Matpic
mathematics, in pictures
Concept

Exhaustive search

Settling a question by generating every candidate and testing each one. It needs no cleverness and gives a definite answer, and it is available only when the candidates can be listed and the list is short enough to finish.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

Trisect every angle, and an equilateral triangle appears. A triangle with angles 78°, 54°, 48°, its six angle trisectors, and the triangle whose corners are where the trisectors nearest each side meet. That inner triangle is equilateral, which is Morley's theorem.

Three trisectors and a triangle nobody expected

Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.

geometry · morley 3 rounds on chains of 4 and 5. Two chains of dots with pebbles placed in turn, and the transcript of a play: Spoiler picks an element of one chain, Duplicator answers in the other, and the pebbles must keep the same order.

A game that decides what can be said

Two players take turns pointing at elements of two structures; if the second can survive k rounds, then no sentence with k quantifiers tells the structures apart — a statement about infinitely many formulas, settled by a finite search.

logic · ef games 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.

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.

computation · de bruijn

Named alongside it

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

AngleConstructionCounterexampleCountingCyclic wordDe bruijn sequenceDecision procedureElementary equivalenceEncodingEquilateral triangleEulerian circuitExpressive power

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