Nets: which flat shapes fold into which solid

0:00Here is a cube. Cut along enough of its edges and the whole surface opens out and lies flat. That flat piece is called a net. A net is not a drawing of the solid. It is the solid's surface, moved. Nothing was stretched and nothing was torn, so every distance along the surface is still what it was. Which means folding it back up is not a guess. It is an operation.

0:28It either works, or it fails for a reason you can point at. We need three words first, and they are the obvious three. A face is one of the flat surfaces the solid is made of. An edge is where two faces meet. A vertex is a point where edges meet. Count them on a cube. Six faces. Twelve edges: four round the top, four round the bottom, four uprights. Eight vertices.

0:59Six, twelve, eight, counted off the solid rather than remembered. A tetrahedron gives four, six and four. An octahedron gives eight, twelve and six. And for all three, faces plus vertices, less edges, comes to two. Two families are worth defining properly. A prism has two identical polygons facing each other, and flat pieces joining them all the way round. A pyramid has one polygon and one extra point off it, with an edge from that point to every corner.

1:38Now count without drawing. Take a prism whose ends have n sides. Faces: the two ends, plus one for each side of an end. That is n plus two. Edges: n round each end, and n uprights. Three n. Vertices: n at each end. Two n. So a prism on ten-sided ends has twelve faces, thirty edges and twenty vertices. A pyramid on n sides has n plus one faces, two n edges and n plus one vertices. On ten sides, eleven, twenty and eleven.

2:16And the check: put n equal to four into the prism and you get six, twelve and eight. The cube. Back to the flat piece. Six squares, joined edge to edge. Watch one square at a time as it folds up. Front. Right. Back. Left. The row of four has wrapped into a band, and the two squares left over drop onto the top and the bottom. Six squares, six faces, one each. That is the whole test.

2:47The little tabs on a cardboard box are for the glue. The net is the surface, and the surface has no tabs. When you draw one, the arithmetic checks it before you cut, because the faces come in three matching pairs. A box five by three by one has a surface of forty-six. One six by three by two has seventy-two. Now six flat pieces. Every one of them is six squares joined edge to edge.

3:20Which of them fold into a cube? Four of them work. Two rows of three, overlapping in a single column, folds. A staircase of three dominoes, each stepped one across from the one before, folds. A row of four with one square above the third and one below the third folds. And a row of four with one above the third and one below the second folds too. The other two do not. All six had exactly six squares, so the count is not what tells them apart.

3:56So what actually goes wrong? Take the first failure. A row of four with two squares sitting above its left end. The row wraps into a band, and the first of the two lands on the top. The second comes down onto a face of the band that already has a square on it. Two squares, one face. And the arithmetic agrees: five faces get covered, and one is left bare.

4:24It is tempting to call that a missing face. Do not. The bare face is the consequence. The fault is the overlap. The other failure does the same thing, with two squares stacked below the second column. A row of four always wraps into a band of four side faces. That part never fails. So take a row of four, put one square anywhere above it, and one square anywhere below it.

4:54The one above becomes the top. The one below becomes the bottom. It folds. Every time. Six of a cube's eleven nets are of exactly that shape. Which leaves the harder question. Why eleven? Eleven is not a fact standing on its own. It is a fact about a convention. Two flat pieces count as the same net if you can turn one round, or flip it over, and land on the other.

5:25Change the convention and the number changes with it. With it fixed, the count is done properly by listing. There are thirty-five different shapes you can build from six squares joined edge to edge, once turns and flips are not counted as different. Then fold every one of them. Eleven close up into a cube. The other twenty-four do not. Not eleven because somebody said so. Eleven because thirty-five were tried.

5:58Here is that number again, from an argument that never lays anything flat. Cutting a cube open is choosing which of its edges to cut. Its faces meet along twelve joins. What stays joined has to reach every face and must not run round in a circle, so you keep five of the twelve. There are three hundred and eighty-four ways to do that. But a cube has forty-eight turns and flips of its own, and two cuts related by one of those give the same net.

6:32Sort the three hundred and eighty-four into groups on that basis, and eleven groups remain. Two arguments, no machinery in common, the same eleven. Run it on a regular tetrahedron and you get two. Run it on an octahedron and you get eleven again. Not a coincidence. An octahedron's eight faces meet in exactly the pattern a cube's eight corners do. Now the tetrahedron. Four equilateral triangles, and four candidates.

7:08The first is a big triangle cut into four by joining the midpoints of its sides. Fold the outer three up and they meet at a point. A net. The third is four triangles in a straight strip. Roll it round and it closes. That is the other one. The second is a strip of three with a fourth bent back off the end. Four triangles, the right number. And two of them land on the same face. Same fault as the squares.

7:40The fourth candidate has five triangles for a four-faced solid, which no amount of folding will fix. Two nets out of four, and the two failures fail for completely different reasons. Two of these surfaces have no flat faces at all, and they still have nets, because they were rolled up rather than folded. Slit a tin down its side and the curved part unrolls into a rectangle. One side of that rectangle is the height. The other is not the diameter, and it is not the radius.

8:18It is the whole way round the base. Two pi r. So the surface of a tin is that rectangle and two circles. Radius three and height five gives forty-eight pi. Now the cone. Slit it up its slant and unroll it. Every point of the base circle is the same distance from the apex, so once it lies flat they all sit on a circle of that radius. You get a piece of a disc. Not a triangle.

8:48How much of the disc? The radius divided by the slant. The slant is always longer, so it is never the whole of it. Last, a ball. Try to wrap one in paper with no wrinkle, no gap and no overlap. You cannot, and the reason is the one we started with. Folding moves paper without stretching it, so it cannot change how much angle sits at a point. Flat paper has a full turn, three hundred and sixty degrees, at every point inside it.

9:23At a cube's corner, three squares meet. Two hundred and seventy degrees arrive where three hundred and sixty would lie flat. The corner is ninety degrees short, and that shortfall is what the fold used up. A tetrahedron's corner is short by a hundred and eighty. An octahedron's by a hundred and twenty. Add the shortfalls over a whole solid and you get seven hundred and twenty degrees. Cube, tetrahedron, octahedron, every prism, every pyramid.

9:58A ball has no corners. It is smooth everywhere, so there is nothing for the fold to take up. A net is not a picture of a surface. It is the surface, and a surface either goes flat or it does not.