Nets and Folding

Nets of polyhedra. The solids are the same ones as on the Polyhedra page; the prisms and antiprisms are built in your browser with every edge 1 long. A net is made by choosing which edges stay as folds (they must link every face without a loop, which mathematicians call a spanning tree), then laying each face flat against the one it hangs from. The page tries several ways of choosing: “steepest-edge” cuts (every corner except the highest, looking from some direction, is cut along its steepest edge upwards), trees grown outwards from different faces, and random trees. It keeps only nets in which no two faces overlap, and drops any net that matches one already found after turning or flipping it over. The known totals of different nets for the Platonic solids (2, 11, 11, 43,380 and 43,380) are standard results, counted in full by Francis Buekenhout and Monique Parker in “The number of nets of the regular convex polytopes in dimension ≤ 4” (Discrete Mathematics 186, 1998).

Before publishing, every solid in the picker (all 53) was checked with a script: each of the 424 nets offered was tested for overlaps twice (by separating lines and by clipping each pair of faces against each other: largest shared area 3 × 10⁻¹⁴), then folded all the way up with the same folding code the 3D view uses. Every corner came back together to within 4 × 10⁻¹², and one rigid motion put every folded face onto the matching face of the solid to within 3 × 10⁻¹² (1 × 10⁻⁴ against the stored corners, which are rounded to four decimals). Trying every one of the spanning trees of the tetrahedron (16), cube (384) and octahedron (384) and sorting the nets that result gives exactly 2, 11 and 11 different nets, matching the known totals.

Draw your own net. The check “rolls” the solid across your drawing: the first cell sits on one face, and stepping over each edge into the next cell tips the solid over that edge, keeping track of which edge of the face lies where. A drawing folds when its cells are all joined edge to edge, no two cells land on the same face, and cells that share an edge land on faces that really meet along that edge. The solids’ faces, edges and fold angles (180° minus the angle between two faces: 90° for the cube, about 109.47° for the tetrahedron, 70.53° for the octahedron and 41.81° for the icosahedron) are worked out from their corners, not typed in.

To check it, a separate test listed every shape that can be made from 6 squares (35 of them) and found exactly 11 that fold into a cube; every shape of 4 triangles (3) and found 2 tetrahedron nets; and every shape of 8 triangles (66) and found 11 octahedron nets. These match the well-known counts and the nets under “Load a net”. Each of those nets, and the icosahedron net, was then folded flat-to-closed and its corners met to within a millionth of an edge. For all 5,698 shapes of up to 7 squares, 6 triangles (tetrahedron), 9 triangles (octahedron) and 12 triangles (icosahedron), the folded 3D model and the rolling check gave the same answer every time.

Hexaflexagons. The simulator models the real paper. Each triangle has the exact positions of its three corners on a triangle grid, and each of the hexagon’s six sectors holds a stack of triangles in order. A pinch flex is done the way hands do it: three alternate radii fold back so the hexagon becomes a three-pointed star of six “wings”, each wing opens out along its outer edge, and the opened wings are laid flat round the new centre. Every step is checked: hinged triangles must still share their edge, no fold may cross another or wrap round a hinge, and the result must be a proper hexagon. A flex happens only if all of that works out, and only in one way. The flexagons themselves are built by folding the strip exactly as the steps below say, so the strips, the diagrams and the simulator all come from the same folds.

To check it, every position each flexagon can reach was explored: 36 for the trihexaflexagon and 108 for the hexahexaflexagon, counting each turn and each side up separately. All of them are physically sound, every face shown is a whole face, and the faces are exactly the 3 or 6 that the usual numbering gives (front 1 2 3, back 4 4 5 5 6 6 for the hexahexaflexagon). A second, separate model, written as Conway’s “pat” algebra with no coordinates at all, gave the same result for every position and 80,000 random flexes, turns and corner moves: the same faces, layer order and turn of every triangle. Each flex sends the front face to the back, the trihexaflexagon cycles 1, 3, 2, and a Tuckerman traverse of the hexahexaflexagon visits 1, 3, 6, 1, 3, 2, 4, 3, 2, 1, 5, 2 (faces 1 to 3 three times each, 4 to 6 once), exactly as in Martin Gardner’s 1956 Scientific American column.