A cube has exactly 11 distinct nets. There are 35 ways to join six squares edge to edge, but only 11 of them fold up into a cube, counting shapes that match after a rotation or a flip as the same net. On this page all 11 open out one at a time from the solid, and once the full set is on screen the sequence starts again.
The faces are colour-coded, and the display panel can also print a number on each one. A colour is decided by which face lands on the bottom as the cube rolls to the next square, so every net shows all six colours exactly once and you can see which colours end up opposite each other. The nets group by the length of their central strip: six are 1-4-1, a strip of four with one square on each side; three are 2-3-1; and the staircase 2-2-2 and the 3-3 make one each.
Use the shape panel to switch to a regular tetrahedron or a regular octahedron. The tetrahedron has two nets; the octahedron has eleven, the same count as the cube. They match because the cube and the octahedron are duals, so their faces join up in the same pattern. The angle a net folds back through differs for each solid: 90 degrees for the cube, about 109.47 for the tetrahedron and about 70.53 for the octahedron, each one the supplement of that solid's dihedral angle.
A regular dodecahedron and a regular icosahedron are on the list as well. Each of them has 43380 nets, far too many to lay out side by side, so a single familiar one is used for each: two rosettes of a central pentagon ringed by five petals for the dodecahedron, and a strip of ten equilateral triangles with a cap of five at each end for the icosahedron. The counts match because these two are duals as well. A net folds back through about 63.43 degrees for the dodecahedron and about 41.81 for the icosahedron.
Cutting the corners off a regular solid gives an Archimedean solid. The cut is placed where the exposed face comes out a regular polygon: a third of the way along each edge for the solids with triangular faces, 1/(2+√2) for the cube and 1/(2+φ) for the regular dodecahedron. A tetrahedron yields 4 triangles and 4 hexagons, a cube 8 triangles and 6 octagons, an octahedron 6 squares and 8 hexagons, a dodecahedron 20 triangles and 12 decagons, and an icosahedron 12 pentagons and 20 hexagons. That last one is the football. Along the edges where two original faces still meet, the fold angle of the original solid survives the cut.
The list goes beyond the regular solids. The cuboctahedron and the icosidodecahedron are quasi-regular solids, made by cutting the corners off a regular solid at the midpoints of its edges. Every edge is the seam between the same two kinds of face, so there is only one fold angle. The rhombic dodecahedron and the rhombic triacontahedron are their duals, and all of their faces are congruent rhombi. The rhombic dodecahedron in particular fills space with no gaps: it is the shape of a garnet crystal and of the base of a honeycomb cell. A net folds back through about 54.74 degrees for the cuboctahedron, about 37.38 for the icosidodecahedron, exactly 60 for the rhombic dodecahedron and exactly 36 for the rhombic triacontahedron.
A cylinder and a cone are on the list as well. The cylinder net is a rectangle as wide as the base circumference and as tall as the cylinder, with a circle touching it at the top and another at the bottom. The cone net is a sector and a circle, and the central angle of the sector is 2π times the base radius divided by the slant height, so a radius exactly half the slant gives a semicircle. The curved sides are not flat, so they are cut into thin strips and wedges and rolled up a little at a time. The display panel changes how fast the nets open.
A triangular prism, a square pyramid and a frustum are on the list as well. The prism net is a strip of three rectangles with a triangle at each end of the middle one; the pyramid net is a square with a triangle on each of its four sides. A frustum is a cone cut short, so its lateral net is the sector of that cone with the tip sector removed, part of an annulus.