Convex Polyhedra with Regular Polygonal Faces David McKillop Making Math Matter Inc.

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?


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?




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


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?


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.


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?




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.


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.


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)



1 1

Uniform Antiprisms

Uniform Prisms

Regular Polyhedra

Convex Deltahedra

2

1

5


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.


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2

1

Archimedean Solids

Uniform Antiprisms

Uniform Prisms

Regular Polyhedra

Convex Deltahedra


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)

Making Math Matter Inc.Making Math Matter Inc.

ArchimedeanSolids

UniformAntiprisms

UniformPrisms

RegularPolyhedra

JohnsonSolids

ConvexDeltahedra

13

5

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Convex Polyhedra With Regular Polygonal Faces


Convex Polyhedra with Regular Polygonal Faces David McKillop Making Math Matter Inc.

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