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 2 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

Named alongside it

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

Polyhedron Angle defect Area Congruence Curvature Dissection Euler characteristic Gauss bonnet Impossibility Invariant Operation set Platonic solids

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