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mathematics, in pictures
Concept

Surface

A space that looks like a piece of the plane near every one of its points. The closed ones are classified completely by orientability and a single whole number, which makes them the largest family of spaces anybody can list.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A solid where V − E + F is 0. a slab with one hole through it, drawn as a wireframe. Its 32 vertices, 64 edges and 32 faces give an alternating sum of 0 rather than 2.

The solid where the answer is not two

A slab with a hole through it has flat faces, straight edges and sixteen corners, and its alternating sum is zero. It is not a trick and not a degenerate case — it is the object that shows the theorem had a hypothesis nobody had written down.

topology · Euler characteristic Three spaces, cut into pieces. 3 panels, drawn from: a sphere split into two caps meeting along a circle, two circles joined at a point, and a square whose opposite edges are glued into a torus. Each carries the fundamental group that van Kampen's theorem computes for it.

Cutting a space to find its group

A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.

topology · Homotopy The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones.

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

dynamics · Strange attractor The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1.

The surface a knot bounds

Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.

topology · Knots

Named alongside it

The objects these essays reach for when they reach for this one.

Euler characteristicGenusBoundaryCantor setChaosConnectednessCounterexampleFractal dimensionFree groupFundamental groupGeneratorGluing

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