Matpic
mathematics, in pictures
Concept

Closure

The smallest set containing a starting collection and shut under a stated set of operations, so applying one never leaves it.

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

keeps 0keeps 1monotoneself-dualaffineenough alone?∧ and✓✓✓··no∨ or✓✓✓··no¬p not p···✓✓no↑ nand·····yes⊕ exclusive or✓···✓no→ implication·✓···no6 connectives against Post's five classes — a tick means the connective stays insidenand escapes all five, and is therefore enough alone

One connective is enough

Of the sixteen ways to combine two truth values, exactly two can build all the others by themselves. Which two is not obvious, and the reason turns out to be five properties that a connective either has or escapes.

logic · truth functions 01−12√3⁄2−√3⁄2two points, 1 line and 2 circles, 4 new pointseach new point was checked to lie on two of the objects drawn before it

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

computation · constructible numbers ℚdim 12ℚ(√2)dim 22ℚ(√2, √3)dim 41√2√3√61√2√3√61√2√3√6√22√62√3√3√633√2√62√33√262 square roots taken, one at a time, and the degree doubles at each: 1 → 2 → 4the 4×4 table is the closure check — every product of basis elements landed on a whole-numbermultiple of another

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

computation · constructible numbers +01αα+101αα+101αα+110α+1ααα+101α+1α10×01αα+101αα+1000001αα+10αα+110α+11α× mod 40000012302020321the 4 elements of GF(4), built as polynomials in α modulo α+1 + α² — every product of two non-zero elements is non-zerobeside it the integers mod 4, where 2 × 2 = 0 and 1 element has no reciprocal

The field with four elements

The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.

computation · finite fields

Named alongside it

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

Constructible numberField extensionStraightedge and compassAffine functionBasisCharacteristicCircleConnectiveCounterexampleDegree of an extensionDualityFinite field

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