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mathematics, in pictures
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Counting the same thing twice

One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.
11122113233231142535344353524115 fractions, all in lowest terms, none of them twiceand left to right they are already in order Number

Every fraction, exactly once

Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.

 124361251020153060× 2 →× 3 ↑2^2 × 3 × 5 — 3 × 2 × 2 = 12 divisors Number

The shape of a number's divisors

Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

 (−1, 0)3/4 → 7, 24, 252/3 → 5, 12, 131/2 → 3, 4, 52/5 → 21, 20, 291/3 → 8, 6, 101/4 → 15, 8, 17each line of rational slope meets the circle a second time at a rational pointclearing the denominators turns that point into a Pythagorean triple, and every triple arises this way Number

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

 1 string1 string5 strings5 strings5 strings5 strings5 strings5 strings32 strings fall into 8 necklaces32 strings in all: 2 constant ones, and 6 rings of 5so 32 − 2 = 5 × 6, and p divides a^p − a with nothing left over Number

Necklaces that prove a theorem

Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.

 0612391017134511281401234012mod 3 ↓mod 5 →every one of the 15 pairs is reached, exactly onceso a remainder mod 3 and a remainder mod 5 together name one number mod 15 Number

Two dials at once

Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.

 slope 7/117 points below the line, 8 above, and 7 + 8 = 5 × 3 = 15(7 | 11) = −1 from the count below; (11 | 7) = +1 from the count above Number

Counting one rectangle, twice

Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.

 5 + 4 + 2 + 14 + 3 + 2 + 2 + 1read downboth are partitions of 12: the same dots, counted along the rows and then down the columnsand turning the diagram over a second time gives back what it started as Number

A diagram turned on its side

Write a partition as rows of dots, then read the columns instead. Every theorem in this essay is that one move, and the move proves things that no formula suggests.

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