Matpic
mathematics, in pictures
Concept

Rigid motion

A transformation of the plane or space preserving every distance: a translation, a rotation, a reflection, or a composition of those. Which of them a construction is allowed decides what it can reach, and forbidding rotation gives dissection a second invariant beside area.

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

A dissection that never comes apart. The three pieces of the triangle-to-rectangle dissection drawn at 4 moments of the swing. Each top piece turns about a pin at the end of the slice it stands on, and the pieces stay joined throughout.

A dissection that never comes apart

The plane theorem lets the pieces be picked up and put down anywhere. Require instead that they stay joined at their corners and swing, and the theorem survives — which was open for a century and is a much stronger statement about the same cuts.

computation · Scissors congruence The quantity a cut cannot change and a turn can. 4 polygons, each with the spikes of its translation invariant drawn round a dial: the length of the edges facing each direction, less the length of those facing the opposite way. It vanishes everywhere for 3 of them.

Slid, but never turned

Every construction on this ladder turns its pieces. Forbid the turn — allow the pieces to be slid and nothing else — and equal area stops being enough, for a reason that is a single number attached to each direction and that a cut cannot change.

computation · Scissors congruence

Named alongside it

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

AreaCongruenceDissectionOperation setPolygonConstructionHinged dissectionImpossibilityInvariant

All concepts