Blog

Turning the machine on: evaluating Gauss' integral and proving links exist

The pullback computation: from cohomology to Gauss' double integral

From degree to integral: de Rham cohomology and the winding number

Capturing the notion of linking in topology

Cusps, limit sets, and the boundary between discrete and indiscrete.

Isometrically embedding flat tori into Euclidean space via the Hopf fibration.

Adding this axiom allows simplification of the previous axioms

Imposing this forces the integral of the rationals' characteristic function to be zero

Showing its independence from the previous axioms

An explicit pair of integrals that satisfies the basic axioms but disagree on a function.

Seeking similarities and differences when trying to understand quantum mechanics

Asking whether the axioms pin down every integral or leave room for choice.

Ready-to-use formulas for geodesics on any surface given a metric.

The quadratic formula as an isometry between models of the hyperbolic plane.

Geodesics on graphs from the condition that acceleration is normal.

The purely intrinsic approach, using only the induced metric.

A coordinate-free reference for forms, Hodge stars, and the Laplacian.

Coordinates and radii for building dodecahedral honeycombs.

Edge lengths, dihedral angles, and volumes as functions of the vertex angle.

Measuring change along a flow without choosing coordinates.

Step-by-step Lie derivative computations in the plane.

How symmetries of a metric yield first integrals of the geodesic equations.

A self-contained proof using only the Fundamental Theorem of Calculus.

Solutions of y'=y satisfy the law of exponents.

Three simple axioms for integration that force the FTC to be true.

The functional equation alone pins down the antiderivative.

Positivity, monotonicity, and convexity from E(x+y)=E(x)E(y).

The ODE for a bead sliding down a wire.

A smooth one-parameter family connecting two Thurston geometries.

Separation of variables as a systematic tool for finding near-harmonic functions.

A recipe for correcting almost-harmonic functions into genuine solutions.

The fundamental solution is the integral of reciprocal surface area.

What happens when you compose harmonic functions with arbitrary maps.

A short proof using naturality of the exterior derivative and Hodge star.

Explicit fundamental solutions in Euclidean, spherical, and hyperbolic space.

Generalizing to arbitrary dimension via the Hopf degree.

Polar coordinates arise naturally from the universal cover of the punctured plane.

Why dθ is closed but not exact, and what that has to do with winding.

How a local diffeomorphism and holonomy map produce an atlas.

Unlike in Euclidean geometry, springs oscillate in hyperbolic space.

Why you can freely pass derivatives through convolutions in distribution theory.

Why the topology on test functions requires all-derivatives convergence.

Extending the pentagon and hexagon analysis to seven-sided polygons.

Turning abstract moduli data into concrete coordinates for rendering.

Alternating side lengths as natural coordinates on moduli space.

A surprise appearance of the golden ratio in hyperbolic geometry.

From abstract side-length data to Möbius-placed coordinates.

On the immensity of Earth's history and why it matters.

Cooking, walking, and looking around in a world of constant negative curvature.

A farewell to Cassini on the eve of its final descent.

What the 3-sphere looks like from the inside.

Decomposing the 3-sphere into a family of linked circles.

The topology of cubic polynomials and the braid group.

The moduli space of planar lattices is a trefoil knot complement.

How symmetry and translation affect the frequency spectrum.

Decomposing functions the same way you decompose vectors — just in infinite dimensions.

Building intuition for convolution from physical examples.

Why springs and circuits naturally live in the complex plane.

Starting from repeated multiplication and arriving at a universal constant.

What happens when you raise e to an imaginary power.

Filling rectangles in a unit square to see why the sum works.

Developing the gradient from directional derivatives and contour maps.

Deriving the divergence formula from the idea of net outward flow.

Four triangles, two squares, one identity.

Completing the square, literally.