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Chapter 5 : Regular Polytopes and Fold-Outs
Study Questions and Projects
Introduction
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How many regular polyhedra in three-space can you identify?
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What kinds of symmetries do the various polyhedra (and the polygons in the plane) exhibit?
The Greek Geometry Game
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How did the Greeks construct a regular triangle, using only a straightedge and a compass? A hexagon? A dodecagon?
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Why is the Greek method not sufficient to construct an enneagon?
The Search for Regular Polyhedra
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What are the requirements for polyhedra to be regular?
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What are the five regular polyhedra?
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How do we know that there are no more than five regular polyhedra in three-space?
Duals of Regular Polyhedra
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What is the "dual" of a polyhedron? What are the examples of duality in three-space?
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What does it mean for a polyhedron to be "self-dual"? Which polyhedron in three-space is self-dual?
The Search for Regular Polytopes
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How is a polytope in four-space constructed? What is the general pattern for a polytope in n-dimensional space?
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Besides the cube, which regular polyhedra (and in what quantities) might fit around an edge?
The Four Simplex
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What is a simplex, and how can it be constructed in three-space? In four-space? In n-dimensional space?
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What are the two different ways of projecting a four-simplex into three-space?
The Hypercube Dual or Sixteen-Cell
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What is the four-dimensional analogy to the self-dual tetrahedron in three-space?
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What are the properties of the 16-cell (number of vertices, polyhedra, etc.), and how do we determine them?
Polytopes in Five or More Dimensions
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Why can we construct the three polytopes mentioned above in every dimension?
The Regular 600-Cell and Its Dual
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What are the component regular polyhedra of the 600-cell? Of its dual?
Fold-Out Patterns in Different Dimensions
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How would the three-dimensional fold-out pattern for the four-simplex look?