Polyhedron Net Maker

A paper model looks like a job for scissors and patience. Most of it is settled earlier, on the flat sheet, by two decisions: which edges of the solid get cut, and how far each remaining edge has to bend.

Get both right and the last tab meets its edge without a fight. Get the first one wrong and two faces land on the same patch of paper. Get the second one wrong and the model closes with a gap.

A net is a list of cuts

A net is what a polyhedron becomes when its surface is cut along some edges and opened flat, with every face still attached to at least one neighbor. The idea is old. Albrecht Durer printed nets in his 1525 treatise on measurement with compass and ruler, one of the first presentations of the device.

Not every set of cuts works. Cut all the edges around one face and that face drops out of the sheet. Leave a corner with no cut edge at all and the faces around it cannot lie flat, because on a convex solid they add up to less than a full turn. So every corner needs a cut, and the cuts must never close a loop.

Those two rules describe a familiar object. The cut edges form a spanning tree of the solid’s corners: a connected network that reaches every vertex and contains no circuit. The edges that are left are the fold lines, and they join the faces into a tree of their own. Choosing a net is choosing a tree.

That makes the bookkeeping simple. A tree on V corners has V - 1 edges, so that is the number of cuts. The faces hang together by F - 1 folds. Take a dodecahedron, with 20 corners, 30 edges and 12 faces. Any one-piece net of it has 19 cuts, 11 folds, and nothing else, because 19 and 11 make 30.

The sum had to come out. Add the cuts and the folds for any convex solid and you get V - 1 plus F - 1 edges, which must be all of them. Rearranged, that is V - E + F = 2, the polyhedral formula. Leonhard Euler wrote about it in 1750 and went on to publish it, and it has since been proved in a great many different ways.

A flat pattern is a good place to watch the formula work. A cube has 8 corners, 12 edges and 6 faces, so a cube net has 7 cuts and 5 folds. A regular icosahedron, with 12 corners and 20 faces, has 11 cuts and 19 folds, the dodecahedron’s numbers swapped, since the two solids trade corners for faces. The count fixes how many cuts there are. It does not fix which ones. For a cube the possible choices come down, after turning and flipping, to 11 nets: the familiar cross and ten others. The octahedron has the same number.

Two families, one made from the other

The solids most people want to build fall into two families. The Platonic solids use one kind of regular polygon with every corner alike, and there are exactly five: tetrahedron, cube, octahedron, dodecahedron and icosahedron. Euclid argued that the list is complete at the end of the last book of the Elements.

Allow two or more kinds of regular polygon, keep every corner the same, and the list grows to 13 solids, the Archimedean family. Archimedes described them, his account was lost, Renaissance artists found most of them again, and Kepler reconstructed the whole set in 1619. Each is named by the faces around one corner: 5.6.6 for the truncated icosahedron of the soccer ball, 3.4.3.4 for the cuboctahedron.

They are not thirteen separate inventions. Slice the corners off a Platonic solid and each corner becomes a new face while each old face gains sides. That is truncation, and cutting just deep enough for neighboring cuts to meet is the special case called rectification. Push the faces of a solid apart and fill the gaps, and more appear, the rhombicuboctahedron among them. The snub cube and snub dodecahedron come from moving the faces outward while giving each a twist, then filling the spaces with equilateral triangles.

There is a tidier way to see the whole family. Stand a point between three mirrors arranged like the symmetry planes of a cube or a dodecahedron, and its reflections are the corners of a solid. Put the point on a mirror or between them, keep it the same distance from every mirror it does not touch, and the reflections give one member of the family after another. This is the Wythoff construction.

Prisms and antiprisms also have regular faces and identical corners, in endless supply. They are left out of the Archimedean list because their symmetry is of a lower kind. Beyond them lie the Johnson solids, convex and regular-faced but with corners that differ, such as a square pyramid with four equilateral sides.

The angle that closes the model

The dihedral angle is the angle between two faces along their shared edge, measured inside the solid. It can be read from the directions the two faces point: the cosine of the angle between two planes is the dot product of their unit normals. For the five Platonic solids the values are 70.529 degrees for the tetrahedron, 90 for the cube, 109.471 for the octahedron, 116.565 degrees for the dodecahedron and 138.190 for the icosahedron.

On the flat sheet two neighboring faces lie in one plane, which is a dihedral angle of 180. Folding brings that down to the true value, so each fold turns by 180 minus the dihedral angle. A cube’s folds turn 90 degrees. An icosahedron’s turn under 42, which is why its net looks nearly finished after a gentle curl.

Here is a case to try. Choose the dodecahedron with a 30 mm edge. The tool reports 12 faces, 30 edges and 20 corners, a surface of 185.8 square centimeters and a volume of 206.9 cubic centimeters, and a net that fits one A4 sheet. Each fold turns 63.435 degrees. Move the fold slider and every pentagon swings on its hinge by that amount, parents carrying children, until the open edges meet.

That meeting is the real test of a net. A drawing can have the right faces in the wrong arrangement and still look plausible. Folding it in arithmetic, hinge by hinge, and checking that every corner returns to its place on the solid is the proof that the cuts and the angles agree. One hinge wrong by a single degree is enough to leave corners visibly apart.

Overlaps, and a question nobody has settled

A tree guarantees that the faces stay connected. It does not guarantee that they stay out of each other’s way. Laid flat, two branches of a net can swing around and cover the same paper, and such a pattern cannot be cut from one sheet. This happens with convex solids too. Overlapping unfoldings of convex polyhedra have been found, a skinny tetrahedron among them. Unfold the truncated icosidodecahedron, with its 62 faces, as one long strip and from 50 of those faces as a start the strip runs over itself. Starting somewhere else, or branching instead of winding, clears it.

Whether such a fix always exists is an open problem. In 1975 the British mathematician G. C. Shephard asked whether every convex polyhedron can be cut along edges and unfolded to a single flat piece that does not overlap. Nobody has proved it and nobody has found a counterexample, though the common hunch is yes. Rules for picking the tree have been put forward as always safe, and a 2006 thesis built convex solids on which every unfolding of several such kinds overlaps, which disproved some of those conjectures.

Drop convexity and the answer is known to be no. There are polyhedra with 24 convex faces, and others with 36 triangular faces, that cannot be unfolded along edges at all. So a net maker can promise only what it has checked: for a given solid it can search for a tree that works, and say so plainly when it has had to split the pattern into pieces.

Tabs, numbers and a clean fold

Every cut edge shows up twice on the outline of a net, once on each of the two faces it used to join. Gluing needs an overlap, and one flap per edge is enough. In the one-tab method a tab sticks out from one of the two faces, folds back, and is glued to the inside of the other. A dodecahedron therefore needs 19 tabs, one for each cut.

Where the tab goes matters less than whether it fits. In a tight corner there may be little room: where a square, a hexagon and a decagon of a 4.6.10 corner lie flat together, only 6 degrees remain. A tab there has to be trimmed to a sliver or moved to the other copy of the edge. Matching numbers on both copies tell the builder which edges meet.

Other methods exist. Two tabs per edge, glued to each other, make a rib inside every edge and can be held with a clip while the glue sets. Cutting bare polygons and taping them from the inside is faster. Glued tabs give the strongest models, at the cost of the occasional fingerprint. The folding itself is plain craft. Score each fold line before bending it, so the paper creases on the line and not beside it. Close the tightest corners first and leave a large face for last, so a finger can still reach inside.

What the arithmetic cannot supply is the paper. A drawn net has no thickness, and card does, so real edges sit a little proud and the last seam always wants coaxing. The method also stops at regular faces and at solids small enough to search. Within those limits, a pattern that folds shut in the numbers will fold shut on the desk.