Matpic
mathematics, in pictures
Concept

Symmetry

A transformation that leaves an object looking exactly as it did before.

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

x²1.5x1.5x1.5²x + 1.5a squarex² + 3x + 1.5²= (x + 1.5)²x = 4, so the total is30.25

Completing the square, by completing a square

The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.

algebra · completing the square the cutter's measure, segment by segmentthe chooser's measure, segment by segment52530201553010552030cut at 4/9the left piecethe right piecethe cutterthe chooser5050130/3≈ 43.33170/3≈ 56.67the cutter's piecethe chooser's piecethe cutter's running total crosses 50 inside segment 3, 2/3 of the way through it, so the cut is at 4/9both pieces are worth exactly 50 to the cutter; the chooser takes the right one at 170/3 ≈ 56.67 andgains 20/3 ≈ 6.67 over half

One cuts and the other chooses

The oldest rule in fair division promises each of two people at least half the cake by their own measure, and it keeps that promise exactly. It does not promise what the word "fair" is usually asked to carry, and the gap opens the moment the two measures disagree across the cut.

applied · fair division

Named alongside it

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

Algebra tilesAreaCompleting the squareCounting argumentDissectionDivide and chooseEnvy freenessFair divisionImaginary unitMeasureProportionalityQuadratic polynomials

All concepts