Convex Polyhedra with Regular Polygonal Faces

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Convex Polyhedra with Regular Polygonal Faces

• David McKillop

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Visualization and Logical Thinking

• Close your eyes and visualize a regular
  octahedron
• Visualize its faces How many? What shapes?
• Visualize its vertices Where are they located?
  How many? Is there vertex regularity?
• Visualize its edges Where are they located? How
  many?
• Visualize one of its nets What do you see?

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Visualization and Logical Thinking

• Close your eyes and visualize how you constructed
  a regular icosahedron
• Visualize its faces How many? What shapes?
• Visualize its vertices Where are they located?
  How many? Is there vertex regularity?
• Visualize its edges Where are they located? How
  many?
• Visualize one of its nets What do you see?

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Regular Polyhedra

• There are only 5 of these 3-D shapes regular
  tetrahedron, cube, regular octahedron, regular
  dodecahedron, regular icosahedron
• Each shape has only one type of regular polygon
  for its faces
• They have vertex regularity
• All angles formed by two faces (dihedral angles)
  are equal

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Visualization and Logical Thinking

• Close your eyes and visualize a uniform
  decagon-based prism
• Visualize its faces How many? What shapes?
• Visualize its vertices Where are they located?
  How many? Is there vertex regularity?
• Visualize its edges Where are they located? How
  many?
• Visualize one of its nets What do you see?

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Uniform Prisms

• Except for the uniform square prism (cube), there
  are two regular polygons of one type as bases (on
  parallel planes) and the rest of the faces are
  squares
• They have vertex regularity, usually 4,4,n but
  uniform triangular prism is 3,4,4
• A net of a uniform n-gonal prism is easily
  visualized as a regular n-gon with a square
  attached to each side and another n-gon attached
  to the opposite side of one of the squares, OR as
  a belt of n squares with an n-gon attached on
  opposite sides of the belt.

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Visualization and Logical Thinking

• Close your eyes and visualize how you would
  construct a uniform hexagonal antiprism
• Visualize its faces How many? What shapes?
• Visualize its vertices Where are they located?
  How many? Is there vertex regularity?
• Visualize its edges Where are they located? How
  many?
• Visualize one of its nets What do you see?

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Uniform Antiprisms

• Except for the uniform triangular antiprism
  (regular octahedron), there are two regular
  polygons of one type as bases (on parallel
  planes) and the rest of the faces are equilateral
  triangles
• They have vertex regularity, usually 3,3,3,n
• A net of a uniform n-gonal antiprism is easily
  visualized as two regular n-gons with an
  equilateral triangle attached to each side and
  these two configurations joined, OR as a belt of
  2n equilateral triangles with an n-gon attached
  on opposite sides of the belt.

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How are these sets of polyhedra alike? Different?
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Deltahedra

• Any 3-D shape constructed using only equilateral
  triangles is called a deltahedron
• There are an infinite number of deltahedra
  however, there is a finite number of convex
  deltahedra.

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No. of Faces No. of Vertices Vertex Configuration No. of Edges
4 4 3,3,3 6
6 5 2_at_3,3,3 3_at_3,3,3,3 9
8 6 3,3,3,3 12
10 7 5_at_3,3,3,3 2_at_3,3,3,3,3 15
12 8 4_at_3,3,3,3 4_at_3,3,3,3,3 18
14 9 3_at_3,3,3,3 6_at_(3,3,3,3,3 21
16 10 2_at_3,3,3,3 8_at_3,3,3,3,3 24
20 12 3,3,3,3,3 30
dipyramids
The Convex Deltahedra
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The Convex Deltahedra

• All faces are equilateral triangles
• They all have an even number of faces
• There are only 8 of them
• Only 3 of them have vertex regularity the
  regular tetrahedron, octahedron, and icosahedron
• 3 of them are dipyramids (6, 8, and 10 faces)

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How are these sets of polyhedra alike? Different?
5
2
1
1
1
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The Archimedean Solids

• Two or three different regular polygons as faces
• Always 4 or more of any regular polygon
• There are only 13 of these solids
• They have vertex regularity
• They are very symmetrical, looking the same when
  rotated in many directions

Why are uniform prisms and uniform antiprisms NOT
Archimedean solids?
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How are these sets of polyhedra alike? Different?
2
1
1
1
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Johnson Solids

• Have only regular polygons as faces (1 or more
  different types)
• They do NOT have vertex regularity
• There are only 92 of them (5 of them are convex
  deltahedra)

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Convex Polyhedra With Regular Polygonal Faces
87
5
13
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