4'5 More Platonic Solids

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4.5 More Platonic Solids

• Wednesday, March 3, 2004

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Some Relationships

• Faces of cube Vertices of Octahedron
• Vertices of cube Faces of Octahedron

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Duality

• Process of creating one solid from another
• Faces - - - Vertices

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Euler's polyhedron theorem

• V F - E 2

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Archimedean Solids

• Allow more than one kind of regular polygon to be
  used for the faces
• 13 Archimedean Solids (semiregular solids)
• Seven of the Archimedean solids are derived from
  the Platonic solids by the process of
  "truncation", literally cutting off the corners
• All are roughly ball-shaped

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Truncated Cube
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Some Relationships

• New F Old F Old V
• New E Old E Old V x number of faces that meet
  at a vertex
• New V Old V x number of faces that meet at a
  vertex

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Stellating

• Stellation is a process that allows us to derive
  a new polyhedron from an existing one by
  extending the faces until they re-intersect

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Two Dimensions The Pentagon
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Octagon
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How Many Stellations?

• Triangle and Square
• Pentagon and Hexagon
• Heptagon and Octagon
• N-gon?

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What About 3-Dimensions?

• Assignment
• Page 286 III.2 Stellated Solids