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Concept

Hinged dissection

A dissection of one shape into pieces that are joined at hinges, so that the pieces swing as a chain from one shape into another. Dudeney's four-piece dissection of a triangle into a square is the best-known example.

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 Dudeney's four pieces, as a triangle and as a square. Dudeney dissection of the equilateral triangle of side 2 (area 1.732051) into a square of side 1.316074: J at (0.25451, 0.44082), K at (0.75451, 1.30685); piece areas 0.40568, 0.68076, 0.39787, 0.24775.

Dudeney's four pieces are the fewest

In 1902 Henry Dudeney cut an equilateral triangle into four pieces that rearrange into a square, and hinged them so the triangle folds into the square in one motion. For a hundred and twenty-two years nobody could say whether three pieces would do. In December 2024 Erik Demaine, Tonan Kamata and Ryuhei Uehara proved they would not. The construction, built exactly here, shows why four is natural; the proof shows why it is necessary.

computation · Scissors congruence

Named alongside it

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

DissectionAreaCase analysisCongruenceConstructionLower boundOperation setPolygonRigid motionScissors congruence

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