convex polyhedra with regular polygonal faces david mckillop making math matter inc
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Convex Polyhedra with Regular Polygonal Faces
David McKillop
Making Math Matter Inc.
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@{3,3,3}; 3@{3,3,3,3} 9
8 6 {3,3,3,3} 12
10 7 5@{3,3,3,3}; 2@{3,3,3,3,3} 15
12 8 4@{3,3,3,3}; 4@{3,3,3,3,3} 18
14 9 3@{3,3,3,3}; 6@(3,3,3,3,3} 21
16 10 2@{3,3,3,3}; 8@{3,3,3,3,3} 24
20 12 {3,3,3,3,3} 30
The Convex DeltahedraMaking Math Matter Inc.
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?
1 1
Uniform Antiprisms
Uniform Prisms
Regular Polyhedra
Convex Deltahedra
2
1
5
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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?Making Math Matter Inc.
How are these sets of polyhedra alike? Different?
11
2
1
Archimedean Solids
Uniform Antiprisms
Uniform Prisms
Regular Polyhedra
Convex Deltahedra
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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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ArchimedeanSolids
UniformAntiprisms
UniformPrisms
RegularPolyhedra
JohnsonSolids
ConvexDeltahedra
13
5
87
Convex Polyhedra With Regular Polygonal Faces
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