# Jitterbug Defined Polyhedra:The Shape and Dynamics of Space

by

Robert W. Gray

Oct. 25-26, 2001

## Introduction

The study of the basic polyhedra is both a study in the properties of the space we live in as well as a source for basic design. Unfortunately, the relationships among the Platonic polyhedra (as well as some other basic polyhedra) are not taught as “basic knowledge” in the grade schools, nor in the Colleges and Universities. I will present an introduction to the basic polyhedra, showing how they are related to each other.

A new 120 triangular faced polyhedron will be introduced. It will be shown that this 120 Polyhedron provides a unifying vertex coordination for all of the polyhedra to be introduced.

The "Jitterbug", a dynamic polyhedron, will be demonstrated. It will be shown that the Jitterbug motion provides a dynamic means for defining these polyhedra and is, therefore, of fundamental importance to the dynamics of space itself.

The Golden ratio will be seen to occur throughout the polyhedra's relationships. This is both a fascinating "coincidence" of space as well as a visually pleasing source for basic design work.

## Topics

 Review: 5 Platonic (Regular) Polyhedra Review: Terminology and equations How the polyhedra are related one to another Rotating cubes The 120 Polyhedron The "Jitterbug" motion The Jitterbug motion defines the polyhedra Interesting Designs Appendix I: Vertex Coordinates Appendix II: Basic Data For The 120 Polyhedron Appendix III: A Comment on the Golden Ratio Appendix IV: Planes and Common Angles Defined by the 120 Polyhedron

## Review: 5 Platonic (Regular) Polyhedra  Tetrahedron Octahedron Cube  Dodecahedron Icosahedron

## Review: Terminology And Equations

### n-Fold Symmetric Rotation Axes

Three kinds of symmetry rotation axes (see above illustrations for examples):

1. Vertex to Vertex,
2. Mid-edge to Mid-edge,
3. Face to Face.

If a rotation of a polyhedron about a particular axis by an angular amount

0° < d < 360°
leaves the polyhedron in its initial position, then the axis is an n-fold symmetric rotation axis, where
n = 360° / d
and the polyhedron is said to be n-fold symmetric.

For example, the Tetrahedron's mid-edge to mid-edge axes: A rotation by 180° puts the Tetrahedron in the same positions as no rotation at all. Therefore, for this polyhedron and for this rotation axis, the Tetrahedron is 360° / 180° = 2-fold symmetric.

### Allspace Filling

Neither the Tetrahedron nor the Octahedron can fill all of space, without intersection, such that

1. only Tetrahedra appear in space
2. only Octahedra appear in space.

The Tetrahedron and the Octahedron can combine face to face to fill all space with Tetrahedra and Octahedra.

The Cube can fill all space by itself.

Neither the Dodecahedron nor the Icosahedron can fill all space, singly or in combination.

### Dual Polyhedra

To create the "dual" of a polyhedron, replace faces with vertices, and vertices with faces. (The following illustrations show the polyhedra scaled so that the dual polyhedra's edges intersect each other.)  Duals: Cube and Octahedron Duals: Dodecahedron and Icosahedron Self-Duals: Two Intersecting Tetrahedra

When scaled as shown, the Cube and Octahedron dual pair define the Rhombic Dodecahedron (shown in green). The Rhombic Dodecahedron fills all space.

 Cube and Octahedron Duals Define Rhombic Dodecahedron

The Icosahedron and the (regular) Dodecahedron dual pair define the rhombic Triacontahedron (shown in green).

 Icosahedron and Dodecahedron Duals Define Rhombic Triacontahedron

The 2 intersecting Tetrahedra self dual pair define the Cube (shown in green).

 Tetrahedra Self Duals Define Cube