Solid Geometry Notebook
A net is what remains when a polyhedron is cut along some of its edges and pressed flat — and folding is the reverse journey from sheet to solid.
Cut enough edges of a convex polyhedron to let the surface open like a box, and the faces spread into one connected arrangement of polygons: a net. The uncut edges become fold lines; the cut edges become the boundary of the sheet.
The cube is the standard exercise, and it is fully solved. Six squares can be joined edge-to-edge in 35 distinct ways — the hexominoes — and exactly eleven of those arrangements fold into a cube. The other twenty-four fail in one of two ways: two squares try to occupy the same face, or some face is left open.
| Strip pattern | Nets |
|---|---|
| 1-4-1 — four in a row, one above, one below | 6 |
| 1-3-2 — a triple, a single, a pair | 3 |
| 2-2-2 — the zigzag | 1 |
| 3-3 — two offset triples | 1 |
On narrow screens, swipe or scroll the plate sideways.
Two quick tests filter most failures before any folding starts. No cube net contains a 2×2 block, because the block's four squares share one point and four faces cannot meet at a single cube vertex. And no net has a straight run of five squares, because the cube has only four faces around any band.
Folding raises the reverse question: given a flat sheet with creases, what solid does it make? One sheet can sometimes fold into more than one polyhedron, and whether every convex polyhedron has some non-overlapping edge-cut net remains an open problem in general.
The idea is everywhere in practice. Cartons, sheet-metal housings, gift wrap, and sewing patterns are nets with tabs; the area of the flat blank equals the surface area of the finished solid, which is why net problems and surface-area problems are the same problem in two costumes.
Further reading