📦 Polyhedron Nets (Unfolding 3D Solids)

📦 What It Demonstrates

A polyhedron net is a flat, two-dimensional pattern of connected polygons that can be folded along shared edges — like cardboard creases — to form a closed three-dimensional solid with no gaps and no overlaps. This simulator computes a genuine net for each of the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) directly from their real 3D vertex coordinates: it finds every face by testing which points are coplanar on the convex hull, builds a graph of which faces share an edge, and then grows a spanning tree through that graph starting from one root face. Every other face is attached to its parent by hinging it flat around their shared edge, so the whole solid unrolls into one connected, non-overlapping flat pattern.

Moving the Fold Amount slider interpolates every face, independently, between two exact rigid poses: its flat, net-space position (fold = 0%) and its true folded 3D position and orientation (fold = 100%). Each pose is stored as a rotation plus a translation, and the rotations are blended using quaternion spherical interpolation (slerp) while the translations are linearly interpolated, so every face sweeps smoothly and continuously along its own hinge path as the solid folds shut or springs open.

How to Use

• Choose a Solid — tetrahedron, cube, octahedron, dodecahedron or icosahedron.
• Drag the Fold Amount slider: 0% shows the flat net, 100% shows the fully folded solid, and anything in between shows the hinges opening or closing.
• Press Play to animate the fold amount back and forth automatically, or Reset to return to the fully folded solid.
• Enable Auto-rotate to spin the model so you can see the fold from every angle, most useful once it is mostly folded.
• Enable Show face labels to number each face — watch the same number travel from its position in the flat net to its final position on the 3D solid.

Did You Know?

A given solid can usually be unfolded in many different valid ways, depending on which spanning tree of faces you choose to cut along. The humble cube — despite being the simplest Platonic solid after the tetrahedron — has exactly 11 distinct nets up to rotation and reflection, a fact first rigorously enumerated in recreational and combinatorial geometry and now a classic result quoted in mathematics education. Some are the familiar "cross" pattern, others are much less obvious zig-zag arrangements. The icosahedron, with its 20 triangular faces, has an astonishing 43,380 distinct nets, while some non-convex polyhedra are conjectured to have no valid net at all — an open problem in geometry known as Shephard's conjecture for convex solids (proven true) versus general polyhedra (still unresolved for some shapes).

About this simulation

This simulator computes real polyhedron nets for the five Platonic solids from their actual 3D vertex coordinates — it finds coplanar faces on the convex hull, builds a face-adjacency graph, grows a spanning tree from a root face, and hinges every other face flat around its shared edge. Folding is animated by blending, per face, between its flat net pose and its true 3D pose using quaternion slerp for rotation and linear interpolation for position, then projecting the result with a simple hand-rolled 3D-to-2D projection and painter's-algorithm face sorting.

Frequently asked questions