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mathematics, in pictures
Concept

Rearrangement

Moving the pieces of a shape without changing any of them, so that whatever the pieces measure is unchanged. It is what a dissection proof does, and the whole force of such a proof is that area survives it while shape does not.

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

Euclid's proof, without moving anything. The square on a leg and its share of the square on the hypotenuse are each exactly twice the same triangle, so they are equal. Nothing in the figure is cut or rearranged; the triangle is only looked at from the other side.

Euclid proves it without moving anything

The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.

geometry · Pythagoras The same terms, with the signs alternating. The partial sums of 1 - 1/2 + 1/3 - 1/4 + …, out to 24 terms. They close on 0.69315 from both sides at once, and the gap between consecutive sums is the next term, so the answer is trapped.

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

analysis · Harmonic series

Named alongside it

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

Absolute convergenceAlternating seriesAltitudeAreaCommutativityConditional convergenceCongruenceConvergenceDissectionDivergenceHarmonic seriesHarmonics

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