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✴️ Kepler-Poinsot Star Polyhedra

Interactive Kepler-Poinsot star polyhedra simulator. Rotate the small stellated dodecahedron, great dodecahedron, great stellated dodecahedron and great icosahedron, built by stellating icosahedra and dodecahedra with golden-ratio geometry.

Geometry3DAdvanced60 FPS

✴️ What It Demonstrates

The Kepler–Poinsot polyhedra are the only four regular star polyhedra: non-convex solids whose faces are regular polygons or star polygons (pentagrams), arranged identically around every vertex, just like the Platonic solids — except their faces intersect each other. Their Schläfli symbols use the fraction 5/2 to denote a pentagram ("star pentagon", formed by the diagonals of a regular pentagon): small stellated dodecahedron {5/2,5}, great dodecahedron {5,5/2}, great stellated dodecahedron {5/2,3} and great icosahedron {3,5/2}. Kepler described the first two in 1619; Poinsot found the other two in 1809, and Cauchy proved in 1813 that these four complete the list.

This simulator builds each solid geometrically from an icosahedron or dodecahedron using golden-ratio (φ) stellation: pentagonal faces are extended outward through their edge-midpoint directions to form five-pointed star faces, and pentagon "vertex-figure" faces are assembled directly from the underlying icosahedron's vertex neighbours. The construction is a faithful approximation for visualization — real stellation diagrams use exact face-plane intersections, which this demo simplifies with a fixed φ-based scale factor.

How to Use

Pick a solid from the Solid dropdown.
Toggle Auto-rotate and adjust its Speed, or drag the canvas to rotate manually.
Toggle Thick lines and Colour by face for different rendering styles.
Read the Statistics panel for the true published V, E, F counts and Euler characteristic of each solid.

Did You Know?

Because their faces self-intersect, the naive Euler formula V − E + F = 2 fails for these solids: the small stellated dodecahedron and great dodecahedron give V − E + F = 12 − 30 + 12 = −6, while the great stellated dodecahedron and great icosahedron give 20 − 30 + 12 = 2 and 12 − 30 + 20 = 2 respectively — matching by coincidence, not by the usual topological argument. The correct generalized formula uses each face's density (how many times it winds around the solid's centre); accounting for density restores a consistent topological picture for all four. Kepler-Poinsot solids also appear in the small stellated dodecahedron's dual relationship with the great dodecahedron — each pair shares an edge arrangement.

About this simulation

This simulator constructs the four Kepler–Poinsot star polyhedra from an icosahedron/dodecahedron base using golden-ratio (φ) stellation, then renders them as rotating wireframes on a 2D canvas with a hand-written 3D projection. Real published V, E, F values (not the simplified render mesh) are shown in the statistics panel.

🔬 What it shows

Pentagon faces of a dodecahedron (or icosahedron vertex-figure pentagons) extended outward through the golden ratio to form five-pointed star spikes, producing self-intersecting non-convex solids — a simplified but geometrically motivated stellation.

🎮 How to use

Choose a solid from the dropdown. Enable Auto-rotate and set Speed, or drag to rotate manually. Toggle Thick lines and Colour by face for different rendering styles. The Statistics panel lists the true Schläfli symbol, V, E, F and Euler characteristic for each solid.

💡 Did you know?

Louis Poinsot described two of the four star polyhedra in 1809, unaware that Kepler had already found the other two nearly two centuries earlier in 1619 while studying star polygons and planetary orbits.

Frequently asked questions

What is a pentagram, and why does {5/2} mean one?

A pentagram is the five-pointed "star pentagon" traced by the diagonals of a regular pentagon. In Schläfli notation {n/d}, the fraction 5/2 means "connect every second vertex of a 5-sided figure," which produces the star shape instead of the ordinary pentagon {5}.

Why are these called "regular" if they self-intersect?

Regularity in the Schläfli sense only requires that all faces are congruent regular polygons (or star polygons) and all vertices are surrounded identically — it says nothing about convexity. The four Kepler-Poinsot solids satisfy this strict local regularity even though their faces pass through each other globally.

Why does V − E + F not equal 2 for these solids?

Euler's formula V − E + F = 2 assumes the solid's surface is topologically a simple sphere. Self-intersecting star polyhedra have faces that wind around the centre more than once (a property called density), which breaks that assumption and produces different values of V − E + F.

How is the great dodecahedron built here?

Its twelve pentagonal faces are the "vertex figures" of an icosahedron: for each of the icosahedron's 12 vertices, the five neighbouring vertices are coplanar (by symmetry) and form a regular pentagon, which becomes one face of the great dodecahedron.

Are these solids found in nature or art?

Not in natural crystals, since true 5-fold symmetry is forbidden in periodic crystal lattices, but stellated polyhedra are popular in mathematical art, papercraft models, and appear on the crest of some universities and in Escher-inspired artwork celebrating polyhedral geometry.

⚙ Under the hood

Interactive Kepler-Poinsot star polyhedra simulator. Rotate the small stellated dodecahedron, great dodecahedron, great stellated dodecahedron and great icosahedron, built via golden-ratio stellation, and see why V-E+F does not equal 2 for these self-intersecting solids.

Canvas 2DStar PolyhedraStellationGolden RatioNon-Convex

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install