Math 1536

1
Math 1536

• 6.1 Regular Polyhedra
• 6.2 Semiregular Polyhedra

2
Review

• Before break, we studied (in 2D)
• Regular tilings
• Semiregular tilings
• Dual tilings
• Schläfli symbols
• Vector formulas for angles and area

3
Looking ahead

• For the rest of the semester, well be studying
  the analogous 3-D concepts.
• So the ideas should be familiar, but some of the
  problems will be challenging to visualize.

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Polyhedron

• A polyhedron is a closed 3-dimensional surface
  made up of polygons, which line up along the
  edges of the polygons so that the vertices match.

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Platonic Solids

• There are only five regular polyhedra. (We built
  these!)
• tetrahedron, cube, octahedron, dodecahedron,
  icosahedron
• These are sometimes called the Platonic Solids.

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Eulers Formula

• For any polyhedron, F V E 2where F
  number of faces V number of vertices E
  number of edges

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Eulers Formula (example)

• The tetrahedron hasfour faces,four
  vertices,and six edges.4 4 6 2

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Vertex Valence

• The valence of a vertex is the number of edges
  that shoot out from that vertex.
• In a cube, for example,every vertex has a
  valence of 3.

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Face Valence

• The valence of a face is simply the number of
  edges (sides) that surround the face.
• Since the faces of an icosahedron are triangles,
  every face has a valence of 3.

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The Valence Formula

• For any polyhedron, TVV TFV 2Ewhere
  TVV total vertex valence (?q(v)) TFV total
  face valence (?p(f)) E number of edges

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Spherical Deviation

• The spherical deviation at a vertex is the angle
  that would need to be added to the angles already
  present at that vertex to make a total of 360º.

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Spherical Deviation (example)

• In the octahedron,every vertex touchesfour 60º
  angles, for a total of 240º.
• The spherical deviationat each vertex is 120º.

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Descartes Formula

• (Note the book spells this wrong.)
• For any polyhedron, TSD 720ºwhere TSD
  total spherical deviation (??(v))

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Truncation

• Truncation is the process of slicing off
  vertices of a polyhedron in order to create a new
  (different) polyhedron.

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Truncation (example)

• Start with a regular tetrahedron. Choose a
  vertex, and make a mark 1/3 of the length along
  each edge that extends from that vertex. Cut off
  the corner at that vertex, then do the same thing
  for each of the other vertices.

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Truncation (example)

• The result is the truncated tetrahedron, and is
  pictured in the margin on page 114.

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Truncated tetrahedron

• Notice that the truncated tetrahedron is not
  regular (it has two different kinds of faces).
• But every vertex has the same Schläfli symbol
  3.6.6. When every vertex has the same symbol,
  the polyhedron is semiregular.

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Group Work

• In groups of 2 or 3 people, complete exercises 2
  and 3 on page 113. (Your drawings dont have to
  be perfect.)
• Also find the Schläfli symbol for each of the
  resulting polyhedra.
• Do not hand these in!