Convex Polyhedra with Regular
Polygonal Faces
David McKillop
Making Math Matter Inc.
Visualization and Logical Thinking
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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?
Making Math Matter Inc.
Visualization and Logical Thinking
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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?
Making Math Matter Inc.
Regular Polyhedra
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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
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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?
Making Math Matter Inc.
Uniform Prisms
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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 ngon attached on opposite sides of the belt.
Making Math Matter Inc.
Visualization and Logical Thinking
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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?
Making Math Matter Inc.
Uniform Antiprisms
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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.
Making Math Matter Inc.
How are these sets of
polyhedra alike? Different?
Regular
Polyh edra
Uniform
Prisms
1
1
Uniform
Antiprisms
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Deltahedra
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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
4
4
{3,3,3}
6
8
5
6
2@{3,3,3}; 3@{3,3,3,3}
10
12
7
8
5@{3,3,3,3}; 2@{3,3,3,3,3}
14
16
9
10
3@{3,3,3,3}; 6@(3,3,3,3,3}
2@{3,3,3,3}; 8@{3,3,3,3,3}
21
24
20
12
{3,3,3,3,3}
30
{3,3,3,3}
4@{3,3,3,3}; 4@{3,3,3,3,3}
No. of
Edges
6
9
12
15
18
The Convex Deltahedra
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The Convex Deltahedra
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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)
Making Math Matter Inc.
How are these sets of
polyhedra alike? Different?
Conv ex
Deltahedra
Regular
Polyh edra
Uniform
Prisms
5
2
1
1
1
Uniform
Antiprisms
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The Archimedean Solids
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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?
Conv ex
Deltahedra
Regular
Polyh edra
Uniform
Prisms
1
2
1
1
Archimedean
Solids
Uniform
Antiprisms
Making Math Matter Inc.
Johnson Solids
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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)
Making Math Matter Inc.
Convex Polyhedra With
Regular Polygonal Faces
Johnson
Solids
Conv ex
Deltahedra
87
5
Regular
Polyhedra
Uniform
Prisms
Uniform
Antiprisms
Archimedean
Solids
13
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