Modular Origami Unit Calculator
Units for a modular polyhedron from faces and vertex degree — checked by Euler, and reported for all three module conventions.
This counts units for an edge-modular design, where one module spans one edge — sonobe and its relatives. Other module families sit on faces or at vertices instead, and the tool reports all three counts rather than guessing which you are folding. Nothing is uploaded.
The unit count is the edge count, and Euler checks it
Every edge is shared by exactly two faces, so the edge count is the faces times the sides per face, halved — no lookup required. And the check is not approximate: vertices minus edges plus faces is exactly 2 for every solid in the table, which is what Euler guarantees for anything topologically a sphere.
| Solid | Faces | Edges | Vertices | V − E + F |
|---|---|---|---|---|
| Tetrahedron | 4 | 6 | 4 | 2 |
| Cube | 6 | 12 | 8 | 2 |
| Octahedron | 8 | 12 | 6 | 2 |
| Dodecahedron | 12 | 30 | 20 | 2 |
| Icosahedron | 20 | 30 | 12 | 2 |
So a cube takes 12 edge units, an icosahedron 30, and a tetrahedron six. Those are the numbers modular folders recognise, and they come out of two lines of counting rather than a table anyone had to memorise.
Which is why a unit count does not name a solid
A cube and an octahedron both take 12 units; a dodecahedron and an icosahedron both take 30. They are duals — one's faces are the other's vertices — and duals always share an edge count, so "a thirty-unit ball" is genuinely ambiguous before anyone has been unhelpful about it.
| Solid | On the edges | On the faces | At the vertices |
|---|---|---|---|
| Tetrahedron | 6 | 4 | 4 |
| Cube | 12 | 6 | 8 |
| Octahedron | 12 | 8 | 6 |
| Dodecahedron | 30 | 12 | 20 |
| Icosahedron | 30 | 20 | 12 |
And the convention matters as much as the solid does. The same icosahedron is 30 units on the edges, 20 on the faces and 12 at the vertices — three right answers to the same question, which is why two people can both be correct about a unit count and be building different things.
How to use
- Enter the number of faces and the sides on each face.
- Enter how many faces meet at a vertex.
- Read the edges, vertices and the Euler check.
- Take the unit count for your module convention.
Frequently asked questions
How many units does a modular ball take?
For an edge-modular design, one per edge — so the answer is the edge count. A cube takes 12, an icosahedron 30, a tetrahedron 6. Sonobe and its relatives all work this way.
How do I count the edges?
Faces times sides per face, halved, because every edge is shared by exactly two faces. No table needed: twenty triangular faces gives sixty face-sides, which is thirty edges.
What is the Euler check for?
Vertices minus edges plus faces is exactly 2 for any solid that is topologically a sphere, which all of these are. If your three inputs do not give 2, they do not describe a closed solid and the unit count is just arithmetic.
Why do a cube and an octahedron take the same number?
Because they are duals — the faces of one are the vertices of the other — and duals always share an edge count. The same is true of the dodecahedron and icosahedron, both 30.
So does a unit count tell me what I am building?
No, and this trips people up. Twelve units could be a cube or an octahedron; thirty could be a dodecahedron or an icosahedron. A unit count names an edge count, and edge counts are shared.
What if my modules sit on the faces?
Then the count is the faces, not the edges, and the tool reports all three conventions side by side. The same icosahedron is 30 units on the edges, 20 on the faces and 12 at the vertices — three right answers.
How many sheets of paper is that?
Divide the units by how many modules you cut from a sheet, and round up. Squares cut four to a sheet turn thirty units into eight sheets, with two modules spare.
Does this send anything anywhere?
No. Every figure is computed in your browser, and nothing is uploaded or stored.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.