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mathematics, in pictures
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The same thing twice — page 4

Two constructions that look unrelated and turn out to be the same object wearing different clothes.
The de Bruijn graph on 2 letters and words of 3, and the cycle through it. A graph whose vertices are short words and whose arrows are words one letter longer, with a closed walk using every arrow exactly once marked, and the cyclic sequence it spells. Computation

Every word once, around a cycle

A cyclic string of eight bits holds all eight three-bit words, each exactly once — and the reason such a thing exists is that the constraint linking overlapping windows is itself the construction.

 A table of shares written as a lottery over 3 whole assignments. A doubly stochastic table of shares, and beneath it the permutation matrices and weights that add up to it exactly, each drawn as a grid with one marked cell per row. Applied

A lottery over whole assignments

A table of shares in which every person's shares add to one task and every task is exactly covered is never anything more than a mixture of whole assignments — and finding the mixture is a matter of taking one complete assignment out at a time.

 The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon. Algebra

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

 The dihedral group of a 4-sided shape, drawn as a map. A Cayley graph: one dot per motion of the shape, with one arrow per generator, so that multiplying by a generator is following an arrow of that colour. Algebra

The group drawn as a map

A multiplication table says everything about a group and shows nothing; lay the same information out as one dot per element and one arrow per generator, and multiplying becomes walking, distance becomes a word length, and the group acquires a shape.

 The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis. Algebra

A tower whose degrees multiply

Treat a field containing another as a vector space over it, and the size of an extension becomes a dimension — one that multiplies along a tower, so that three impossible constructions become arithmetic about which numbers divide which.

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