. Decadic Integers:  The strange realm of 10-adic integers  (composite radix).
  463. The field of p-adic numbers  is the  quotient field  of the ring of p-adic integers.
  464. Dividing two p-adic numbers  looks like "long division", only backwards...
  465. The p-adic metric can be used to define p-adic numbers analytically.
  466. The reciprocal of a p-adic number  computed by successive approximations.
  467. Hasse's local-global principle  was established for the quadratic case in 1920.
  468. Integers which double  when their digits (in base g) are rotated.


  469. Pseudoprimes to base aPoulet numbers  are pseudoprimes to base 2.
  470. Weak pseudoprimes to base a :  Composite integers  n  which divide  (an-a).
  471. Counting the bases  to which a composite number is a pseudoprime.
  472. Strong pseudoprimes to base a  are less common than Euler pseudoprimes.
  473. Rabin-Miller Test:  An efficient and trustworthy  stochastic  primality test.
  474. The product of 3 primes  is a pseudoprime when all  pairwise  products are.
  475. Wieferich primes  are scarce but there are (probably) infinitely many of them.
  476. Super-pseudoprimesAll  their composite divisors are pseudoprimes.
  477. Maximal super-pseudoprimes  have no super-pseudoprime multiples.

    Factoring into Primes

  478. Jevons Number.  Factoring  8616460799  is now an  easy  task.
  479. Challenges  help tell  special-purpose  and  general-purpose  methods apart.
  480. Special cases  of  a priori  (partial) factorizations may help number theorists.
  481. Trial division  may be used to weed out the small prime factors of a number.
  482. Ruling out factors  can speed up trial divison in special cases.
  483. Recursively defined sequences  (over a  finite  set)  are  ultimately periodic.
  484. Pollard's r (rho) factoring method  is based on the properties of such sequences.
  485. Pollard's p-1 Method  finds prime factors  p  for which  p-1  is  smooth.
  486. Williams' p+1 Method  is based on the properties of Lucas sequences.
  487. Lenstra's Elliptic Curve Method  is a generalization of Pollard's p-1 approach.
  488. Dixon's method:  Combine small square residues into a solution of   x 2 º y 2

    Quadratic Reciprocity

  489. Motivation:  On the prime factors of some quadratic forms...
  490. Quadratic residues:  Half of the nonzero residues modulo an odd prime  p.
  491. Euler's criterion:  A quadratic residue raised to the power of  (p-1)/2  is 1.
  492. The Legendre symbol  (a|p)  can be extended to values of p besides odd primes.
  493. The law of quadratic reciprocity  states a simple but surprising fact.
  494. Gauss' Lemma  expresses a  Legendre symbol  as a product of many  signs.
  495. Eisenstein's Lemma:  A variation of  Gauss's lemma  allowing a simpler proof.
  496. One of many proofs of the  law of quadratic reciprocity.
  497. Artin's Reciprocity.

    Continued Fractions  (and related topics)

  498. What is a continued fraction?  Example:  The expansion of p.
  499. The convergents of a number are its best rational approximations.
  500. Large partial quotients allow very precise approximations.
  501. Regular patterns in the continued fractions of some irrational numbers.
  502. For almost all numbers, partial quotients are ≥ k with probability  lg(1+1/k).
  503. Elementary operations on continued fractions.
  504. Expanding functions as continued fractions.
  505. Engel expansion of a positive number.  A nondecreasing sequence of integers.
  506. Pierce expansions of numbers between 0 and 1.  Strictly increasing sequences.

    Recreational Mathematics

  507. Counterfeit Coin Problem: In 3 weighings, find an odd object among 12, 13, 14.
  508. General Counterfeit Penny Problem: Find an odd object in the fewest weighings.
  509. Seven-Eleven: Four prices with a sum and product both equal to 7.11.
  510. Equating a right angle and an obtuse angle, with a clever false proof.
  511. Choosing a raise: Trust common sense, beware of  fallacious accounting.
  512. 3 men pay $30 for a $25 hotel room, the bellhop keeps $2... Is $1 missing?
  513. Chameleons: A situation shown unreachable because of an invariant quantity.
  514. Sam Loyd's 14-15 puzzle also involves an invariant quantity (and two orbits).
  515. Einstein's riddle: 5 distinct house colors, nationalities, drinks, smokes and pets.
  516. Numbering n pages of a book takes this many digits (formula).
  517. The Ferry Boat Problem (by Sam Loyd): To be or not to be ingenious?
  518. Hat overboard !   What's the speed of the river?
  519. All digits once and only once: 48 possible sums (or 22 products).
  520. Crossing a bridge: 1 or 2 at a time, 4 people (U2), different paces, one flashlight!
  521. Managing supplies to reach an outpost 6 days away, carrying enough for 4 days.
  522. Go south, east, north and you're back... not necessarily to the North Pole!
  523. Icosapolis: Numbering a 5 by 4 grid so adjacent numbers differ by at least 4.
  524. Unusual mathematical boast for people born in 1806, 1892, or 1980.
  525. Puzzles for extra credit: From Chinese remainders to the Bookworm Classic.
  526. Simple geometrical dissection:  A proof of the Pythagorean theorem.
  527. Early bird saves time by walking to meet incoming chauffeur.
  528. Sharing a meal: A man has 2 loaves, the other has 3, a stranger has 5 coins.
  529. Fork in the road: Find the way to Heaven by asking only one question.
  530. Proverbial Numbers: Guess the words which commonly describe many numbers.
  531. Riddles: The Riddle of the Sphinx and other classics, old and new.

    The Mathematical Games of Martin Gardner

  532. FlexagonsHexaflexagons  were popularized by Martin Gardner in 1956.
  533. Polyominoes:  The 12 pentominoes and other tiles invented by  Sol Golomb.
  534. Soma:  7 nonconvex solids consisting of  3 or 4  cubes make a larger cube.
  535. Tessellations by convex pentagons.  The contributions of  Marjorie Rice.
  536. Kites and Darts.  The  aperiodic  tilings of Roger Penrose.
  537. Ambigrams:  Calligraphic spellings which change when rotated or flipped.
  538. The Game of Life.  John Conway's  endearing  cellular automaton  (1970).
  539. Rubik's Cube:  Ernõ Rubik (1974)  D. Singmaster (1979)  M. Gardner (1981).

    Mathematical "Magic" Tricks

  540. 1089:  Subtract a 3-digit number and its reverse, then add this to  its  reverse...
  541. Mass media mentalism  by  David Copperfield  (1992).
  542. Grey Elephants in Denmark: "Mental magic" for one-time classroom use.
  543. The 5-card trick of Fitch Cheney:  Tell the fifth card once 4 are known.
  544. Generalizing the 5-card trick and  Devil's Poker...
  545. Kruskal's Count.
  546. Paths to God.
  547. Stacked Deck.
  548. Magic Age Cards.
  549. Ternary Cards.
  550. Magical 21  (or 27).
  551. Boolean Magic.
  552. Perfect Faro Shuffles.

    Mathematical Games (Strategies)

  553. Dots and Boxes: The "Boxer's Puzzle" position of Sam Loyd.
  554. The Game of Nim: Remove items from one of several rows. Don't play last.
  555. Grundy numbers are defined for all positions in impartial games.
  556. Moore's Nim: Remove something from at most (b-1) rows. Play last.
  557. Normal Kayles: Knocking down one pin, or two adjacent ones, may split a row.
  558. Grundy's Game: Split a row into two unequal rows. Whoever can't move loses.
  559. Wythoff's Game: Remove counters either from one heap or equally from both.

    Ramsey Theory

  560. The pigeonhole principle:  What must happen with fewer holes than pigeons...
  561. n+1 of the first 2n integers  always include two which are coprime.
  562. Largest sets of small numbers with at most  k  pairwise coprime integers.
  563. Ramsey's Theorem:  Monochromatic complete subgraphs of a large graph.
  564. Infinite alignment among infinitely many lattice points in the plane?  Nope.
  565. Infinite alignment in a lattice sequence with bounded gaps?  Almost...
  566. Large alignments in a lattice sequence with bounded gaps.  Yeah!
  567. Van der Waerden's theorem:  Long monochromatic arithmetic progressions.


  568. Ford circles are nonintersecting circles touching the real line at rational points.
  569. Farey series:  The rationals from 0 to 1, with a bounded denominator.
  570. The Stern-Brocot tree  contains a single occurrence of every positive rational.
  571. Any positive rational  is a unique ratio of two consecutive Stern numbers.
  572. Pick's formula gives the area of a lattice polygon by counting lattice points.

    History, Nomenclature, Vocabulary, etc.

    History :
  573. Earliest mathematics on record. Before Thales was Euphorbe...
  574. Indian numeration became a positional system with the introduction of zero.
  575. Roman numerals are awkward for larger numbers.
  576. The invention of logarithms: John Napier, Bürgi, Briggs, Saint-Vincent, Euler.
  577. The earliest mechanical calculator(s), by W. Shickard (1623) or Pascal (1642).
  578. The Fahrenheit Scale: 100°F  was meant to be the normal body temperature.
  579. The revolutionary innovations  which brought about new civilizations.
    Nomenclature & Etymology :
  580. The origin of the word "algebra", and also that of "algorithm".
  581. The name of the avoirdupois system:  Borrowed from French in a pristine form.
  582. Long Division: Cultural differences in writing the details of a division process.
  583. Is a parallelogram a trapezoid? In a mathematical context [only?], yes it is...
  584. Naming polygons. Greek only please; use hendecagon not "undecagon".
  585. Chemical nomenclature: Basic sequential names (systematic and/or traditional).
  586. Fractional Prefixes: hemi (1/2), sesqui (3/2) or weirder hemipenta, hemisesqui...
    • Matches, phosphorus, and phosphorus sesquisulphide.
  587. Zillion. Naming large numbers.
  588. Zillionplex. Naming huge numbers.

    Style and Usage

  589. Abbreviations:  Abbreviations of scholarly Latin expressions.
  590. Typography of long numbers.
  591. Intervals  denoted with square brackets (outward for an excluded extremity).
  592. Dates  in the simplest ISO 8601 form  (with  customary  time stamps or not).
  593. The names of operands in common numerical operations.
  594. The word  respectively  doesn't have the same syntax as "resp."

    Setting the Record Straight

  595. The heliocentric Copernican system was known two millenia before Copernicus.
  596. The assistants of Galileo Galilei and the mythical experiment at the Tower of Pisa.
  597. Switching calendars: Newton was not born the year Galileo died.
  598. The Lorenz Gauge is an idea of Ludwig Lorenz (1829-1891) not H.A. Lorentz.
  599. Special Relativity was first formulated by H. Poincaré (Einstein a close second).
  600. The Fletcher-Millikan "oil-drop" experiment was not the sole work of Millikan.
  601. Collected errata  about customary physical units.
  602. Portrait of Legendre:  The  mathematician  was confused with the politician.
  603. Dubious quotations:  Who  really  said that?

    Ancient Knowledge

  604. Obliquity of the ecliptic:  An evolving quantity first measured by Eratosthenes.
  605. Vertical wells at Syene are completely sunlit only once a year, aren't they?
  606. Eratosthenes sizes up the Earth:  700 stadia per degree of latitude.
  607. Latitude and longitude:  The spherical grid of meridians and parallels.
  608. Itinerary units:  The  land league  and the  nautical league.
  609. Amber, compass and lightning:  Glimpses of electricity and magnetism.


  610. The "work done" on a point-mass  equals the change in its  kinetic energy.
  611. Relativistic work done  and the corresponding change in  relativistic energy.
  612. Relativistic thermodynamics:  The case of a point endowed with internal heat.
  613. Spacecraft speeds up upon reentry into the upper atmosphere.
  614. Lewis Carroll's monkey climbs a rope over a pulley, with a counterweight.
  615. Two-ball drop can make a light ball bounce up to 9 times the dropping height.
  616. Normal acceleration  is the square of speed divided by the radius of curvature.
  617. Roller-coasters  must rise more than half a radius above any  loop-the-loop.
  618. Conical pendulum:  A hanging bob whose trajectory is an horizontal circle.
  619. Ball in a BowlPure rolling  increases the period of oscillation by 18.3%.
  620. Hooke's Law:  Motion of a mass suspended to a spring.
  621. Speed of an electron estimated with the Bohr model of the atom.
  622. Thermal expansion coefficients:  The cubical coefficient is 3 times the linear one.
  623. Waves in a solid: P-waves (fastest), S-waves, E-waves (thin rod), SAW...
  624. Rayleigh Wave: The quintessential surface acoustic wave (SAW).
  625. Hardest Stuff:  Diamond is no longer the hardest material known to science.
  626. Hardness  is an elusive  nonelastic  property, distinct from  stiffness.
  627. Hot summers, hot equator! The distance to the Sun is not the explanation.
  628. Kelvin's Thunderstorm:  Using falling water drops to generate high voltages.
  629. The Coriolis effect:  A dropped object falls to the east of the plumb line.
  630. Terminal velocity  of an object falling in the air.
  631. Angular momentum and torque.  Spin and orbital angular momentum.

    Motion of Rigid Bodies  (Classical Mechanics)

  632. Rotation vector  of a moving rigid body (and/or "frame of reference").
  633. Angular momentum  equals  moment of inertia  times  angular velocity.
  634. Moments about a point or a plane  are convenient mathematical fictions.
  635. Moment of inertia of a spherical distribution  and of an homogeneous ellipsoid.
  636. Perpendicular Axis Theorem:  Axis of rotation perpendicular to a  thin plate.
  637. The Parallel Axis Theorem  gives the moment of inertia about an off-center axis.
  638. Moment of inertia of a thick plate,  as obtained from the  parallel axis theorem.
  639. Momenta of homogeneous bodies:  List of examples.
  640. Rigid pendulum  moving under its own weight about a fixed horizontal axis.
  641. Reversible pendulum  swings with the same period around two distinct axes.

    Newtonian Gravity

  642. All physical theories  have a limited range of validity.
  643. Gravity vs. Electrostatics:  Straight comparisons.
  644. Airy weighs the Earth  by timing a pendulum at the bottom of a mine.
  645. Rigid equilateral triangle  formed by three gravitating bodies.
  646. The five Lagrange points  of two gravitating bodies in circular orbit.


  647. Huygens' Principle  is a convenient fiction to describe the propagation of waves.
  648. Diffraction  occurs when when a wave emanates from a bounded source.
  649. Young's  double-slit  experiment  demonstrates the wavelike nature of light.
  650. Celerity  is the speed with which  phase  propagates.
  651. Standing waves  feature stationary nodes and antinodes.
  652. Chladni patterns:  The lines formed by nodes in an oscillating  plate  (or surface).
  653. Snell's Law (1621)  gives the angle of refraction (or says nothing is refracted).
  654. Birefringence  and the discovery of  polarization  (Erasmus Bartholinus, 1669).
  655. Brewster's angle  is the incidence which yields a 100% polarized reflection.
  656. Fresnel equations  give the reflected or refracted intensities of polarized light.
  657. Stokes parameters:  A standard description of the  state of polarization.

    Colors & Dispersion

  658. Dispersion relation:  pulsatance vs. wave number (or frequency vs. wavelength).
  659. Group velocity  is the traveling speed of a beat phenomenon.
  660. Rayleigh scattering  makes the sky blue and sunsets red.
  661. Index of refraction of water  for light of different colors.
  662. A spherical drop  bounces red light up to 42.34° and violet light up to 40.58°.

    Analytical Mechanics  &  Classical Field Theory

  663. Fermat's principle  (least time)  for light (c.1655) predates Newton's mechanics.
  664. Maupertuis principle  of  least action  (1744).
  665. Virtual Work:  A substitute for Newton's laws which factors out constraint forces.
  666. Phase Space:  A   phase  describes all there is to a classical system.
  667. Either velocities or momenta  can be used (with positions) to specify a  phase.
  668. Relativistic point-mass:  Lagrangian, Hamiltonian and momentum in free space.
  669. Charge in a magnetic field:  The canonical momentum isn't the linear momentum.
  670. Lagrangian mechanics:  The  Lagrangian  is a function of positions and velocities.
  671. Hamiltonian mechanics:  The  Hamiltonian  depends on positions and momenta.
  672. Poisson brackets:  An abstract synthetic view of analytical mechanics.
  673. Liouville's theorem:  The  volume  in Hamiltonian phase space doesn't change.
  674. Noether's Theorem:  Conservation laws express the symmetries of physics.
  675. Field Theory:  Lagrangian mechanics on a continuum of values (and derivatives).

    Electromagnetism  (Maxwell's Equations)

  676. Clarifications by Heaviside & Lorentz:  Vector calculus & microscopic view.
  677. The vexing problem of units  is a thing of the past if you stick to SI units.
  678. The Lorentz force  on a test particle defines the local electromagnetic fields.
  679. Electrostatics:  The study of the electric field due to static charges.
  680. Electric capacity  is an electrostatic concept  (adequate at low frequencies).
  681. Electrostatic multipoles:  The multipole expansion of an electrostatic field.
  682. Birth of electromagnetism (1820):  Electric currents generate magnetic fields !
  683. Biot-Savart Law:  The  static  magnetic induction due to steady currents.
  684. Magnetic scalar potential:  A multivalued function whose gradient is induction.
  685. Magnetic monopoles do not exist :  A law stating a fact not yet disproved.
  686. Ampère's law:  The law of static electromagnetism devised by Ampère in 1825.
  687. Faraday's law:  A varying magnetic flux  induces  an electric circulation.
  688. Self-induction:  The induction received by a circuit from its own magnetic field.
  689. Ampère-Maxwell law:  The dynamic generalization (1861) of  Ampère's law.
  690. Putting it all together:  Historical paths to Maxwell's  electromagnetism.
  691. Maxwell's equations  unify electricity and magnetism dynamically  (1864).
  692. Continuity equation:  Maxwell's equations imply  conservation of charge.
  693. Waves anticipated by Faraday, Maxwell & FitzGerald were observed by Hertz.
  694. Electromagnetic energy density  and the flux of the Poynting vector.
  695. Planar electromagnetic waves:  The simplest type of electromagnetic waves.
  696. Electromagnetic potentials  are postulated to obey the  Lorenz gauge.
  697. Solutions to Maxwell's equations,  as  retarded  or  advanced  potentials.
  698. Electrodynamic fields  corresponding to  retarded  potentials.
  699. The gauge of retarded potentials:  is it  really  the Lorenz gauge?
  700. Electric and magnetic dipoles:  Dipolar solutions of Maxwell's equations.
  701. Static distributions of magnetic dipoles  can be simulated with steady currents.
  702. Static distributions of electric dipoles  can be simulated with static charges.
  703. Sign reversal  in the fields of uniformly distributed magnetic or electric dipoles.
  704. Fields at the center of uniformly magnetized or polarized spheres  (of any size).
  705. Relativistic dipoles:  A moving magnet develops an electric moment.
  706. Power radiated by an accelerated charge:  The Larmor formula (1897).
  707. Lorentz-Dirac equation  for the motion of a point charge is of  third  order.

    Electromagnetic Dipoles

  708. Molecular electric dipole moments  were first studied by Peter Debye (1912).
  709. Force exerted on a dipole  by a  nonuniform  field.
  710. Torque on a dipole  is proportional to its cross-product into the field.

    Magnetism,  Electromagnetic Properties of Matter

  711. Magnetization and polarization  describe densities of dipoles  bound  to matter.
  712. Gauge invariance:  Many magnetizations and polarizations create the same field.
  713. Maxwell's equations in matter:  Electric displacement D,  magnetic strength H.
  714. Electric susceptibility  is the propensity to be polarized by an  electric field.
  715. Electric permittivity and magnetic permeability  are related to susceptibilities.
  716. Paramagnetism:  Weak  positive  susceptibility.
  717. Diamagnetism:  The Lorentz force turns orbital moments against the external B.
  718. Magnetic levitation:  How to skirt the theorem of Samuel Earnshaw (1842).
  719. Pyrolytic carbon:  The most diamagnetic sunstance known, at room temperature.
  720. Bohr & Van Leeuwen Theorem:  Diamagnetism and paramagnetism cancel ?!
  721. Thermodynamics of dielectric matter:  dU = E.dD + ...
  722. Ferromagnetism:  Permanent magnetization without an external magnetic field.
  723. Antiferromagnetism:  When adjacent dipoles tend to oppose each other...
  724. Ferrimagnetism:  With two kinds of dipoles, partial cancellation may occur.
  725. Magneto-optical effect  discovered by Faraday on September 13, 1845.
  726. Ohm's Law:  Current density is proportional to electric field:  j = s E.

    Motors and Generators

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


  732. Observers in motion:  A simple-minded derivation of the Lorentz Transform.
  733. Adding up velocities:  The combined speed can never be more than c.
  734. Fizeau's empirical relation  between refractive index  (n) and  Fresnel drag.
  735. The Harress-Sagnac effect  used to measure rotation with fiber optic cable.
  736. Combining relativistic speeds:  Using rapidity, the rule is transparent.
  737. Relative velocity of two photons:  Defined unless both have the same direction.
  738. Minkowski spacetime:  Coordinates of 4-vectors obey the Lorentz transform.
  739. The Lorentz transform expressed vectorially:  A so-called  boost  of speed  V.
  740. Wave vector:  The 4-dimensional gradient of the phase describes propagation.
  741. Doppler shift:  The relativistic effect is not purely radial.
  742. Kinetic energy:  At low speed, the relativistic energy varies like  ½ mv 2.
  743. Photons and other massless particles:  Finite energy at speed  c.
  744. The de Broglie celerity  (u)  is inversely proportional to a particle's speed.
  745. Compton diffusion:  The result of collisions between photons and electrons.
  746. The Klein-Nishina formula:  gives the  cross-section  in Compton scattering.
  747. Compton effect is suppressed  quantically for visible light and bound electrons.
  748. Elastic shock:  Energy transfer is  v.dp.  (None is seen from the barycenter.)
  749. Photon-photon scattering  is like an  elastic collision of two photons.
  750. Cherenkov Effect:  When the speed of an electron exceeds the celerity of light...
  751. Constant acceleration  over an entire lifetime will take you  pretty far...

    General Relativity

  752. The Harress-Sagnac effect  seen by an observer rotating with the optical loop.
  753. Relativistic rigid motion  is an  equilibrium  modified at the speed of  sound.
  754. In the Euclidean plane:  Contravariance and covariance.
  755. In the Lorentzian plane:  Contravariance and covariance revisited.
  756. Tensors of rank n+1  are linear maps that send a vector to a tensor of rank n.
  757. Signature  of the quadratic form defined by a given metric tensor.
  758. Covariant and contravariant coordinates  of tensors of rank  n, in 4 dimensions.
  759. The metric tensor and its inverse.  Lowering and raising indices.
  760. Partial derivatives  with respect to  contravariant  or  covariant  coordinates.
  761. Christoffel symbols:  Coordinates of the partial derivatives of the basis vectors.
  762. Covariant derivativesAbsolute differentiation  with the  nabla  operator  Ñ.
  763. Contravariant derivatives:  The lesser known flavor of absolute derivatives.
  764. The antisymmetric part of Christoffels symbols  form a fundamental  tensor.
  765. Totally antisymmetric spacetime torsion  is described by a  vector field.
  766. Levi-Civita symbols:  Antisymmetric with respect to any pair of indices.
  767. Einstein's equivalence principle  implies  vanishing  spacetime torsion.
  768. Ricci's theorem:  The covariant derivative of the metric tensor vanishes.
  769. Curvature:  The  Ricci tensor  is obtained by contracting the  Riemann tensor.
  770. The Bianchi identity  show that the  Einstein tensor  is divergence free.
  771. Stress tensor:  Flow of energy density is density of [conserved] momentum.
  772. Einstein's Field Equations:  16 equations in covariant form (Einstein, 1915).
  773. Free-falling bodies:  Their trajectories are  geodesics  in curved spacetime.
  774. The "anomalous" precession of Mercury's perihelion  is entirely  relativistic.
  775. The Schwarzschild metric:  The earliest exact solution to Einstein's equations.
  776. What is mass?
  777. Electromagnetism:  Covariant expressions, using tensors.
  778. Kaluza-Klein theory of electromagnetism  involves a  fifth dimension.
  779. Harvard Tower Experiment:  The slow clock at the bottom of the tower.
  780. Shapiro time delay:  The effect on radar signals of gravitational time dilation.

    String Theory and other "Theories of Everything"

  781. Unification:  Consistency is required.  Actual high-energy unification is not.
  782. Kaluza-Klein Theory:  Postulating one extra dimension for electromagnetism.
  783. Gabriele Veneziano:  The magic of Euler's  beta and gamma functions.
  784. Leonard Susskind (1940-):  The basic idea of a fundamental string.
  785. Joël Scherk (1946-1979) & John Schwarz:  Rediscovering  gravity.
  786. Michael Green & John Schwarz:  Hoping for a  Theory of Everything.
  787. String QuintetFive  different consistent string theories!
  788. M-Theory:  Ed Witten's 11-dimensional brainchild, unveiled at  String '95.
  789. The brane world scenarios  of  Lisa Randall  and  Burt Ovrut.

    Physics of Gases and Fluids

  790. The Magdeburg hemispheres  are held together by more than one ton of force.
  791. The ideal gas laws  of Boyle, Mariotte, Charles, Gay-Lussac, and Avogadro.
  792. Joule's law:  The internal energy of a perfect gas depends only on its temperature.
  793. The Van der Waals equation and other interesting equations of state.
  794. Virial equation of state.  Virial expansion coefficients.  Boyle's temperature.
  795. Viscosity is the ratio of a shear stress to the shear strain rate it induces.
  796. Permeability and permeance: Vapor barriers and porous materials.
  797. Resonant frequencies of air in a box.
  798. The Earth's atmosphere. Pressure at sea-level and total mass above.
  799. The first hot-air balloon  (Montgolfière)  was demonstrated on June 4, 1783.

    Transport Propertities of Matter

  800. Viscosity:  The transport of microscopic momentum.
  801. Brownian motion  and  Einstein's estimate of molecular sizes.
  802. Thermal Conductivity:  The transport of microscopic energy.
  803. Diffusivity:  The transport of  chemical concentration.
  804. Speed of Sound:  Reversible transport of a pressure disturbance in a fluid.

    Filters and Feedback

  805. Complex pulsatance (s)  is damping constant (s) plus imaginary pulsatance (iw).
  806. Complex impedance:  Resistance and reactance.
  807. Quality Factor (Q).  Ratio of maximal stored energy to dissipated power.
  808. Nullators and norators:  Strange dipoles for analog electronic design.
  809. Corner frequency  of a simple  first-order  low-pass filter.  -3 dB bandwidth.
  810. Second-order  passive low-pass filter, with inductor and capacitor.
  811. Sallen key filters:  Active filters do not require inductors.
  812. Lowpass Butterworth filter of order n :  The flattest low-frequency response.
  813. Linkwitz-Riley crossover filters  are used in modern active audio crossovers.
  814. Chebyshev filters:  Ripples in either the passband or the stopband.
  815. Elliptic (Cauer) filters  encompass all Butterworth and Chebyshev types.
  816. Legendre filters  maximal roll-off rate with a monotonous frequency response.
  817. Gegenbauer filters:  From Butterworth to Chebyshev, via Legendre.
  818. Phase response  of a filter.
  819. Bessel-Thomson filters:  Phase linearity and group delay.
  820. Gaussian filters:  Focusing on time-domain communication pulses.
  821. Linear Phase Equiripple:  Ripples in group delay to improve on Bessel filters.
  822. DSL filter  allows POTS below 3400 Hz and blocks digital data above 25 kHz.

    Fantasy Engineering: Just for fun.

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

    Steam Engines

  826. The aeolipile:  This ancient steam engine demonstrates jet propulsion.
  827. Edward Somerset of Worcester (1601-1667):  Blueprint for a steam fountain.
  828. Denis Papin (1647-1714):  Pressure cooking and the first piston engine.
  829. Thomas Savery (c.1650-1715):  Two pistons and an independent boiler.
  830. Thomas Newcomen (1663-1729) and John Calley:  Atmospheric steam engine.
  831. Nicolas-Joseph Cugnot (1725-1804):  The first automobile  (October 1769).
  832. James Watt (1736-1819):  Steam condenser and  Watt governor.
  833. Richard Trevithick (1771-1833) and the first railroad locomotives.
  834. Sadi Carnot (1796-1832):  Carnot's cycle and the theoretical  efficiency limit.
  835. Sir Charles Parsons (1854-1931):  The modern steam  turbine, born in 1884.


  836. The elementary concept of temperature.  The zeroth law of thermodynamics.  Lord Kelvin  1824-1907 Hermann von Helmholtz  1821-1894
  837. Conservation of energy:  The first law of thermodynamics.
  838. Increase of Entropy:  The second law of thermodynamics.
  839. State variablesExtensive  and  intensive  quantities.
  840. Entropy  is  missing information, a measure of  disorder.
  841. Nernst Principle  (third law):  Entropy is zero at zero temperature.
  842. Thermodynamic potentials  can be convenient alternatives to  internal energy.
  843. Latent heat  (L)  is the heat transferred in a change of  phase.
  844. Calorimetric coefficients, adiabatic coefficient  (g)  heat capacities, etc.
  845. Cryogenic coefficients:  Lower temperature with an  isenthalpic  expansion.
  846. Relativistic considerations:  A moving body appears  cooler.
  847. Inertia of energy  for an object at nonzero temperature.
  848. Stefan's Law:  A black body radiates as the fourth power of its temperature.
  849. The "Fourth Law":  Is there really an upper bound to temperature?
  850. Hawking radiation:  On the entropy and temperature of a black hole.
  851. Partition function:  The cornerstone of the statistical approach.

    Demons of Classical Physics

  852. Laplace's Demon:  Deducing past and future from a detailed snapshot.
  853. Maxwell's Demon:  Trading information for entropy.
  854. Shockley's Ideal Diode Equation:  Diodes don't violate the Second Law.
  855. Szilard's engine & Landauer's Principle: The thermodynamic cost of  forgetting.

    Statistical Physics

  856. Lagrange multipliers  are associated to the constraints of a maximizing problem.
  857. Microcanonical equilibrium:  All states of an isolated system are equiprobable.
  858. Canonical equilibrium:  In a heat bath, probabilities involve a Boltzmann factor.
  859. Grand-canonical equilibrium  when  chemical  exchanges are possible.
  860. Bose-Einstein statistics:  One state may be occupied by  many  particles.
  861. Fermi-Dirac statistics:  One state is occupied by  at most one  particle.
  862. Boltzmann statistics:  The  low-occupancy limit  (most states are unoccupied).
  863. Maxwell-Boltzmann distribution  of molecular speeds in an  ideal gas.
  864. Partition function:  The cornerstone of the statistical approach.

    Quantum Mechanics

  865. Quantum Logic:  The surprising way quantum probabilities are obtained.
  866. Swapping particles  either  negates the quantum state  or  leaves it unchanged.
  867. The Measurement Dilemma:  What makes  Schrödinger's cat  so special?
  868. Matrix Mechanics:  Neither measurements nor matrices can be switched at will.
  869. Schrödinger's Equation:  A nonrelativistic quantum particle in a classical field.
  870. Noether's Theorem:  Conservation laws express the symmetries of physics.
  871. Kets  are Hilbert vectors (their duals are bras) on which observables operate.
  872. Observables  are operators explicitely associated with physical quantities.
  873. Commutators are the quantities which determine  uncertainty relations.
  874. Uncertainty relations  hold whenever the commutator does not vanish.
  875. Spin  is a form of angular momentum without a classical equivalent.
  876. Pauli matrices:  Three 2 by 2 matrices with  eigenvalues  +1 and -1.
  877. Quantum Entanglement:  The  singlet  and  triplet  states of two electrons.
  878. Bell's inequality  is violated for the  singlet  state of two electron spins.
  879. Generalizations of Pauli matrices  beyond spin ½.
  880. Density operators  are quantum representations of imperfectly known states.

    Quantum Field Theory

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

    Ancient Recipes and Modern Chemistry

  884. Black Powder:  An ancient explosive, still used as a propellant (gunpowder).
  885. Predicting explosive reactions:  A useful but oversimplified rule of thumb.
  886. Thermite  generates temperatures hot enough to weld iron.
  887. Enthalpy of Formation:  The tabulated data which gives energy balances.
  888. Gibbs Function  (free energy):  Its sign indicates the direction of spontaneity.
  889. Labile  is not quite the same as  unstable.
  890. Inks:  India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.
  891. Redox Reactions:  Oxidizers are reduced by accepting electrons...
  892. Gold ChemistryAqua regia ("Royal Water") dissolves gold and platinum.
  893. Who is the "father" of modern chemistry?

    Medicine by the Numbers

  894. International Unit  (IU) is an arbitrarily-defined rating of  biological activity.
  895. Concentration  is an amount (either mass or moles) per volume.
  896. Glycosylated hemoglobin  (HbA1c) relates to  average  blood glucose (bG).

    Cosmology 101

  897. Kant's Island Universes:  The Universe is filled with  separate  galaxies.
  898. The Cosmological Principle: The Universe is homogeneous and isotropic.
  899. The Big Bang:  An idea of Georges Lemaître  mocked by Fred Hoyle.
  900. The Cosmic Microwave Background (CMB): Its spectrum and density.
  901. Cosmic redshift (z):  Light emitted in a Universe which was (1+z) times smaller.
  902. Hubble Law:  The relation between redshift and distance for comoving points.
  903. Omega (W): The ratio of the density of the Universe to the critical density.
  904. Look-Back Time:  The time ellapsed since observed light was emitted.
  905. Distance:  In a cosmological context, there are several flavors to the concept.
  906. Comoving points are reference points following the expansion of the universe.
  907. The Anthropic Principle: An obvious explanation which may not be the final one.
  908. Dark matter & dark energy: Gravity betrays the existence of some  dark  stuff.
  909. The Pioneer Effect: The anomalous escape of the Pioneer spaceprobe.

    Stars and Stellar Objects

  910. Nuclear fusion  is what powers the stars.
  911. Brown dwarves  fail to ignite fusion.  They glow from gravitational contraction.
  912. The Jeans mass  above which a gas at temperature T collapses gravitationally.
  913. Main sequence:  The evolution of a typical star.
  914. Eta Carinae  and  hypergiants.  The most massive stars possible.
  915. Betelgeuse  and red supergiants.
  916. Rigel  and blue supergiants.
  917. Planetary nebulae:  Aftermaths of stellar explosions.
  918. White dwarfs:  The ultimate fate of our Sun and other small stars.
  919. Neutron stars:  Remnants from the supernova collapse of medium-sized stars.
  920. Stellar black holes:  They form when supermassive stars run out of nuclear fuel.
  921. Stellar X-ray source:  A small  accretor  in tight orbit around a  donor  star.

    The Solar System

  922. Astronomical unit:  The precise definition of a standard unit of length.
  923. The solar corona  is a very hot region of rarefied gas.
  924. Solar radiation:  The Sun has radiated away about 0.03% of its mass.
  925. The Titius-Bode Law: A numerical pattern in solar orbits?
  926. The 4 inner rocky planets:  Mercury, Venus, Earth, Mars.
  927. EarthThis  is home.
  928. The asteroid belt:  Planetoids and bolids between Mars and Jupiter.
  929. The 4 giant gaseous planets: Jupiter, Saturn, Uranus, Neptune.
  930. The discovery of NeptuneUrbain Le Verrier  scooped John Couch Adams.
  931. Pluto  and other  Kuiper Belt Objects  (KBO).
  932. Sedna  and other planetoids beyond the  Kuiper Belt.
  933. What's a planet?  Anything besides the 6 ancient planets, Uranus & Neptune?
  934. Heliosphere and Heliopause:  The domain where solar wind exerts its influence.
  935. Oort's Cloud  is a cometary reservoir at the fringe of the Solar System.

    Practical Formulas

  936. Easy conversion between Fahrenheit and Celsius scales:  F+40  =  1.8 (C+40).
    Automotive :
  937. Car speed is proportional to tire diameter and engine rpm, divided by gear ratio.
  938. Car acceleration. Guessing the curve from standard data.
  939. "0 to 60 mph" time, obtained from vehicle mass and actual average power.
  940. Thrust  is the power to speed ratio (measuring speed along thrust direction).
  941. Power of an engine as a function of its size:  Rating internal combustion engines.
  942. Optimal gear ratio  to maximize top speed on a flat road  (no wind).
    Surface Areas :
  943. Heron's Formula (for the area of a triangle) is related to the Law of Cosines.
  944. Brahmagupta's Formula gives the area of a quadrilateral, inscribed or not.
  945. Bretschneider's Formula: Area of a quadrilateral of known sides and diagonals.
  946. The (vector) area of a quadrilateral  is  half  the cross-product of its diagonals.
  947. Parabolic segment:  2/3 the area of a circumscribed parallelogram or triangle.
    Volumes :
  948. Content of a cylindrical tank (horizontal axis), given the height of the liquid in it.
  949. Volume of a spherical cap, or content of an elliptical vessel, given liquid height.
  950. Content of a cistern (cylindrical with elliptical ends), as a function of fluid height.
  951. Volume of a cylinder or prism, possibly with tilted [nonparallel] bases.
  952. Volume of a conical frustum:  Formerly a staple of elementary education...
  953. Volume of a sphere...  obtained by subtracting a cone from a cylinder !
  954. The volume of a tetrahedron  is the determinant of three edges, divided by 6.
  955. Volume of a wedge of a cone.
    Averages :
  956. Splitting a job evenly between two unlike workers.
  957. Splitting a job unevenly between two unlike workers.
  958. Alcohol solutions are rated by volume not by mass.
  959. Mixing solutions to obtain a predetermined intermediate rating.
  960. Special averages: harmonic (for speeds), geometric (for rates), etc.
  961. Mean Gregorian month: either 30.436875 days, or 30.458729474253406983...
  962. The arithmetic-geometric mean  is related to a  complete elliptic integral.
    Geodesy and Astronomy :
  963. Distance to ocean horizon line is proportional to the square root of your altitude.
  964. Distance between two points on a great circle at the surface of the Earth.
  965. The figure of the Earth. Geodetic and geocentric latitudes.
  966. 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.

    Perimeter of an Ellipse

  967. Circumference of an ellipse: Introducing exact series and approximate formulas.
  968. Ramanujan I and Lindner formulas:  The journey begins...
  969. Ramanujan II:  An awesome approximation from a mathematical genius (1914).
  970. Hudson's Formula and other  Padé approximations.
  971. Peano's Formula:  The sum of two approximations with cancelling errors.
  972. The YNOT formula  (Maertens, 2000.  Tasdelen, 1959).
  973. Euler's formula is the first step in an exact expansion.
  974. Naive formulap  ( a + b )  features a  -21.5% error for elongated ellipses.
  975. Cantrell's Formula:  A modern attempt with an overall accuracy of 83 ppm.
  976. From Kepler to Muir.  Lower bounds and other approximations.
  977. Relative error cancellations in symmetrical approximative formulas.
  978. Complementary convergences of two series yield a nice foolproof algorithm.
  979. Padé approximants  are used in a whole family of approximations...
  980. Improving Ramanujan II  over the whole range of eccentricities.
  981. The Arctangent Function as a component of several approximate formulas.
  982. Abed's formula uses a parametric exponent to fine-tune the approximation.
  983. Zafary's formula.
  984. Rivera's formula gives the perimeter of an ellipse with 104 ppm accuracy.
  985. Better accuracy from Cantrell, building on his own previous formula
  986. Rediscovering  a well-known exact expansion due to Euler (1773).
  987. Exact expressions for the circumference of an ellipse:  A summary.

    Surface Area of an Ellipsoid

  988. Surface Area of a Scalene Ellipsoid:  The general formula isn't elementary.
  989. Thomsen's Formula:  A simple symmetrical approximation.
  990. Approximate formulas  for the surface area of a scalene ellipsoid.
  991. Nautical mile:  "Average"  minute of latitude  on an oblate spheroid.

    The Unexplained

  992. The Magnetic Field of the Earth.
  993. Life (1):  The mysteries of evolution.
  994. Life (2):  The origins of life on Earth.
  995. Life (3):  Does extraterrestrial life exist?  Is there intelligence out there?
  996. Nemesis: A distant companion to the Sun could explain extinction periodicity.
  997. Current Challenges to established dogma.
  998. Unexplained artifacts and sightings.

    Open Questions (or tough answers)

  999. The Riemann Hypothesis:   {Re(z) > 0   &   z(z) = 0}   Þ   {Re(z) = ½}.
  1000. P = NP ?   Can we  find  in polynomial time whatever we can  check  that fast?
  1001. Collatz sequences go from n to n/2 (iff n is even) or 3n+1. Do they all lead to 1?
  1002. The Poincaré Conjecture  was proven by  Grisha Perelman  in 2002.

    Mathematical Miracles

  1003. The only magic hexagon.
  1004. The law of small numbers applied to conversion factors.
  1005. Quadratic formulas yielding long sequences of prime numbers.
  1006. The area under a Gaussian curve  involves the square root of  p
  1007. Exceptional simple Lie groups.
  1008. Monstrous Moonshine in Number Theory.


  1009. Oldest unsolved mathematical problem:  Are there any odd perfect numbers?
  1010. Magnetic Field of the Earth: The south side is near the geographic north pole.
  1011. From the north side,  a counterclockwise angle is positive  by definition.
  1012. What initiates the wind?  Well, primitive answers were not so wrong...
  1013. Why "m" for the slope of a linear function  y = m x + b ? [English textbooks]
  1014. The diamond mark on US tape measures corresponds to 8/5 of a foot.
  1015. Naming the largest possible number, in n keystrokes or less (Excel syntax).
  1016. The "odds in favor" of poker hands: A popular way to express probabilities.
  1017. Reverse number sequence(s) on the verso of a book's title page.
  1018. Living species: About 1400 000 have been named, but there are many more.
  1019. Dimes and pennies: The masses of all current US coins.
  1020. Pound of pennies: The dollar equivalent of a pound of pennies is increasing!
  1021. Nickels per gallon: Packing as much as 5252.5523 coins per gallon of space.

    Geography, Geographical Trivia

  1022. The volume of the Grand Canyon  would be 2 cm (3/4") over the entire Earth.
  1023. The Oldest City in the World: Damascus or Jericho?
  1024. USA (States & Territories): Postal and area codes, capitals, statehoods, etc.

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

  1025. Keyboard and modifier keys.  Lesser-used functions require several keystrokes.
  1026. Physical units:  A very nice afterthought, with some unfortunate rough edges.
  1027. Real analytical functions  may present discontinuity  cliffs  in the complex realm.
  1028. 68000 Assembly Programming:  A primer without the help of an assembler.
  1029. The clock frequency of your calculator:  How to measure it with 0.1% accuracy.
  1030. BASIC Programming.  TI's built-in interpreted language is convenient but slow.

    Money, Currency, Precious Metals

  1031. Inventing Money: Brass in China, electrum in Lydia, gold and silver staters...
  1032. Prices of Precious Metals:  Current market values (Gold, Silver. Pt, Pd, Rh).
  1033. Medieval sysyem:  12  deniers  to a  sol.
  1034. Ancien Régime  French monetary system.
  1035. British coinage  before decimalization.
  1036. Exchange rates  when the  euro  was born.
  1037. Worldwide circulation  of currencies.

    The Counterfeit Penny Problem

  1038. Counterfeit Coin Problem: In 3 weighings, find an odd object among 12, 13, 14.
  1039. General Counterfeit Penny Problem: Find an odd object in the fewest weighings.
  1040. Explicit tables  for detecting  one  odd marble among  41,  in  4  weighings.
  1041. Find-a-birthday:  Detect an odd marble among 365, in 6 weighings.
  1042. Error-correcting codes for ternary numeration.
  1043. If the counterfeit is known to be heavier, fewer weighings may be sufficient.

    Calendars & Chronology

  1044. Fossil calendars: 420 million years ago, a lunar month was only 9 short days.
  1045. Julian Day Number (JDN) Counting days in the simplest of all calendars.
  1046. The Week has not always been a period of seven days.
  1047. Egyptian year of 365 days: Back to the same season after over 1500 years.
  1048. Heliacal rising of Sirius: Sothic dating.
  1049. Coptic Calendar: Reformed Egyptian calendar based on the Julian year.
  1050. The Julian Calendar: Year starts March 25. Every fourth year is a leap year.
  1051. Anno Domini: Counting roughly from the birth of Jesus Christ.
  1052. The Gregorian Calendar: Multiples of 100 not divisible by 400 aren't leap years.
  1053. Counting the days between dates, with a simple formula for month numbers.
  1054. Age of the Moon, based on a mean synodic month of  29.530588853 days.
  1055. Easter Sunday is defined as the first Sunday after the Paschal full moon.
  1056. The Muslim Calendar:  The Islamic (Hijri) Calendar (AH = Anno Hegirae).
  1057. The Jewish Calendar:  An accurate lunisolar calendar, set down by Hillel II.
  1058. Zoroastrian Calendar.
  1059. The Zodiac:  Zodiacal signs and constellations.  Precession of equinoxes.
  1060. The Iranian Calendar.  Solar Hejri [SH]  or  Anno Persarum  [AP].
  1061. The Chinese Calendar.
  1062. The Japanese Calendar.
  1063. Mayan System(s)Haab (365), Tzolkin (260), Round (18980), Long Count.
  1064. Indian Calendar:  The Sun goes through a zodiacal sign in a solar month.
  1065. Post-Gregorian CalendarsPainless  improvements to the secular calendar.
  1066. Geologic Time Scale:  Beyond all calendars.

    Roman Numerals  (Archaic, Classic and Medieval)

  1067. Roman Numeration: The basics and the precise rules (including medieval ones).
  1068. Larger Numbers, like 18034...
  1069. Extending the Roman system.
  1070. IIS (or HS) is for sesterce (originally, 2½ asses, "unus et unus et semis").


  1071. Standard jokes.
  1072. Limericks.
  1073. Proper credit may not always be possible.
  1074. Trick questions can be legitimate ones.
  1075. Ignorance is bliss:  Why not read all that mathematical stuff  faster ?
  1076. Silly answers to funny questions.
  1077. Why did the chicken cross the road?  Scientific and other explanations.
  1078. Humorous or inspirational quotations by famous scientists and others.
  1079. Famous Last Words:  Proofs that the guesses of experts are just guesses.
  1080. Famous anecdotes.
  1081. Parodies, hoaxes, and practical jokes.
  1082. Omnia vulnerant, ultima necat:  The day of reckoning.
  1083. Funny Units: A millihelen is the amount of beauty that launches one ship.
  1084. Funny Prefixes: A lottagram is many grams; an electron weighs 0.91 lottogram.
  1085. The Lamppost Theory:  Look only where there's enough light to find anything.
  1086. Anagrams: Rearranging letters may reveal hidden meanings ;-)
  1087. Mnemonics: Remembering things and/or making fun of them.
  1088. Acronyms: Funny ones and/or alternate interpretations of serious ones.
  1089. Usenet Acronyms: If you can't beat them, join them (and HF, LOL).

    Scientific Symbols and Icons

  1090. The equality symbol ( = ).  The "equal sign" dates back to the 16th century.
  1091. "Lines" among symbols: Vinculum, bar, solidus, virgule, slash, macron, etc.
  1092. The infinity symbol ( ¥ ) introduced in 1655 by John Wallis (1616-1703).
  1093. Transfinite numbers:  Mathematical symbols for the multiple faces of infinity.
  1094. Chrevron symbols:  Intersection (highest below)  or  union (lowest above).
  1095. Disjoint union.  Square "U" or  inverted  p  symbol.
  1096. Blackboard boldDoublestruck  symbols are often used for sets of numbers.
  1097. The integration sign ( ò ) introduced by Leibniz at the dawn of Calculus.
  1098. The end-of-proof box (or tombstone) is called a halmos symbol  (QED).
  1099. Two "del" symbols  for partial derivatives, and  Ñ  for Hamilton's nabla.
  1100. The Staff of Aesculapius:  Medicine and the 13th zodiacal constellation.
  1101. The Caduceus:  Scepter of Hermes, symbol of  commerce  (not medicine).
  1102. The Tetractys: Mystical Pythagorean symbol, "source of everflowing Nature".
  1103. The Borromean Rings: Three interwoven rings which are pairwise separate.
  1104. The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.

    Unabridged Answers (monographs and complements):

  1105. Sagan's number:  The number of stars, compared to earthly grains of sand.
  1106. The Sand Reckoner:  Archimedes fills the cosmos with grains of sand.
  1107. About Zero.
  1108. Wilson's Theorem.
  1109. Counting Polyhedra:  A tally of polyhedra with n faces and k edges.

    Hall of Fame:

  1110. Numericana's list of distinguished Web authors in Science...  Links to their sites.
  1111. Giants of Science:  Towering characters in the history of Science.
  1112. Two legendary Solvay conferences  defined modern physics, in 1911 and 1927.
  1113. Physical Units: A tribute to the late physicist Richard P. Feynman (Nobel 1965).
  1114. The many faces of Nicolas Bourbaki  (b. January 14, 1935).
  1115. Lucien Refleu  (1920-2005).  "Papa" of 600 mathematicians.  [ In French ]
  1116. Roger Apéry  (1916-1994)  and the irrationality of  z(3).
  1117. Hergé (1907-1983):  Tintin and the Science of Jules Verne (1828-1905).
  1118. Escutcheons of Science (Armorial):  Coats of arms of illustrious scientists.

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

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