Geometry · Calculus
Manifolds, differential forms, and multivariable calculus
How do you do calculus on the surface of a sphere, and what does \(dx\) really mean?
Mathematics beyond the classroom
Beautiful mathematics you probably won't see in school.
Explore surprising ideas from number theory, geometry, calculus, physics, and beyond — with visual and interactive explanations.
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A good example of what Hidden Phenomena is aboutTopology · Calculus
Möbius strips and differential equations
Start with the strange one-sided geometry of a Möbius strip, then watch the same phenomenon reappear when solving differential equations.
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Three accessible ways into the siteFractal patterns hidden inside Pascal's triangle
A familiar combinatorial triangle turns into a self-similar geometric pattern when you look at divisibility.
What is \(\sqrt{-1}\) modulo \(5\)?
Revisit the quadratic formula in modular arithmetic and learn when square roots exist in finite worlds.
What is the factorial of 3.5?
Euler's approximation trick extends factorials beyond whole numbers — and unexpectedly brings in \(\pi\).
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Newest firstAlgebra
Syzygies and higher groupoids
A route from ordinary equations to the idea that mathematical objects can be equal in more than one way.
Physics
Mechanics II: symmetry is conservation
Why symmetries of a physical system produce conservation laws.
Physics
Mechanics I: nature is lazy
A geometric way to understand motion through the principle of least action.
Topology · Calculus
Möbius strips and differential equations
A one-sided surface and a differential equation turn out to share the same obstruction.
Number theory
Fractal patterns hidden inside Pascal's triangle
Color Pascal's triangle by divisibility and a fractal suddenly appears.
Algebra
How to solve cubic and quartic equations II: cubics
Continue beyond the quadratic formula and see how cubic equations can be solved explicitly.
Algebra
How to solve cubic and quartic equations I: quartics
Why equations of degree four still admit a formula in radicals.
Calculus
What is the factorial of 3.5?
Extend factorials to non-integers and discover why \(\pi\) appears.
Number theory · Calculus
Infinitely many primes end in a 7
Why there must be infinitely many primes in certain arithmetic progressions.
Number theory · Analysis
The Riemann hypothesis (or, how to earn $1 million)
How a complex-valued function encodes the mysterious distribution of prime numbers.
Number theory
Why study Diophantine equations?
Use modular arithmetic to turn difficult integer equations into finite puzzles.
Number theory
What is \(\log(3)\) modulo \(7\)?
Exponentials and logarithms still make sense in modular arithmetic — but with a twist.
Number theory
What is \(\sqrt{-1}\) modulo \(5\)?
When does a number have a square root modulo a prime?
Number theory · Calculus
Using calculus to do number theory
A discrete analogue of Newton's method lets us lift solutions of congruences.
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