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Concept

Dihedral angle

The angle between two faces of a solid, measured across the edge they share. A cut through a polyhedron either preserves such an angle or splits it into two that add back to it, which is why it appears in the invariant that blocks a dissection.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The gap at a corner of the cube. The 3 faces meeting at one corner of the cube, unfolded onto the page. They leave a gap of 90.0 degrees, and the 8 gaps come to 720 degrees in total.

Seven hundred and twenty degrees of gap

Unfold the faces around any corner of a solid and they do not close up. The gap left over is different at every corner and on every solid, and the gaps always add to two full turns.

topology · Euler characteristic A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

computation · Scissors congruence Volume and one more number decide what a solid can be cut into. A table of a cube, a prism, a sixth of a cube, a regular tetrahedron, a regular octahedron and a collection of two tetrahedra with one octahedron, giving each one's volume, its Dehn invariant computed from its measured dihedral angles, and whether it can be cut into a box of equal volume.

The obstruction that was the only one

Dehn showed in 1901 that a cube cannot be cut into a regular tetrahedron of the same volume, because a number built from edges and angles disagrees. For sixty-four years nobody knew whether that number was the whole story. Sydler proved in 1965 that it is: volume and Dehn's number together decide every case.

computation · Scissors congruence

Named alongside it

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

PolyhedronDissectionImpossibilityInvariantOperation setVolumeAngle defectAreaCongruenceConstructionCurvatureEuler characteristic

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