Hyperbolized Platonic Solids

I created these unique 3D models as part of a scientific study on particle packing. I thought they were too beautiful not to share!

What are Platonic Solids?

Platonic solids are geometric shapes where each face is an identical, regular polygon. They include:

• Tetrahedron (4 faces)
• Hexahedron (cube, 6 faces)
• Octahedron (8 faces)
• Dodecahedron (12 faces)
• Icosahedron (20 faces)

As the number of faces increases, these solids become more spherical but always remain strictly convex.

What is Hyperbolization?

By applying a geometric transformation known as 'hyperbolization,' I could manipulate the convexity of these solids. This process allows each solid to morph continuously from a perfect sphere to a highly angular version of itself.

To maintain the volume constant during these transformations, I rescaled the solids. This makes the changes appear more fluid and allows for easy comparison between different stages of transformation.

Some of these hyperbolized shapes might look familiar. For example, a decorated hyperbolic dodecahedron has been used as the logo for a well-known symbolic math package for many years. Check it out here: Hyperbolization of a Dodecahedron.

Available Files

• STL Files: Snapshots of the hyperbolized solids for alpha values of -1, -0.5 (sphere), 0 (original), 0.5, and 1.0 (most angular).
• Movies: Continuous transformations of each solid for alpha values ranging from -1 to 1.

I hope you find these models as fascinating and useful as I do!