Math

Nets of the solids

Cut a solid along some of its edges and open it flat. The uncut edges must keep every face connected without a loop - a spanning tree of the face graph - and every such tree is a net. The solid opens along its hinges by composed rotations, the net's area is checked against the surface, every pair of faces is clipped to prove nothing overlaps, and the nets are counted up to the solid's symmetries: two for the tetrahedron, eleven for the cube, eleven for its dual.

By Mykhailo Makarov

Actual application image

What happens

Cut a cube along seven of its twelve edges - the right seven - and it opens flat into one of eleven nets; the wrong seven leave two pieces or a loop. The uncut edges must form a spanning tree of the faces, and every spanning tree is a net, though some overlap themselves when laid flat. The unfolding is a hinge motion: each face swings about the edge it shares with its parent until the dihedral angle is gone. The cube has eleven nets, the icosahedron forty-three thousand. Pick a solid, choose which edges to cut, and fold it open and closed.

Equations used in the module

Equations
a net: the uncut edges form a spanning tree of the faces
unfolding: each face turns about its shared edge by π − dihedral angle
dihedrals: tetrahedron 70.5°, cube 90°, octahedron 109.5°, dodecahedron 116.6°, icosahedron 138.2°
nets of a cube: 11;  of an icosahedron: 43 380

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