Koch Extrusions on Platonic Solids

December 3, 2019
Joshua Holder

This project re-implements and extends Yuliya Semibratova’s 2013 MA198 IGL project Higher Dimensional Koch Curves. I have used Yuliya’s code for Koch extrusions on a tetrahedron in Python and re-implemented it in an interactive online web page using Javascript/WebGL. I further extended functionality to two more cases, the octahedron and icosahedron. After seeing Koch extrusions on a tetrahedron seemingly converge into a cube and extrusions on a cube seemingly converge into an octahedron, it was conjectured that the other Platonic solids would also converge to a Platonic solid. However, after completing the octahedron and icosahedron it was apparent that neither converge into a Platonic solid. A remaining possible explanation for the seeming convergence of the tetrahedron and cube are the dihedral angles between the faces of the polyhedra. The extrusions on a tetrahedron produce a tetrahedron on each face, and it’s possible that the dihedral angle between the extrusions create the cube shape.

Interactive Applications showing Koch Extrusions on Platonic Solids
Koch Extrusions on a tetrahedron
Tetrahedron

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Koch Extrusions on an octahedron
Octahedron

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Koch Extrusions on an icosahedron
Icosahedron

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