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Algebra
Mobius Transformations of the Night Sky
Kummer's Objection
Irreducibility Criteria
Tensors, Contravariant and Covariant
Proving Algebraic Inequalities
Bit-String Orbits Under Rotate-XOR
Piano Keys
The Crystallographic Restriction
Leaning Ladders
Finite Subgroups of the Mobius Group
String Algebra
Asymptotic Approach to 2D Arrays
Iterated Logarithmic Functions
Eigenvalue Problems and Matrix Invariants
Iterated Means
Fibonacci, 1/89, And All That
Square Roots by Pencil and Paper
A Quasi-Periodic Sequence
SumsofPowers inTermsofSymmetricFunctions
Generalized Mediant
Recurrences For Harmonic Sums
Bernoulli Numbers and Harmonic Series
Quasi-Groups
Ancient Square Roots
Pythagoras On Dot and Cross Products
Infinite Products and a Tangent Fan
Exponential Partitions
GeometricDot Products and DigitReversals
Radical Expression For cos(2pi/7)
Compressor Stalls& MobiusTransformations
TheShortestPathToTrigonometricIdentities
Convoluting Arrays
HarmonicSums ofIntegers with k Binary1's
Gauss' Lemma Without Divisibility Arguments
Reflecting on the Geometric Mean
Inverse Square Weighted Interpolation
Platonions
Annuities With Inflation
Polynomials From Pascal's Triangle
On Eulerian Numbers
Intersections of Polynomials
Angular Angst
From the Geometric Series to Stirling Numbers
Orthomagic Square of Squares
Automedian Triangles and Eigen Vectors
ContinuFractions&CharacterisRecurrences
Phased Summations
Arranging the Solutions of f(x+y+z)=xyz
Tangent To PI
Cross-Linear Interpolation
The Super-Symmetric Mean
Cyclic Redundancy Checks
Permutation Loops
Loxodromic Aliasing
Determining the GaloisGroupof aPolynomial
Formal-Numeric Series
Napoleonic Vectors
Expulsion Sets
Bi-Rational Substitutions Giving Squares
Cyclic Divisibility
Harmonia Mensurarum
Iterative Isoscelizing
Galois Groups
Hermitian Matrices
Linear Fractional Transformations
The Resultant and Bezout's Theorem

Number Theory

Symmetric Pseudoprimes
Identities for Linear Recurring Sequences
The Half-Totient Tree
Limit Cycles of xy (mod x+y)
A Knot of Congruences
Fermat's Last Theorem for Cubes
Sequence Partitionable Into Powers of 2or3
Iterated Sum-of-Digits Function
If ab+1, ac+1, bc+1 are squares,...
Zeisel Numbers
TheJewel ofArithmetic:QuadraticReciprocity
Highly Wilsonian Primes
Quadratic Congruences
Squares in Arithmetic Progression (mod p)
Numeri Idonei
Sequences With No Arithmetic Progressions
A Primality Criterion
Can n Divide !n ?
Diophantine Walk-a-thon
Barlow's Observation
Divisors of an n-term Geometric Series
More Results on the Form xy (mod x+y)
Sums of Three Cubes
Congruences Involving the Totient Function
Why z IsNotaPrimePower in z^p = x^p+ y^p
Summations and Recurrences
Evidence for Goldbach
Product Divisible By Sum of Squares
Pseudoprimes For x^2 - 4x - 9
Sums of Consecutive NthPEqualtoanNthP
On Case 1 of Fermat's Last Theorem
Coprime Partitions
The 450 Pound Problem (x^3 + y^3 = 6z^3)
Is This a Prime?
Reverse Greed for Unit Fractions
Sum of Divisors Equals a Power of 2
Sublime Numbers
On x^2 + y^3 = z^6
The Distribution of Perfection
Fermat's Last Theorem forQuadratic Integers
Square Triangular Numbers
Recurrences and Pell Equations
In Defense of Base-Related Problems
On x^3 - x + y^3 - y = z^3 - z
On the Density of Some Exceptional Primes
Double Equations fromTriangles inSquares
Differently Perfect
A Special Property of 151
Simple Complex Quadratic Fields
The Euclidean Algorithm
Casting Out Nines
Detecting Squares
Numbers Expressible As (a^2 - 1)(b^2 - 1)
Concordant Forms
Solving Magic Squares
Unit Fractions and Fibonacci
The 2 Ohm Problem
Reflective and Cyclic Sets of Primes
Unit Fraction Partitions
N = (x^2 + y^2)/(1+xy) Is A Square
MinimizingDenominatorsinUnitFractionExp
On General Palindromic Numbers
Pythagorean Graphs
Series Within Parallel Resistor Circuits
Sequences Based On Continued FractionForPI
SmallQuadrilateralWDistinctIntegerSides
Schizophrenic Numbers
Coherent Arrays of Squares
Anti-Carmichael Pairs
Magic Square of Squares
One In The Chamber
Odd, Greedy, and Stubborn(Unit Fractions)
Lucas's Primality Test With Factored N-1
Highly Heronian Ellipses
How Leibniz Might Have Anticipated Euler
Wagon Trains and Sticky Wickets
Factoring Zeta
Accidental Melodies
Four Squares From Three Numbers
MeanderingConvergenceofaDirichletSeries
Geodesic Diophantine Boxes
Least Significant Non-Zero Digit of n!
Discordance Impedes Square Magic
AUniqueIntersectionPatternforPlaneCurves

History
The Thought of a Thought - Edgar Allan Poe
The Prismoidal Formula
Gerber's Gravity
Carnot, Organizer of Transversals
Magnetism and Earnshaw's Theorem
Dialogues
Wonder En Is Gheen Wonder
Legendre's Prime Number Conjecture
The Helen of Geometers
Berkeley and the Infidel
Franklin's Magic Squares
Fermat's Infinite Descent
Negative Numbers
The Metamorphosis of Judas Iscariot
Natural and Violent Motions
Catch of the Day (153 Fishes)
The Guest Star
From Euclid to Gregory
What Happened to Dingle?
Napoleon's Theorem
Why Was Michelson Surprised?
Zulu Hitler
Who Invented Relativity?
The End of My Latin
Omar Khayyam On Cubics
Stationary Paths
Anomalous Precessions
Constructing the Heptadecagon
Kepler, Napier, and the Third Law
De Mora Luminis
Translating Aristotle
The Longitude of Lewis and Clark
Constructing the Principles
An Anomaly In Translation

Calculus and Differential Equations

The Laplace Equation&HarmonicFunctions
Curvature, Intrinsic and Extrinsic
Radial Paths in Schwarzschild Spacetime
AnalyticFunctionsTheMagnusEffect&Wings
Fourier Transforms and Uncertainty
Series Solutions of the Wave Equation
The Limit Paradox
Proof That PI is Irrational
Simple Proof that e is Irrational
Root-Matched Recurrences For DiffEQs
Why Calculus?
High Order Integration Schemes
Do We Really Need Eigen Values?
Leibniz's Rule
ARemovablSingularityinLeadLagCoefficients
Convergence of Series
Sum of n^2 / (n^3 + 1), n=1 to inf
Tilting Pencils
Curvature of Linear Interpolation
Ptolemy's Orbit
Distances In Bounded Regions
Integrating Factors
Ratio Populations
Perpendicular Regression Of A Line
Potential Flow and d'Alembert's Paradox
Round-Robin Ties
Refractions on Relativity
Irrationality of Quadratic Sums
Time for a Rocking Chair
GraAccelerationinSchwarzschildCoordinates
Mean Distance from Vertex to Interior ofPFig
Mean Partial Sums of Non-ConvergentSeries
Constant Headings and Rhumb Lines
The Wave Equation and Permutation ofRays
Huygens' Principle
Recurrence Relations for OrdDiff Equations
The Curvatures of Hypersurfaces
Poisson Processes and Queues
Lagrangian and Hamiltonian Mechanics
Lead-Lag Frequency Response
Differential Operators&Divergence Theorem
Precession in a Circle
Higher-OrderWaveEquations&MatterWaves
Complete Solutions of Linear Systems

Probability and Statistics
Boolean Expansion as Linear Interpolation
At Least One Girl
The Collector's Problem
Collecting k Complete Sets of Coupons
Generalized Birthday Problem(N Items inMB)
Independence and Negation
Dice Rolling A Given Sum
Redundant Systems With A Common Threat
Meeting Probabilities
A Better Lottery?
Bayesian Balls
Balls In Bins
Combining Probabilities
A Barrel of Colors and Null Outcomes
Open-Loop& Closed-Loop Markov Models
The e in Petersburg
Weibull Analysis
Two Properties of Markov Models
Hail Encounters
Random Triangles
Probabilities and Velocities
Two In the Shop
Differences Between Normal Samples
Benford's Law
On 13 Bridges
Bernoulli Trials
Sequences of Characters in Random Strings
Botches, Failures, and Successes
A Random Walk Through PI
The Shape of Coincidence
Markov Models of Dual-Redundant Systems
Then There Were None
Generating Random Numbers In Order
Markov's Marbles
Expected Area of Random Polygon in aCircle
World's Oldest Man Dies

Combinatorics

Partitions into Distinct Parts
Dedekind's Problem
The Four Color Theorem
Polyomino Enumerations
Powers of Primes Dividing Factorials
The Dartboard Sequence
Meeting Combinations and Interval Graphs
Binary Games
Additive and Multiplicative Partitions
414298141056 Quarto Draws Suffice!
Coloring The Edges of an Icosahedron
Uniform Squares
Cumulative Permutation Sequences
Center of Gravity With Integer Coordinates
The Factorial Number System
The Statistics of Counterintelligence
Partition Transform Cycles
Clouds, Shy Squares, and Diophantus
Computing the Partitions of n
Generating the Monotone BooleanFunctions
The Determinants of 4x4 Magic Squares
The Importance of Being Urnest
Balls In Bins With Limited Capacities
Complete Bit-Cubes
A Procrustean Protocol
Partitions into Consecutive Integers
Tables of 1 to N with Equal Columns
Regular Polygonal Arrangements
The Pigeon Hole Principle
Two Proofs ofImpossibilityforFive-PointStar
GolfTournaments, Polyhedra,&LatinSquares
Derangements and Generalizations
The Coin in Three Fountains
Impossible Strings for Cellular Automata
Permutation Puzzles
Christmas Ornaments
Ringing of the Changes (The Shape of 4!)
f(f(x))=exp(x) Generates the Multinomials
Progress on Dedekind's Problem

Physics
The Hydrogen Atom
The Speed of Sound
Born Rigidity, Acceleration, and Inertia
Doppler Shift for Sound and Light
Formation and Growth of Black Holes
Locality and Temporal Asymmetry
Imaginary Explanations
Entangled Events
Force, Curvature, and Uncertainty
Null Coordinates
Retrograde Moon?
A Primer On Special Relativity
Two Geophysical Coincidences
A Mass and Spring System in Motion
The Lemniscate and Spacetime
Geophysical Altitudes
Interference or Refraction?
Sandford's Elevator
The Inertia of Twins
Photons, Neutrinos, And Their Anti-Particles
Accelerated Travels
Is Entropy Quantized?
Free-Fall Equations
Reasoning With Reversibility
Immovable Spacetime
Relative Rain
Intrinsic Wobble of Spinning Objects
Is The Universe Closed?
Polarization and Spin
Factoring Convex Figures
Proper Time in Circular Orbits
Accelerating In Place
Invariance, Covariance, and Contravariance
Falling Into& Hovering Near a Black Hole
Cylindrical Dimensions
The Projection Postulate in Q Mechanics
Packing Universes In Spacetime
The Phase Effect of Acceleration
Weighing the Moon
A More Practical Arrangement
Reversibility and Entropy
Orbits and Perturbations
An Exact Solution
Bending Light
Rigidly Re-Orienting An Extended Body
Normal Shock Waves
Hypocycloidal Engine
The Stars In The Sky
Entangled Choices
Quantum Entanglement and Bell's Theorem
A Rotating Disk in Translation
Energy, Entropy, Enthalpy
True, Equivalent, and Calibrated Airspeeds
The Moon Always Falls Toward the Sun
Twin Times
Hero and Fermat on Receding Mirrors
Does a Uniformly AcceleraCharge Radiate?
Thomas Precession
Frequency and Energy

Set Theory and Foundations

Representing Sets of Pure Order
Zeno's Paradox of Motion
Global Reversibility of Cellular Automata
Guessing Faster Than Light
Fuzzy Logic
Fractal Logic
Infinite Descent versus Induction
Zeno and Uncertainty
Is Arithmetic Consistent?
Reconstructing Brouwer
Euclid's Plan and Proposition 6
Constructible Points and Coverable Points
Cantor's Diagonal Proof
Conditions for Relativity
Function 34776
An Infinite Wall
Interfering With PI
Choice Without Context?
Certainty In Math and Physics
Von Neumann's 5th Postulate
Attractive Forces From Quantum Exchanges
Newton's Cosmological Queries
Timely Confessions
Galileo's Forgotten Leap
Gamaliel's Principle
The Gestalt of Determinism
Preserving the Wave
On Gauss's Mountains
This Business of Interpretation
Unreasonable Effectiveness
A Quintessence of So Subtle a Nature
MathPages Wanted List
TheY10KProblem,Pascal'sWager,&Petersberg
Newtonian and RelativisticConservationLaws
Interleaving Ad Infinitum
White Ravens and Black Swans
SpacetimCoordinaSystems&ReferenceFrames
Cellular Automata
Continuity and the Void
Potential Energy, Inertia,&QuantumCoherence
Harmonia Praestabilita

Animated (Java) Illustrations

The Stars In The Sky

Kinematics of a Hypocycloidal Engine

Bell-Ringing Solid

Kepler's Rhombic Dodecahedron

Geometry
Sphere Packing In Curved Space
Pascal's Mystic Hexagram
An Optical Illusion
Cramer's Paradox
Old Notes on Curvature
Pent Up Ratios
Main Diagonal Passing Through Interior
Embedding Non-Euclidean Within EGeo
Pappus' Theorem
Volume Under a Triangle
No Equilateral Triangles on a Chess Board
Random Tiling of a Sphere
Path Lengths and Coordinates
The Cube Unfolded
Revisiting the Four-Point Parabola
Morley's Trisection Theorem
Menelaus and Ceva
Non-Periodic Tilings WithN-foldSymmetry
Aperiodic Tilings
Tilings Based on Perrin's Sequence

Music

Washed Away
Recovery
Practice Rooms
Steps
Hymn J
Temporary
Rain
Hymn C
Of Strangers
Student
Mercy
Hymn Y
Missing
1918
Prospect 6
Ideas
The Light Cubed
13 Aprils
Losing Hartford
Carol
Descent
Over There
Eta Carinae
Macbeth