Matpic
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 3 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

Named alongside it

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

Cantor setChaosConnectednessCounterexampleEuler characteristicFractal dimensionFree groupFundamental groupGeneratorGenusGluingGroup presentation

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