Final Answers - Science - NUMERICANA Final Answersby Gérard P. Michon, Ph.D.(unless otherwise stated)g.michon@att.net© 2000 - 2008 by Gérard Michon. All Rights Reserved. All texts and illustrations are copyrighted;short excerpts may only be quoted according toapplicable copyright laws.this sitethe Webfor    home | NumericanaTM | latest backlinks MostPopular Pages :(detailed index below)Symbols and IconsUnits & MeasuresEllipse PerimeterRoman NumeralsMathematical RecreationsMagic TricksGeometry & TopologyFunctions (trig. etc.)TriviaHistory & CulturePractical FormulasIntegersPolyhedraChemistryIndex (this page)Home (numericana.com)Escutcheons of ScienceCalendarsFilter DesignCombinatoricsNumerical ConstantsArea of an EllipsoidPhysicsCounterfeit PennyHumorThe UnexplainedGases and FluidsRichard FeynmanAnalysisPositional NumerationAlgebraCalculusTwo-Player GamesSets & LogicHandheld CalculatorsContinued FractionsPolyhedra TableWilson's TheoremLanczos FormulaPopular FallaciesMiscellanyCosmologyUtility & DecisionsStochastic ProcessesMoneyModular ArithmeticPlanar CurvesSolar SystemRelativityDemons of PhysicsPseudoprimesPrime FactorizationsQuantum TheorySteam EnginesElectromagnetismVector CalculusScales & RatingsScale InvarianceGeographical Triviap-adic NumbersCounting the StarsGearsStyle & UsageHypergeometric SeriesRing TheoryGroup TheoryFields & Skew FieldsVector SpacesQuadratic ReciprocityFantasy EngineeringThermodynamicsAncient SciencePrime NumbersStarsMatrices & DeterminantsSchrödinger's EquationMathematical ProofsGraph TheoryMedicineOpen QuestionsMathematical MiraclesRamsey TheoryDifferential EquationsAngles, Solid AnglesGeneral RelativityRounding NumbersString TheoryMotion of Rigid BodiesOptimizationNewtonian GravityMagnetismMotors & GeneratorsDipolesAnalytical Mechanics  Table of ContentsSee also: Dates of creation of all indexed pagesBegin 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 isbetter to know some of the questions than all of the answers.James Grover Thurber (1894-1961)Measurements and UnitsAll metric prefixes:Current SI prefixes, obsolete prefixes, bogus prefixes...Prefixes for units of information.(Multiples of the bit only.)Density one.Relative and absolute density precisely defined.Acidsyielding a mole of H+ per liter arenormal (1N) solutions.Calories:Thermochemical calorie, gram-calorie (g-cal), IST calorie (and Btu).Horsepowers:hp, electric horsepower, metric horsepower, boiler horsepower.The standard acceleration of gravity (1G)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-cmor 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 = 31557600SI 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.Leagues: Land league, nautical league.Radius of the Earthand circumference at the Equator.Distance between two pointsat the surface of the Earth.The figure of the Earth.Geodetic and geocentric latitudes.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 busheland 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, etc.Other tons: Energy(kiloton, toe, tce), cooling power, thrust, speed...The Art of Rounding NumbersScientific notation: Nonzero numbers given as multiples of powers of ten.So many "significant" digits imply a result of limited precision.Standard deviation gives the precision of a result as a form of uncertainty.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.Scales and Ratings:Measuring without UnitsThe Beaufort scale is now defined in terms of wind speed.The Saffir / Simpson scale for hurricanes.The Fujita scale for tornadoes.The Richter scale for earthquakes andother sudden energy releases.Decibels: A general-purpose logarithmicscale for relative power ratios.Apparent and absolute magnitudes of stars.Scaling and Scale InvarianceThe 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.Numerical Constants:Mathematical & Physical ConstantsPhysical 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 fundamentaland most misunderstood of all numbers.1 and -1: The unit numbers.p ("Pi"):The ratio of the circumference of a circle to its diameter.࡞: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: The base of the exponential function which equals its own derivative.ln(2): The alternating sum of the reciprocalsof 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.Mertens constant: The limit of [1/2 + 1/3 + 1/5 +...+ 1/p] - ln(ln p)Artins's constant is the proportion oflong 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 ±3sPrévost's Constant: The sum of the reciprocals of the Fibonacci numbers.Grossman's Constant: Onerecurrence converges for only one initial point.Ramanujan's Number: exp(p Ö163) is almost an integer.Viswanath's constant: Mean growthin random additions and subtractions.Counting, Combinatorics, ProbabilityAlways change your first guessif you're always told another choice is bad.The Three Prisoner Problempredated Monty Hall and Marilyn by decades.Seating N children at a round tablein (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) choicesof p items among n different types, allowing duplicates.How many new intersectionsof 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.The Markov inequality is used to prove the Bienaymé-Chebyshev inequality.The Bienaymé-Chebyshev inequality is 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 flushamong 26 cards (or any other number).The exact probabilitiesin 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 sequencesof 5 digits (2002 nonincreasing ones).How many ways are thereto make change for a dollar? Closed formulas.The number of rectanglesin an N by N chessboard-type grid.The number of squaresin 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 distancebetween two random points on a segment, a disk, a cube...Average distancebetween two points on the surface of a sphere.Stochastic Processes& Stochastic ModelsPoisson Processes:Random arrivals happening at a constant rate (in Bq).Simulating a poisson processis easy with a uniform random number generator.Markov Processes:When only the present influences the future...The Erlang B Formulaassumes callers don't try again after a busy signal.Markov-Modulated Poisson Processesmay look like Poisson processes."Utility" and Decision AnalysisThe Utility Function:n a dollar lost.Saint Petersburg Paradox:What would you pay to play the Petersburg game?Mathematical ProofsYou should only prove anegative (i.e., a lack of counterexamples).Stochastic proofs establish specific statements with arbitrary certainty.Heuristic arguments merely establish the likelihood of general conjectures.Geometry and Topology (for Polyhedra page, see below)Center of an arcdetermined 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.Circumference of an ellipse:Unabridged discussion.Surface area of an ellipsoidof revolution (oblate or prolate spheroid).Surface area of a general ellipsoid.Surface of an ellipse.Quadratic equations in the planedescribe 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 hypersphereand hper-surface area, in any number of dimensions.Hexahedra. The cube is not the onlypolyhedron with 6 faces.Unabridged discussion.Descartes-Euler Formula:F-E+V=2 but restrictions apply.Angles and Solid Angles:Planar angles separate two directions. In an oriented plane, they are signed.Solid angles are to spherical patches what planar angles are to circular arcs.Circular measures: Angles and solid angles aren't quite dimensionless quantities.Formulas for solid angles subtended by patches with simple shapes.Planar Curves: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.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: The shape of the infinity symbol is a quartic curve.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.Bézier curves arealgebraic 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 include spikes.The quadratrix (or trisectrix)of Hippias can square the circle and trisect angles.The parabola is a curve that's constructible with straightedge and compass.Mohr-Mascheroni constructions use the compass alone (no straightedge).Gears: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.Polyhedra (3D), Polychora (4D), Polytopes (nD)Hexahedra. The cube is not the only polyhedron with 6 faces.Fat tetragonal antiwedge: Chiral hexahedron of least area for a given volume.Enumeration of polyhedra:Tally of polyhedra with n faces and k edges.Counting Polyhedra (1):Table, comments, references...Counting Polyhedra (2): Larger table!The 5 Platonic solids:Cartesian coordinates of the vertices.The 13 Archimedean solids and their duals (Catalan solids).Some special polyhedramay have a traditional (mnemonic) name.Polyhedra in certain familiesare named after one of their prominent polygons.Deltahedrahave equilateral triangular faces. Only 8 deltahedra are convex.Johnson Polyhedra and the associated nomenclature.Polytopesare the n-dimensional counterparts of 3-D polyhedra.A simplex of touching unit spheresmay 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.Graph TheoryAdjacency matrixof a directed graph or a bipartite graph.Silent Circles: An enumeration based on adjacency matrices (Max Alekseyev).Silent Prisms: Another version of the screaming game, for short-sighted people.Tallying allmarkings of one edge per node in which no edge is marked twice.Line graph: The nodes of L(G) are edges of G (connected iff adjacent in G).Transitivity: Vertex-transitive and/or edge-transitive graphs.AlgebraFactorial zero is 1, so is an empty product;an empty sum is 0.Anything raised to the power of 0is 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 overa Riemann surface.Square roots are inherently ambiguous fornegative 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.Matrices and DeterminantsPermutation matrices include the identity matrix and the exchange matrix.Operations on matrices are conveniently defined using Dirac's notation.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 thereciprocal of Catalan numbers.Hadamard matrix: Unit elements and orthogonal columns.Sylvester matrix of two polynomialshas their resultant for determinant.The discriminant of a polynomialis the resultant of itself and its derivative.Trigonometry,Elementary Functions, Special FunctionsNumerical functions:Polynomial, rational, algebraic, transcendental, special...Trigonometric functions: Memorize a simple picture for 3 basic definitions.Solving triangles with thelaw 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(and their square roots) as trigonometric functions of zero!The sine function: How to compute it numerically.Chebyshev economizationsaves billions of operations on routine computations.The Gamma function:Its definition(s) properties and values.Lambert's W functionis used to solve practical transcendental equations.Hypergeometric FunctionsPochhammer's symbol: Upper factorial of k increasing factors, starting with x.Gauss's hypergeometric function: 2+1 parameters (and one variable).Kummer's transformations relatedifferent values of the hypergeometric function.Sum of the reciprocal of Catalan numbers,in closed hypergeometric form.CalculusDerivative:Usually, the slope of a function, but there's a more abstract approach.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 spanning a mile,if its length is just a foot longer.Sagging: A cable which spans 28 m and sags 30 cm is 28.00857 m long.The length of the arch of a cycloidis 4 times the diameter of the wheel.Integrating the cube root of the tangent function.Changing inclinationto 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 higherorder and/or in several variables.Theory of Distributions: Convolution products and their usage.Laplace Transforms:The Operational Calculus of Oliver Heaviside.Integrabilityof a function and of its absolute value.Analytic functions of a linear operator;defining f (D) when D is d/dx...Differential EquationsOrdinary differential equations. Several examples.A singular change of variable isvalid over a domain which may not be maximal.Vertical fall against fluid resistance (including both viscous and quadratic drag).Differential Forms & Vector CalculusGeneralizing the fundamental theorem of calculus.The surface of a loop is a vector determining its apparent area in any direction.Practical identities of vector calculusOptimization: Operations Research, Calculus of VariationsStationary points (or saddlepoints ) are where all partial derivatives vanish.Single-variable optimization: Derivative vanishes unless the variable is extreme.Extrema of a function of two variables must satisfy a second-order condition.Saddlepoints of a multivariate function. One equation to satisfy per variable.Lagrange multipliers: Optimizing an objective function under various constraints.Minimizing the lateral surface area of a cone of given base and volume.Euler-Lagrange equations hold along the path of a stationary integral.Isoperimetric Inequality: The largest area enclosed by a loop of unit perimeter.Plateau's problem extends the calculus of variations from paths to membranes.Connecting blue dots to red dots in the plane, without any crossings...The shortest way to connect 3 dots is to join them to a fourth point.The Honeycomb Theorem: A conjecture of old, proved by Thomas Hales.Counterexamples to Kelvin's conjecture. Tiling space with unit cells of least area.Analysis, Convergence, Series, Complex AnalysisPermuting the terms of a seriesmay change its sum arbitrarily.Uniform convergenceimplies properties for the limit of a sequence of functions.Cauchy sequenceshelp define real numbers rigorously.Defining integrals:Cauchy, Riemann, Darboux, Lebesgue.Cauchy principal value of an integral.Fourier series.A simple example.Infinite sums maysometimes be evaluated with Fourier Series.A double sum is often the product of two sums,which may be Fourier series.At a jump, the sum of 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 Froebeniusabout a regular singularity of a differential equation.Laurent seriesof a function about one of its poles.Cauchy's Residue Theoremis helpful to compute difficult definite integrals.Tame complex functions:Holomorphic and meromorphic functions.Set Theory and LogicThe Barber's Dilemma.Not a paradox if analyzed properly.What is infinity? More than a pretty symbol(¥).There are more real than rational numbers.Cantor's argument.The axioms of set theory:Fundamental axioms and the Axiom of Choice.A set is smaller than its powerset: A simple proof applies to all sets.Transfinite cardinals, transfinite ordinals:Two different kinds of infinite numbers.Surreal Numbers: These include reals, transfinite ordinals, infinitesimals & more.Numbers: From integers to surreals. From reals to quaternions and beyond...Integer Arithmetic, Number TheoryThe number 1 is not prime,as definitions are chosen to make theorems simple.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 multiplemay be obtained without factoring into primes.Standard Factorizations: n4 + 4 is never prime for n > 1 because...Euclid's algorithm gives the greatest common divisorand 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 numbers and Mersenne primes.Fast exponentiation by repeated squaring.Partition function.How many collections of positive integers add up to 15?A Lucas sequencewhose oscillations never carry it back to -1.A bit sequence with intriguingstatistics. Counting squares between cubes.Binet's formulas:N-th term of a sequence obeying a second-order 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 (about Fibonacci squares).Faulhaber's formulagives 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).Dirichlet convolutionis especially interesting for multiplicative functions.Totally multiplicative functions arethe simplest type of multiplicative functions.Euler products and generalized zeta functions.Positional Numeration & Number SystemsModular Arithmetic may be used to find the last digit(s)of very large numbers.Powers of tenexpressed 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 composedof only 7's and 0's.Binary and/or hexadecimal numerationfor floating-point numbers as well.Extract a square root the old-fashioned way.Prime NumbersA prime number is a positive integer with just two 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: There are arbitrarily long arithmetic progressions of primes.The Prime Number Theorem: The probability that N is prime is roughly 1/ln(N).The largest known prime: Historical records, old and new.The Lucas-Lehmer Test checks the primality of a Mersenne number very fast.Formulas giving only primes may not help with new primes.Modular ArithmeticChinese Remainder Theorem: How remainders define an integer (within limits).Modular arithmetic:The algebra of congruences, formally introduced by Gauss.Fermat's little theorem: For any prime p, ap-1 is 1 modulo p,unless p divides a.Euler's totient function: f(n) is the number of integers coprime to n, from 1 to n.Fermat-Euler theorem: If a is coprime to n, then a to the f(n) is 1 modulo n.Carmichael's reduced totient function(l) : A very special divisor of the totient.91 is a pseudoprimeto 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 numbersmay be obtained in various ways.Conjecture:Any odd number coprime to its totient has a Carmichael multiple.Group Theory and SymmetriesMonoids are endowed withan associative operation and a neutral element. The inverse of an element comes in two flavors which 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 groupare 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 ona set E consists of all the bijections of E onto itself.Inner automorphisms: Inn(G) is isomorphic to the quotient of G by 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 groupis generated by its commutators.Direct product of two groups (also called a direct sum for additive groups).Groups of small orders and their 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 (1982).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.Ring TheoryRings are sets endowed with addition, subtraction and multiplication.Nonzero characteristic: The 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.Fields, Galois Fields and Skew FieldsVocabulary: We consider skew fields to be noncommutative. Some don't.Fields are commutative rings whereevery nonzero element has a reciprocal. Wedderburn's Theorem: Finite division rings are commutative (they're fields).Every finite integral domain is afield. A corollary of Wedderburn's theorem.Galois fields are the finite fields. Their orders are powers of prime numbers.The trivial field has a singleelement. It's the only field where 0 has a reciprocal.The splitting field of PÎF[x] is the smallest extension of F where P fully factors.The Nim-Field is algebraically complete. It contains [surreal] infinite ordinals.Ternary multiplication compatible with ternary addition (without "carry").Vector Spaces (over a field) and Modules (over a ring)Vectors were originally justdifferences between points in ordinary space...Abstract vector spaces: Vectors can be added, subtracted and scaled.Banach spaces are complete normed vector spaces.Modules are vectorial structures overa ring of scalars (instead of a field).An algebra is a vector spacewith a scalable and distributive internal product.Clifford algebras are unitalassociative algebras endowed with a quadratic form.Ring of p-adic Integers,Field of p-adic NumbersThe ring of p-adic integers includes 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 is the quotient field of the ring of p-adic integers.Dividing two p-adic numbers looks like "long division", only backwards...The p-adic metriccan be used to define p-adic numbers analytically.The reciprocal of a p-adic number computed by successive approximations.Hasse's local-global principle was established for thequadratic case in 1920.Integers which double when their digits (in base g)are rotated.PseudoprimesPseudoprimes to base a. Poulet numbers are pseudoprimes to base 2.Weak pseudoprimes to base a : Composite integers n which divide (an-a).Strong pseudoprimes to base a are less common than Euler pseudoprimes.Counting the bases to whicha composite number is a pseudoprime.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 are (probably) infinitely many of them.Super-pseudoprimes: All their composite divisors are pseudoprimes.Maximal super-pseudoprimes have no super-pseudoprime multiples.Factoring into PrimesChallenges help tell special-purpose and general-purpose methods apart.Trial division may be usedto weed out the small prime factors of a number.Ruling out factors can speed up trialdivison in special cases.Recursively defined sequences (over a finite set) are ultimately periodic.Pollard's r (rho) factoringmethod is based on the properties of such sequences.Pollard's p-1 Method findsprime factors p for which p-1 is smooth.Williams' p+1 Method is based on theproperties of Lucas sequences.Lenstra's Elliptic Curve Method is a generalizationof Pollard's p-1 approach.Dixon's method: Combine small square residues intoa solution of x 2º y 2Quadratic ReciprocityMotivation: On the prime factors of some quadratic forms...Euler's criterion: Modulo an odd prime p, a square to the power of (p-1)/2 is 1.The Legendre symbol (a|p) can be extended to values of p besides odd primes.The law of quadratic reciprocity states a simple but surprising fact.Gauss' Lemma is the basis of one proofof the law of quadratic reciprocity.Continued Fractions (and related topics)What is a continued fraction? Example: The expansion of p.The convergents of a numberare its best rational approximations.Large partial quotientsallow very precise approximations.Regular patternsin 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 expansion of a positive number. A nondecreasing sequence of integers. Pierce expansions of numbers between 0 and 1. Strictly increasing sequences.Recreational MathematicsCounterfeit Coin Problem:In 3 weighings, find an odd object among 12, 13, 14.Unabridged discussion.General Counterfeit Penny Problem:Find an odd object in the fewest weighings.Unabridged discussion.Seven-Eleven: Four priceswith 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 shown unreachable because of an invariant quantity.Sam Loyd's 14-15 puzzlealso involves an invariant quantity (and two orbits).Einstein's riddle:5 distinct house colors, nationalities, drinks, smokes and pets.Numbering n pagesof 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).Crossing a bridge:1 or 2 at a time, 4 people (U2), different paces, one flashlight!Managing suppliesto reach an outpost 6 days away, carrying enough for 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 bornin 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 tomeet 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:Guess the words which commonly describe many numbers.Riddles:The Riddle of the Sphinx and other classics, old and new.Mathematical "Magic" Tricks1089: Subtract a 3-digit number and its reverse,then add this to its reverse...Grey Elephants in Denmark:"Mental magic" for one-time classroom use.The 5-card trick of Fitch Cheney: Tell thefifth card once 4 are known.Generalizing the 5-card trickand Devil's Poker...Kruskal's Count.Paths to God.Mathematical Games (Strategies)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.Grundy numbersare defined for all positions in impartial games.Moore's Nim:Remove something from at most (b-1) rows. Play last.Normal Kayles:Knocking down one pin, or two adjacent ones, may split a row.Grundy's Game:Split a row into two unequal rows. Whoever can't move loses.Wythoff's Game:Remove counters either from one heap or equally from both.Ramsey TheoryThe pigeonhole principle: What must happen with fewer holes than pigeons...Infinite alignment among infinitely many lattice pointsin the plane? Nope.Infinite alignment in a lattice sequence with boundedgaps? Almost...Large alignments in a lattice sequence with boundedgaps. Yeah!MiscellaneousFord circles are nonintersecting circlestouching the real line at rational points.Farey series: The rationals from 0 to 1, with a bounded denominator.The Stern-Brocot tree contains a single occurrence of every positive rational.Any positive rational is a unique ratio of two consecutive Stern numbers. Pick's formula gives the area of a lattice polygonby counting lattice points.History, Nomenclature, Vocabulary, etc. History :Earliest mathematics on record.Before Thales was Euphorbe...Indian numerationbecame a positional system with the introduction of zero.Roman numerals are awkward for larger numbers. Roman numerals: Unabridged discussion.The invention of logarithms:John Napier, Bürgi, Briggs, Saint-Vincent, Euler.The earliest mechanical calculator(s),by W. Shickard (1623) or Pascal (1642).The Fahrenheit Scale:100°F was meant to be the normal body temperature.Nomenclature & Etymology :The origin of the word "algebra",and also that of "algorithm".The name of the avoirdupois system: Borrowed from French in a pristine form.Long Division:Cultural differences in writing the details of a division process.Is a parallelogram a trapezoid?In a mathematical context [only?], yes it is...Naming polygons.Greek only please; use hendecagon not "undecagon".Chemical nomenclature:Basic sequential names (systematic and/or traditional).Fractional Prefixes:hemi (1/2), sesqui (3/2)or weirder hemipenta, hemisesqui... Matches, phosphorus, and phosphorus sesquisulphide.Zillion. Naming large numbers.Zillionplex. Naming huge numbers.Style and UsageAbbreviations: Abbreviations of scholarly Latin expressions.Typography of long numbers.Intervals denoted with square brackets (outward for an excluded extremity).Dates in the simplest ISO 8601form (with customary time stamps or not).The names of operandsin common numerical operations.The word respectively doesn't have the same syntax as "resp."Setting the Record StraightThe heliocentric Copernican systemwas known two millenia before Copernicus.The assistants of Galileo Galileiand the mythical experiment at the Tower of Pisa.Switching calendars:Newton was not born the year Galileo died.The Lorenz Gauge is an idea ofLudwig Lorenz (1829-1891) not H.A. Lorentz.Special Relativity was first formulatedby H. Poincaré (Einstein a close second).The Fletcher-Millikan "oil-drop" experimentwas not the sole work of Millikan.Collected errata about customary physical units.Portrait of Legendre: Mathematician (Adrien-Marie) or politician (Louis) ?Dubious quotations: Who really said that?Ancient KnowledgeObliquity of the ecliptic: An evolving quantity first measured by Eratosthenes.Vertical wells at Syene arecompletely sunlit only once a year, aren't they?Eratosthenes sizes up the Earth: 700 stadia per degree of latitude.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.PhysicsThe "work done" on a point-mass equals the change in its kinetic energy.Relativistic work done and the corresponding change in relativistic energy.Relativistic thermodynamics: The case of a point endowed with internal heat.Spacecraft speeds up upon reentryinto the upper atmosphere.Lewis Carroll's monkeyclimbs a rope over a pulley, with a counterweight.Two-ball dropcan make a light ball bounce up to 9 times the dropping height.Normal acceleration is the square of speed divided by the radius of curvature.Roller-coasters must risemore 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 electronestimated with the Bohr model of the atom.Thermal expansion coefficients: The cubical coefficient is 3 times the linear one.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.Motion of Rigid Bodies (Classical Mechanics)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 areconvenient mathematical fictions.Moment of inertia of a spherical distribution and of an homogeneous ellipsoid.Perpendicular Axis Theorem: Axis of rotation perpendicular to a thin plate.The Parallel Axis Theorem gives the momentof inertia about an off-center axis.Moment of inertia of a thick plate, as obtained from the parallel axis theorem.Momenta of homogeneous bodies: List of examples.Rigid pendulum moving under its own weight about a fixed horizontal axis.Newtonian GravityAll physical theories have a limited range of validity.Gravity vs. Electrostatics: Straight comparisons.Rigid equilateral triangle formed by three gravitating bodies.The five Lagrange points of two gravitating bodies in circular orbit.Analytical Mechanics & Classical Field TheoryFermat's principle (least time) for light (c.1655) predates Newton's mechanics.Maupertuis principle of least action (1744).Virtual Work: A substitutefor Newton's laws which factors out constraint forces.Phase Space: A phase describes all there is to a classical system.Either velocities or momenta can be used(with positions) to specify a phase.Relativistic point-mass: Lagrangian, Hamiltonian and momentum in free space.Charge in a magnetic field: The canonical momentum isn't the linear momentum.Lagrangian mechanics: The Lagrangian is a function of positions and velocities.Hamiltonian mechanics: The Hamiltonian depends on positions and momenta.Poisson brackets: Anabstract synthetic view of analytical mechanics.Liouville's theorem: The volume in Hamiltonian phase space doesn't change.Noether's Theorem: Conservation laws express the symmetries of physics.Field Theory: Lagrangian mechanics on a continuum of values (and derivatives).Electromagnetism (Maxwell's Equations)Clarifications by Heaviside & Lorentz: Vector calculus & microscopic view.The vexing problem of units is a thing of the pastif you stick to SI units.The Lorentz force on a test particledefines the local electromagnetic fields.Electrostatics: 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 generate magnetic fields !Biot-Savart Law: The static magnetic induction due to steady currents.Magnetic scalar potential: A multivalued function whose gradient is induction.Magnetic monopoles do not exist : A law stating a fact not yet disproved.Ampère's law: The law of static electromagnetism devised by Ampère in 1825.Faraday's law: A varying magnetic flux induces an electric circulation.Self-induction: The induction received by a circuit from its own magnetic field.Ampère-Maxwell law: The 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.Wavesanticipated by Faraday, Maxwell & FitzGerald were observed by Hertz.Electromagnetic energy density andthe flux of the Poynting vector.Planar electromagnetic waves: The simplest type of electromagnetic waves.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.Electric and magnetic dipoles: Dipolar solutions of Maxwell's equations.Static distributions of magnetic dipoles can be simulated with steady currents. Static distributions of electric dipoles can be simulated with static charges.Sign reversal in the fieldsof uniformly distributed magnetic or electric dipoles.Fields at the center of uniformly magnetized or polarizedspheres (of any size).Relativistic dipoles: A moving magnetdevelops 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.Electromagnetic DipolesMolecular electric dipole moments were first studied by Peter Debye (1912).Force exerted on a dipole by a nonuniform field.Torque on a dipole is proportionalto its cross-product into the field.Magnetism, Electromagnetic Properties of MatterMagnetization and polarization describe densities of dipoles bound to matter.Gauge invariance: Many magnetizations and polarizations create the same field.Maxwell's equations in matter: Electric displacement D, magnetic strength H.Electric susceptibility is the propensity to be polarized by an electric field.Electric permittivity and magnetic permeability are related to susceptibilities.Paramagnetism: Weak positive susceptibility.Diamagnetism: The Lorentz forceturns orbital moments against the external B.Magnetic levitation: How to skirt the theorem of Samuel Earnshaw (1842).Pyrolytic carbon: The most diamagnetic sunstance known, at room temperature.Bohr & Van Leeuwen Theorem: Diamagnetism and paramagnetism cancel ?!Thermodynamics of dielectric matter: dU = E.dD + ...Ferromagnetism: Permanent magnetization without an external magnetic 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 proportionalto electric field: j = s E.Motors and GeneratorsHomopolar motor: The first electricmotor, by Michael Faraday (1831).Faraday's disk can generate huge currentsat 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.RelativityObservers 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: Defined unless both have the same direction.Minkowski spacetime: Coordinates of 4-vectors obey the Lorentz transform.The Lorentz transform expressed vectorially: A so-called boost of speed V.Wave vector: The 4-dimensional gradient of the phase describes propagation.Doppler shift: The relativistic effect is not purely radial.Kinetic energy: At low speed, the relativisticenergy varies like ½ mv 2.Photons and other massless particles: Finite energy at speed c.The de Broglie celerity (u) isinversely 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 quantically 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 the speed of an electron exceeds the celerity of light...Constant acceleration over an entire lifetime will take you pretty far...General RelativityThe Harress-Sagnac effect seen by an observer rotating with the optical loop.Relativistic rigid motion is an equilibrium modified at the speed of sound.Covariant Derivatives.Einstein's Field Equations.What is mass?Electromagnetism: Covariant expressions, using tensors.Harvard Tower Experiment: The slow clock at the bottom of the tower.String Theory and other"Theories of Everything"Unification: Consistency is required. Actual high-energy unification is not.Kaluza-Klein Theory: Postulating one 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.Physics of Gases and FluidsThe Magdeburg hemispheres are held togetherby more than one ton of force.The ideal gas laws ofBoyle, Mariotte, Charles, Gay-Lussac, and Avogadro.Joule's law: The internal energy of a perfectgas depends only on its temperature.The Van der Waals equation and other interestingequations of state.Virial equation of state. Virial expansion coefficients. Boyle's temperature.Viscosityis 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.Filters and FeedbackComplex pulsatance (s) isdamping constant (s) plus imaginary pulsatance(iw).Complex impedance: Resistance and reactance.Quality Factor (Q). Ratio of maximal stored energy to dissipated power.Nullators and norators: Strange dipolesfor 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 filtersdo 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 with a 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 ontime-domain communication pulses.Linear Phase Equiripple: Ripplesin group delay to improve on Bessel filters.DSL filter allows POTSbelow 3400 Hz and blocks digital data above 25 kHz.Fantasy Engineering:Just for fun.Raising the Titanic, with (a lot of) hydrogen.Gravitational Subway: From here to anywhere in 42 minutes.In a vacuum tube, a drop to the center of theEarth would take 21 minutes.Steam EnginesThe aeolipile: Thisancient steam engine demonstrates jet propulsion.Edward Somerset of Worcester (1601-1667): Blueprint for a steam fountain.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)and John Calley: Atmospheric steam 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 and the theoretical efficiency limit.Sir Charles Parsons (1854-1931): The modern steam turbine, born in 1884.ThermodynamicsThe 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 can be convenient alternatives to internal energy.Latent heat (L) is the heat transferred in a change of phase.Calorimetric coefficients, adiabaticcoefficient (g) heat capacities, etc.Cryogenic coefficients: Lower temperature with an isenthalpic expansion.Relativistic considerations: 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.Demons of Classical PhysicsLaplace'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:The thermodynamic cost of forgetting.Quantum MechanicsQuantum 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: Neither measurements nor matrices can be switched at will.Schrödinger's Equation: A nonrelativistic quantum particle in a classical field.Noether's Theorem: Conservation laws express the symmetries of physics.Kets are Hilbert vectors(their duals are bras) on which observables operate.Observables are operatorsexplicitely associated with physical quantities.Commutatorsare the quantities which determine uncertainty relations.Uncertainty relations hold whenever the commutator does not vanish.Spin is a form of angular momentumwithout a classical equivalent.Pauli matrices: Three 2 by 2 matriceswith 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.Ancient Recipesand Modern ChemistryBlack 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 indicates the direction of spontaneity.Labile isnot quite the same as unstable.Inks: India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.Redox Reactions: Oxidizers are reduced by accepting electrons...Gold Chemistry: Aqua regia ("Royal Water") dissolves gold and platinum.Who is the "father" of modern chemistry?Medicine by the NumbersInternational 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).Cosmology 101The Cosmological Principle:The Universe is homogeneous and isotropic.The Big Bang: An idea of Georges Lemaître mocked by Fred Hoyle.Cosmic redshift (z): Light emitted in a Universewhich was (1+z) times smaller.Hubble Law: The relation between redshiftand distance for comoving points.Omega (W):The ratio of the density of the Universe to the critical density.Look-Back Time: The time ellapsed sinceobserved light was emitted.Distance: In a cosmological context,there are several flavors to the concept.Comoving points are reference points followingthe expansion of the universe.The Anthropic Principle:An obvious explanation which may not be the final one.Dark matter & dark energy:Gravity betrays the existence of some dark stuff.The Cosmic Microwave Background (CMB):Its spectrum and density.Stars and Stellar ObjectsNuclear fusion is what powers the stars.Brown dwarves fail to ignite fusion. They glow from gravitational contraction.The Jeans mass above which a gas at temperature T collapses gravitationally.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 Sunand other small stars.Neutron stars: Remnants from thesupernova collapse of medium-sized stars.Stellar black holes: They form whensupermassive stars run out of nuclear fuel.Stellar X-ray source: A small accretor in tight orbit around a donor star.The Solar SystemAstronomical 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 domain where solar wind exerts its influence.Oort's Cloud is a cometary reservoir at the fringe of the Solar System.Practical FormulasEasy conversion betweenFahrenheit and Celsius scales: F+40 = 1.8 (C+40).Automotive :Car speedis proportional to tire diameter and 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 power to speed ratio(measuring speed along thrust direction).Power of an engine as a function of its size: Rating 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 Formulagives the area of a quadrilateral, inscribed or not.Bretschneider's Formula:Area of a quadrilateral of known sides and diagonals.The (vector) area of a quadrilateral is half the cross-product of its diagonals.Parabolic segment: 2/3 the area of a circumscribed parallelogram or triangle.Volumes :Content of a cylindrical tank (horizontal axis),given the height of the liquid in it.Volume of a spherical cap, or content of anelliptical vessel, given liquid height.Content of a cistern(cylindrical with elliptical ends), as a function of fluid height.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 solutionsare rated by volume not by mass.Mixing solutionsto obtain a predetermined intermediate rating.Special averages:harmonic (for speeds), geometric (for rates), etc.Mean Gregorian month:either 30.436875 days, or 30.458729474253406983...Geodesy and Astronomy :Distance to ocean horizon lineis proportional to the square root of your altitude.Distance between two pointson a great circle at the surface of the Earth.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.Perimeter of an EllipseCircumference of an ellipse:4 exact series and a dozen approximate formulas!Ramanujan II: An awesome approximation from a mathematical genius (1914).Cantrell's Formula: A modern attempt with an overall accuracy of 83 ppm.Padé approximants are used in a whole family of approximations...Improving Ramanujan II over the whole range of eccentricities.The Arctangent Functionas a component of several approximate formulas.Rivera's formula gives theperimeter of an ellipse with 104 ppm accuracy.Better accuracy fromCantrell, building on his own previous formulaRediscovering a well-known exact expansion due to Euler (1773).Exact expressions for thecircumference of an ellipse: A summary.The UnexplainedThe 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 to the Sun could explain extinction periodicity.Current Challenges to established dogma.Unexplained artefacts, sightings and other records...Open QuestionsThe Riemann Hypothesis: {Re(z) > 0 & z(z) = 0} Þ {Re(z) = ½}.P = NP ? Can we find in polynomial time whatever we can check that fast?Collatz sequences go from n to n/2 (iff n is even)or 3n+1. Do they all lead to 1?Mathematical MiraclesThe 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 pExceptional simple Lie groups.Monstrous Moonshine in Number Theory.TriviaOldest unsolved mathematical problem: Are there any odd perfect numbers?Magnetic Field of the Earth:The 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 ? [English textbooks]The diamond mark on US tape measurescorresponds 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 as much as 5252.5523 coins per gallon of space.Geography, Geographical TriviaThe volume of the Grand Canyon would be 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.Handheld Calculators: TI-92, TI-92+, TI-89, Voyage 200Keyboard and modifier keys. Lesser-used functions require several keystrokes.Physical units: A very nice afterthought, with some unfortunate rough edges.Real 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: How to measure it with 0.1% accuracy.BASIC Programming. TI's built-in interpreted language is convenient but slow.Money, Currency, Precious MetalsInventing Money: Brass in China, electrumin Lydia, gold and silver staters...Prices of Precious Metals: Current market values (Gold, Silver. Pt, Pd, Rh).Exchange rates on the day the euro was born.Worldwide circulation of major currencies.Calendars & ChronologyFossil calendars:420 million years ago, a lunar 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.The 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 Dayis 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 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.HumorStandard 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 quotationsby 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 weighs 0.91 lottogram.The Lamppost Theory: Look only where there's enough light to find anything.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).Scientific Symbols and IconsThe equality symbol ( = ). The "equal sign" dates back to the 16th century."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 multiple 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 Staff of Aesculapius: Medicine and the 13th zodiacal constellation.The Caduceus: Scepter of Hermes, symbol of commerce (not medicine).The Tetractys: Mystical Pythagorean symbol,"source of everflowing Nature".The Borromean Rings: Three interwoven rings which arepairwise separate.The Tai-Chi Mandala: The taiji(Yin-Yang) symbol was Bohr's coat-of-arms.Unabridged Answers (monographs and complements):Surface Area of a General Ellipsoid:Elementary only for ellipsoids of revolution.Roman numerals:Archaic, classic or medieval (including "large" numbers too).Counterfeit Coin Problem: Find an odd coin among n, in k weighings or less.Sagan's number: The number of stars,compared to earthly grains of sand.The Sand Reckoner: Archimedes fills the cosmoswith grains of sand. About Zero.Wilson's Theorem.Counting Polyhedra: A tally of polyhedra with n faces and k edges.l of Fame:Numericana's listof distinguished Web authors in Science... Links to their sites.Giants of Science: Towering characters in the history of Science.Two legendary Solvay conferences defined modern physics, in 1911 and 1927.Physical Units:A tribute to the late physicist Richard P. Feynman (Nobel 1965).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.SupplementaryArmorial: Contemporary scientists, engineers, explorers...Note: The above numbering may change, don't use it for reference purposes.Guest Authors:Lanczos Implementation of the Gamma Function by Paul Godfrey.Public-Domain Texts:Preliminary report (1819) on the BritishWeights and Measures reform of 1824.  Some Recent Reciprocal Links(Pages quoting Numericana's "Final Answers")See also:2008 |2007 |2006 |2005 |2004 |2003 |2002 |2001Bohr's coat-of-arms in Estonian (2008-11-20).Coat-of-arms of Ferdinandvon Zeppelin by Jochen Wilke (2008-11-11).Academic PokerResources ay Big Edge Poker (before 2008-11-06).Introduction toCombinatorial Analysis by Levent Kandiller (2008-11-03).Units ofMeasure (Writing Resources) by Alexander Burns (2008-10-29).LleonardVinci (in Catalan) by Joan Calsapeu (2008-10-17).Arms of Leonardoda Vinci (in Catalan) by JM (2008-10-08).LinearSystems at NCTM's Illuminations (before 2008-10-08).Fun Site by Trisha (2008-07-13).Fork in the road by Sanders (2008-10-02).Daily Links [ 1 ] on Webliminal by Ernest Ackermann (2008-09-12).RediscoveringMathematics (MA 143) by Shai Simonson (bef. 2008-09-11).Wernher von Braun in Catalan [?] (before 2008-08-16).PackingPeople in the Grand Canyon by Mikabee (2008-08-15).Last Year at Marienbad by AngryCandy (2008-08-09).Szilard's engine (in Japanese) by hiroki_f (2008-08-04).Maxwell Equations &Lorentz Force by Zentara (2008-08-01).Kilotons& megatons. United Nations Cyberschoolbus (bef. 2008-07-30).Polyhedra with large number of edges by Andreas Rüdinger (2008-07-28).Apéry'sConstant by Shane (2008-07-28).Ed Pegg, Jr. (anagram) by Rotund Filipino (2008-07-24).Set Theory at GiftedResources.org (before 2008-07-10).Quick Snippets (Grey Matters) by Scott Cram (2008-06-05).How many elephants are there inDenmark? by Ned Kelly (2008-05-31).Visit to the 7-11 by ChuckJerry (2008-05-29).Mathematical Broodingover an Egg by André Heck (before 2008-05-27).The diamond mark atevery 19.2" by kpepin (2008-05-21).Coriolisdeviation in a Tower of Pisa drop by Markogts (2008-05-21).Meaning ofh-bar symbol by Starman (2008-05-19).Number TheoryEducation Resources by Mike at mahalo.com (2008-05-13).Escutcheonsof Science at Forum-Rycerstwa (before 2008-05-13).Billion on The Timber Yard by Noddy330 (2008-05-04).Périmètred'une ellipse (in French) by Valeri Astanoff (2008-05-04).Escudos deCientíficos by Jose Juan Carrion Rangel (2008-04-27).Iron-gall ink in a fountain pen by Ondina (2008-04-14).Funny units by bp56 (2008-04-13).Wilson'sTheorem by Balan (2008-04-09).Perimeter of an Ellipse by Hellburner at ars technica (2008-04-09).Numericana Hall of Fame by Seth Tyler at ISH (before 2008-04-01).Bohr's coat-of-arms (in Latvian) by Ierindnieks (2008-03-28).NamingPolyhedra (in German) by Hugo Pfoertner (2008-03-26).Demons ofPhysics by Ughaibu (2008-03-23).LibraryLinks Metropolitan Community College, MO (bef. 2008-03-22).StevenWeinberg by Pentcho Valev (2008-03-21).Feynman on Physical Units by Henry L. Neal at CAU (bef. 2008-03-18).Famous Scientists: Archimedes by Marie P., 4th Grade (2008-03-10).Quick witted student learns todeadpan by Gregg Chenoweth (2008-02-25).C183 and C209 by Rogue, updated by Fivemack (2008-02-14).Wieferich-related factoring (C183, C209) [ 1] by Zeta-Flux (2008-02-13).The Natural Number e by Stirling Stan (2008-02-10).Perimeter of an ellipse by John Nimphius (2008-02-03).Borromeanrings by Emlaabs (2008-01-30).Blason de Adrien-Marie Legendre by Infofiltrage (2008-01-28).Mathematics at 2000clicks.com (2008-01-26).MultiplyingNimbers by Joshua Zucker (2008-01-17).Hacks (Funny Units) by Dr. Jeremy Frank of NASA (2008-01-15).Cool math-magicsites by John Dougherty (before 2008-01-11).Math Symbol (Equal sign) by Lori Carrigan (before 2008-01-11).Math Symbol (Equal sign) by Donette Carter (before 2008-01-08).The elusive Kevin S. Brown (in Russian) by Z.K. Silagadze (2008-01-06).Archimedes Spiral by David McCombs. (before 2008-01-03).CarmichaelMultiples by Joe Crump. (2008-01-02).The Tai-Chi or Yin-Yang symbol by Barbara (2007-12-31).Surfing Note in Mathematics Teacher by Derek Smith (bef. 2007-12-31).Abell1835 IR1916 (in Spanish) by Horacio C. (2007-12-30).Area of an Ellipsoid (in French) by Cauchy (2007-12-25).Area of an Ellipsoid (in French) by Zapple (2007-12-23).ContrariaSunt Complementa by Kyle Matthew Oliver (2007-12-17).Tycho Brahe by Lupo LeBoucher (2007-12-10).Surface area of anelliptical dish head by Oil Field Trash (2007-12-09).The experiment Galileo didn't do by Gideon Reich (2007-12-01).Computing the partitionfunction (in Polish) by Samolot (2007-11-25).Huge numbers inRoman nnumerals by Maryt/theteach (2007-11-18).Arc length ofa parabola (University of Regina) by Harley (bef. 2007-11-15).Fun withmath?! (true nature of magic) by Neutronjockey (2007-11-14).Grave of Count Rumford by John A. Robertson (before 2007-11-11).Divisors of zero (Math 651, Ohio State) by G.A. Edgar (bef. 2007-11-08).Prominent Scientists. Amman Baccalaureate School, Jordan (2007-11-07).Epitaph ofRoger Apéry by Bertrand Beyern (before 2007-11-06).How manystars in the Andromeda galaxy by Chris Wagoner (2007-11-04).Arms of Adrien-MarieLegendre by Infofiltrage (2007-10-31).Squuzuu = Partition Function by Abbas (2007-10-28).Circumference of an ellipse by Devrath (2007-10-28).HistoricalArmorials at The Heraldry Society (before 2007-10-27).Useful Stuff by J. Paul Gustafson at Messiah College (bef. 2007-10-25).Surfacearea of an ellipsoid by Raxit (2007-10-24).PlanarCurves by Bryan Bishop (before 2007-10-24).Introduction to Group Theory by Anton Gorodetski (before 2007-10-24).Sources / Fonti for the libconic mathematical project (before 2007-10-21).Credits / Meritii : Perimeter of an ellipse for libconic (before 2007-10-21).DCMI Date Working Group Dublin Core Metadata Initiative (bef. 2007-10-21).0^0 = 1 in "The Student Room" by FusionSKD (2007-10-20).Références mathématiques [ 1 ] by MC (before 2007-10-20).The Superfund Three-card Monte by TMFMarathonMan (2007-10-19).Favorite Math Links (in Finnish) by Janne Aukia (2007-10-18).Favorite Links (in Finnish) by Jannep (2007-10-18).Miracle de la Monarchie by JesusFranco on zentropa.info (2007-10-17).Best of the Web about hexahedra, polyhedra & polytopes (bef. 2007-10-17).Infinity (patchwork of clippings) Concord Middle School (bef. 2007-10-14).Baltzar von Platen by Jeremy (2007-10-11).CarmichaelZahlen in Meyers Lexikon (before 2007-10-08).Taaffe Viscount (in German) by Frank Martinoff (2007-10-01).Two odd balls (found by weighing with a balance) by Opalg (2007-09-30).Arms of scientists by Stephen J.F. Plowman, TD FSA Scot (2007-09-30).Volume of aSpheroid by LearnExcel (2007-09-27).Mathematicsand Physics in Bulgarian, by B.B. Bantchev (bef. 2007-09-26).Aurifeuillian Factorization by James Wanless (2007-09-23).Stars vs. SandRecounted in RASNZ newsletter, by Laintal (2007-09-23).Intégrales elliptiques de deuxième espèce by Framboise (2007-09-22).Infinity symbol for Mrs. Martin's Fourth Grade (before 2007-09-18).Fattest tetragonalantiwedge? by Erimo (2007-09-17).Unit(s) forPlanck's constant (in German) by Martin Müller (2007-09-16).Tetragonal antiwedge in Erimo's Journal (2007-09-15).A Mecânica doSéculo XVII in História da Ciência (2007-09-04).Una vaschettaorigami by Giorgio Vecchi (2007-09-04).Really cool mathsteasers in Orange Sunset (2007-09-03).Prime Number in Conservapedia (before 2007-09-01).Sir George GabrielStokes: August 13 in Science History (bef. 2007-08-28).Linki do ciekawychstron Escutcheons of Science (before 2007-08-22).Volume of the Grand Canyon by Orbinspace (2007-08-21).EDC 5V95 Syllabus by Trena L. Wilkerson at Baylor (bef. 2007-08-20).Wernher von Braun at www.spock.com (before 2007-08-20).Perimeter of an Ellipse by WesleySonofCornelius (2007-08-16).Trigonometry Class in "The Classroom" (before 2007-08-16).Useful Reference WebSites by David G. Simpson (before 2007-08-16).Another page Jim mightlike by Rybo on GeoJourney (2007-08-05).Michon's conjecture on Carmichael divisors by Maxal (2007-08-05).Fundamentals of Electronics by Dr. Stephen McArthur (bef. 2007-08-04).GoodWebsites by Forever May Not (2007-08-01).Propertiesof Specific Numbers by Robert P. Munafo (before 2007-07-31).Mathématiquesludiques (in French) by Tropique (2007-07-27).Another nice site by Galactus at MathHelpForum (2007-07-25).Arni'sMindgames by Chandru Arni (before 2007-07-23).Wie das radrichtig benutzt wird (in German) by Moriaan (2007-07-21).PlankschenKonstante (in German) by King_of_Loss (2007-07-20).ComedyShtick and Counterfeit Penny by D. Frisch (2007-07-19).GreyElephants in Denmark by Mark Runge (2007-07-19).Niste rezolvari (in Romanian) by Constantin Obreja (2007-07-17).How to become atheoretical physicist by Bryan Bishop (bef. 2007-07-16).Germanhorsepower by Djmatty (2007-07-14).Vector Calculus by Michael Jansson (before 2007-07-13).Verizongets out of the copper business [1]by Mark Ontkush (2007-07-12).Unit for Planck'sConstant (in German) by Alex (2007-07-06).Die HeisenbergscheUnschärferelation (in German) by pgz (bef. 2007-07-06).Perimeter ofan ellipse (using Adobe Illustrator) by Amy (2007-07-05).Bohr coat of arms by Diane Kulisek (2007-07-04).Counting without knowing how to count by Mathmate (2007-07-02).Patt Hawkey's MathLinks at Parks Junior High School (bef. 2007-06-29).Heraldry Links at ForumRycerstwa hosted by Yahoo! (before 2007-06-19).Perimeter ofan ellipse by Sam at UK Vintage Radio (2007-06-18).Elliptic arc by Xeriar (2007-06-17).Perfect Numbers,Mersenne Primes by Martin Gelbaum (bef. 2007-06-13).Coat-of-arms ofKarl Drais? (in German) by Jochen Wilke (2007-06-12).1 is not a prime number (in Romanian) by Lendar (2007-06-10).Armigerous Scientists at Book of THoTH by MediaMonkey (2007-06-07).MathematicalMagic by Byrd (2007-06-06).MathematicalHumor (update) by Scott Cram (2007-05-24).Liczbach rzymskich: Roman numerals, in Polish (before 2007-05-24).Infinity SymbolCloth by Lady Tricotine (2007-05-21).Etymology ofbehen by Scion (before 2007-05-21).Coalescing black-holebinaries by Damour & Nagar (2007-05-17).Infinite loops by Mike (2007-05-15).Car speed (in Romanian) by Victori (2007-05-11).Elliptical orbit by Ilya (2007-05-06).Gamma Function Citizendium Article (2007-05-05).MonkeyGravity by Yucha Yamaguchi (2007-05-03).CentroVirtual de Divulgación de las Matemáticas (before 2007-05-01).AdditionalInfo by David Fawcette at Columbia State (bef. 2007-05-01).Nauticalmile? by smci at "Yahoo! Answers" (2007-04-28).Heraldry and FamousCoats of Arms in Symbolism by Mtsar (2007-04-26).Archimedes [ 2 ] (before 2007-04-25).Mr. Miller makes the Numericana Hall ofFame (2007-04-25).Tycho Brahe's family... by Vacapinta (2007-04-23).Mental Magic at magicpen.com (before 2007-04-19).Review ofMath Resources in I Like Math Blog (before 2007-04-19).Armsof Wernher von Braun by Ahnungslos (2007-04-15).António Aniceto Monteiro by Jorge (2007-04-14).Stemmi di scienziati by Sebastiano Pasquini (before 2007-04-11).Algebra 2 (Fibonacci sharing) at The Morning Star Academy (2007-04-11).Algebra 1 (Fibonacci sharing) at The Jubilee Academy (2007-04-11).Perimeter of anEllipse at Math Open Reference (before 2007-04-11).Calculus Symbols at MathRamz.com (2007-04-10).Mensa Sverige Private forum, in Swedish (2007-04-08).Galeria (Polish) by Karolina Alicja Kaszuba (before 2007-04-08).Wappeneintrag etc. by Dr. Frank Joachim Reuther (before 2007-04-07).Howmany stars are there? in Sergiu M's Blog (2007-04-05).Ships andHorizon by David A. Smith (2007-04-02).Theoremof the Day (related sites) by Robin Whitty (before 2007-03-31).Vapensamlingar på internet (Swedish) by Awdaniec (2007-03-26).TheKruskal Count by Sir Ross (2007-03-26).Bimetallic Ratio by CJS (2007-03-25).Vapen förberömda/kända/ökända personer by Marcus K (2007-03-20).A Slice of Pi by Alex S. (2007-03-15).Pi Day and noteworthy constants... by Stephen Speicher (2007-03-14).Quart ofquarters by Squink (2007-03-11).Stoichiometry of Black Powder by KevFla2001 (2007-03-03).Arms of JosephLister by Stephen J F Plowman (2007-02-28).Arhimedovaspirala Bandwidth stealing in Hungarian (2007-02-23).Minimal surfacebounding a given volume by Sydney Chivers (2007-02-23).Butylic and butyric by Porlock Junior in Pharyngula blog (2007-02-18).Logaritma (in Turkish) by Eddie (2007-02-15).BasicTrigonometry by Roy (2007-02-07).Visconti,Sforza, Alfa-Romeo (in Swedish) by Awdaniec (2007-02-01).Vapen förberömda personer (in Swedish) by Marcus K. (2007-01-30).Metric value ofa pound (The AnswerBank) by mibn2cweus (2007-01-30).Ryedale's pennysorter by citizenkane (around 2007-01-29).Perimeter of an ellipse by DrLaszloJamf (2007-01-26).Carl von Weizsäcker by Norman Redington (before 2007-01-26).PartitionFunction by Sam Kong (2007-01-24).Counterfeit Coin Challenge (2007-01-23).CherishFreedom in Out From Under (2007-01-18).Bereken de omtrek van een ellips in Dutch Wikipedia (bef. 2007-01-17).MathematicalNews by Umberto Cerruti (before 2007-01-15).Blacktruss diamond markings by Schaula359 (2007-01-14).Coat-of-armsof Margaret Thatcher by Sean J. Murphy (2007-01-11).K.Thomsen's formula for the surface area of an ellipsoid (bef. 2007-01-11).The countability ofrational numbers by Krusty (2007-01-08).Horsepower at CycleChaos (before 2007-01-06).Measurements & Units at Uncover the Net (before 2007-01-03).SchoolNotes of Mr. Keeler, Fort Walton Beach HS (before 2007-01-03).Feynmann's wayfrom Newton to Kepler byBrian Beckman (bef. 2006-12-31).DiffusionQuadratic Residues by RFZ_Quest (2006-12-03).Tycho Brahethe Great Dane by Pmgisme (2006-12-03).Kevin S. Brown,author of MathPages.com by Chris Hillman (2006-12-02).Perimeter of an Ellipse in Dutch Wikipedia (2006-12-01).RomanNumerals by CCC (2006-11-30).Epiphany andHumor by Mohan K.V (2006-11-27).ThreePrisoners Paradox by Scott Cram (2006-11-26).Zoroastrian symbol in the Bible? by Seeker of Truth (2006-11-22).Mathematics Links by Thomas Fink (before 2006-11-22).Currency by Gaeleth (2006-11-21).Mrs. Bennett's page ofLinks Whitefish School District (bef. 2006-11-21).The full 1089 force by Scott Cram (2006-11-19).Mathematicalabbreviation ("Resp.") by Margaret Marks (2006-11-18)."Insanely cool for math/languagedorks" by Bynner (2006-11-15).Fulminating Gold by JWB in Pepys' Diary (2006-11-12).Signs, symbols and icons by Dave Gray (before 2006-11-12).Harmonic series in Physics 121 by Corbin Covault (2006-11-10).Tons galore... in Indonesian Wikipedia (before 2006-10-31).Ideal gas law at www.squinch.org (before 2006-10-30).Everything you ever wanted... by Marco Plumley (bef. 2006-10-25).Subtractive principle inRoman numerals by akaSpider (2006-10-23).Pound of pennies by JLam (2006-10-22).U2 Puzzle (Microsoft employement test) by Steveo1983 (2006-10-22).Infinity symbol (Lemniscus) by Winhill at UKC Forums (2006-10-19).Dirichlet Beta Function by Gerson W. Barbosa (2006-10-16).7-11 puzzle as ahomework assignment by CharlieLuciano (2006-10-15).Weighing12 coins (in Lithuanian) by Hoho (2006-10-10).Omega constant (in Swedish) on Wikipedia (bef. 2006-10-08).Feet, meters or leagues by Jacquelyn, a.k.a. polar griff (2006-10-04).Tycho Brahe'sShakespearean relatives by Banubula (2006-10-01).Tycho Brahe'sShakespearean relatives [ 1 ] by John Baez (2006-10-01).Closingspeed and Sagnac effect by Pupamancur (2006-09-28).Weighing Marbles by Jwwells (2006-09-23).Martin Luther King Jr.High School by Sharon Smith (before 2006-09-16).Math 432, Fall 2006 by Dr. Jim Haglund (before 2006-09-15).Speed of light in a movingliquid (Fizeau, 1851) by Montec (2006-09-14).Websites related to Planck by Skwirlinator (2006-09-10).Coat-of-arms of Wernher vonBraun on Wikipedia (bef. 2006-09-07).Combinatorial Game Theory by Richard J. Nowakowski (bef. 2006-08-31).Lanczosapproximation on Wikipedia (bef. 2006-08-30).TeoríaElectromagnética I by Ing. William Marín Moreno (2006-08-26).Metrologist? by Dr. Russell J. Rowlett (2006-08-10).Ramanujan andEllipses by Ken Takusagawa (2006-08-01).Distractionsmathématiques (in French) by Tropique (2006-07-27).Speed of light in refractivemedia by tsmid, corrected by przyk (2006-07-25).Entropy andMaxwell's Demon by Franklyn (2006-07-23).Counterfeit coinproblem by TxHockeyGuy (2006-07-02).L'énigme dubarbier [ 1] by Tropique (2006-06-30).Chord, Height andLength of an "elliptic bridge" by Thurrdome (2006-06-22).Math Courseby Chris Fenwick, University College London (bef. 2006-06-18).Chinese Remainder Theorem by Mathmate (2006-06-11).MathLinks (in Romanian) by flip2004 (2006-06-04).Feynman "Redo Some Basics Physics" (before 2006-06-03).Lose yourmarbles by pp-guest (2006-06-02).Ancient blackpowder by mrputter (2006-05-28).Brain Teasers by Rita I. Carreira (before 2006-05-21).World Science edited by Jack Lucentini (2006-05-03).Other links are archived.Please,letus know about other links to this site not yet included in our list.   |
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