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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

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Section 3.2 Semiregular Polyhedra

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Subsection 3.2.1 Defining Semiregular Polyhedra

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We now extend our collection of polyhedra by exploring what it means for a polyhedron to be semiregular. Note the resemblance to the definition of semiregular tiling.
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Definition 3.2.1.

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A polyhedron is semiregular if it satisfies the following conditions:
  • Every face is a regular polygon.
  • At least two polygonal faces have a different number of sides.
  • The arrangement of polygons at every vertex of the polyhedron is the same.
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The notation for a vertex arrangement of a regular or semi-regular polyhedron is the same as what we used for tilings. For example, the vertex arrangement for a cube would be 4.4.4 because there are three squares meeting at any vertex of the cube.

Checkpoint 3.2.2. Check your Understanding.

Which regular polyhedron corresponds to the notation 3.3.3.3.3?
Hint.
Does the number 3 correspond to the shape of the face or the number of faces that meet? Definition 3.2.1
Answer.
The icosahedron because five triangles meet at each vertex.
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Subsection 3.2.2 Examples of Semi-regular Polyhedra

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Exploration 3.2.1.

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(a) The Great Rhombicosidodecahedron.
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One example of a semi-regular polyhedron is the Great Rhombicosidodecahedron which is shown in Figure 3.2.3
 1 
Illustration produced by Robert Webb’s Stella software accessible via http://www.software3d.com/Stella.php.
. Check that each visible vertex has the arrangement 4.6.10.
The front side of a great rhombicosidodecahedron. Five decagonal faces, eight hexagonal faces, and eight square faces are clearly visible, but there is a hint of a square and hexagon alonge the bottom edge.
Figure 3.2.3. Great Rhombicosidodecahedron
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(i)
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Three polygons meet at each vertex of the great rhombicosidodecahedron. What is the sum of three angle measures at any one vertex?
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(b) The Icosidodecahedron.
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Another Archimedean solid is the icosidodecahedron Figure 3.2.4
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Illustration produced by Robert Webb’s Stella software accessible via http://www.software3d.com/Stella.php.
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A picture of an icosidodecahedron
Figure 3.2.4. The Icosidodecahedron
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(i)
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What is the vertex arrangement of the icosidodecahedron?
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(ii)
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How many pentagons are needed to create an icosidodecahedron? Be sure to count the ones you cannot see as well.
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(iii)
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How many triangles are needed to create an icosidodecahedron?
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(c) Other Semi-regular Polyhedra.
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Choose one of the following vertex arrangements, 3.6.6, 4.6.6, 3.4.4.4, 3.4.3.4, 3.10.10, or 5.6.6.
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(i)
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Construct the semiregular polyhedron that has that has your chosen vertex arrangement.
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(ii)
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How many faces of each type does your semiregular polyhedron have?
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(iii)
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What is the total number of vertices?
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(iv)
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What is the total number of edges?
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(v)
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As a bonus, repeat this task for other vertex arrangements. If you are up for a challenge, you may choose 3.4.5.4, 3.3.3.3.4, or 3.3.3.3.5.
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Subsection 3.2.3 Families of Semiregular Polyhedra

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Exploration 3.2.2.

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(a) Semiregular Polyhedra with Vertex Arrangement 4.4.n.
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Build the following semiregular polyhedron which has the following vertex arrangements. You will need these solids to answer the remainder of the questions in this task.
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(iv)
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Without building 4.4.8 describe clearly what it would look like.
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(v)
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The polyhedra you just created are called prisms. Use your constructions to complete Table 3.2.5. The first row should list the number of faces, vertices, and edges in a semiregular solid with a 3.4.4-vertex arrangement.
Table 3.2.5. Attributes of Prisms
Shape of base Number of faces F Number of vertices V Number of edges E
Triangle 6
Quadrilateral
Pentagon
Hexagon
Heptagon (7) 9 14 21
Octagon
n-gon
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(b) Semiregular Polyhedra with Vertex Arrangement 3.3.3.n.
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Build the following semiregular polyhedron which has the following vertex arrangements. You will need these solids to answer the remainder of the questions in this task.
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(i)
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3.3.3.3 (a regular polyhedron)
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(v)
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Without building 3.3.3.8 describe clearly what it would look like.
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(vi)
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The polyhedra you just created are called antiprisms. Use your constructions to complete Table 3.2.6.
Table 3.2.6. Attributes of Antiprisms
Shape of base Number of faces F Number of vertices V Number of edges E
Triangle 6
Quadrilateral
Pentagon
Hexagon
Heptagon (7) 16 14 28
Octagon
n-gon
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Subsection 3.2.4 Defining Prisms and Antiprisms

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In this section we discovered two families of semiregular polygons: prisms and antiprisms. We can also create prisms and antiprisms which have irregular faces. In particular, the sides of a prism need not be squares, they could be rectangles or other parallelograms. The sides of an antiprism do not need to be equilateral. Moreover, the bases of a prism or antiprism may be any polygon whatsoever. Some examples of prisms are shown in Figure 3.2.8 and Figure 3.3.1
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Definition 3.2.7.

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A prism is a polyhedron consisting of two congruent parallel polygonal faces joined by sides that are parallelograms, often rectangles.
Prisms
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Figure 3.2.8. An assortment of right prisms
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In Task 3.2.2.b, you built and analyzed some antiprisms, similar to the one on the right in Figure 3.2.10.
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Definition 3.2.9.

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An antiprism is a polyhedron consisting of a pair of parallel congruent polygonal faces joined by sides that are triangles.
pentagonal prism and antiprism
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Figure 3.2.10. A pentagonal prism and pentagonal antiprism
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The two parallel congruent polygonal faces are called the bases of the prism or antiprism. The top base may be called the summit in which case the bottom would be called the base. We specify the type of prism or pyramid by shape of its base so that the prism with a 4.4.n-vertex configuration is called an n-gonal prism. Of course, we can set a polyhedron on any of its faces. Sometimes the base(s) will appear to be at the ends of the solid rather than at the top and bottom.
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Exercises 3.2.5 Exercises

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Building Our Toolbox

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1. Understanding the Definition of Semiregular Polyhedron.
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State the definition of semiregular polyhedron in your own words.
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(b)
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Of the three criteria in Definition 3.2.1, which one explains why a 3×4×5 rectangular prism (box) is not a semiregular polyhedron?
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(c)
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Could a pyramid with a square base be a semiregular polyhedron? Explain.
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2. Observing Euler’s Identity.
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(a)
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You have calculated F+V−E for the five regular polyhedra, several pyramids, and several prisms. What do you notice?
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Skills and Recall

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3. Counting Vertices, Faces, and Edges.
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An octadecagon is an 18-gon.
  1. How many vertices will an octadecagonal prism have?
  2. How many faces will an octadecagonal prism have?
  3. How many edges will an octadecagonal prism have?
  4. How many vertices will an octadecagonal antiprism have?
  5. How many faces will an octadecagonal antiprism have?
  6. How many edges will an octadecagonal antiprism have?
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4. Applying Euler’s Identity.
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A pentagonal cupola is a convex polyhedron with 12 faces and 25 edges. Determine how many vertices a pentagonal cupola must have.
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Extending the Concepts

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5. Dipyramids.
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The regular octahedron looks like two congruent square pyramids glued together. Solids formed in this way are called dipyramids. Thus, the regular octahedron is also called a square dipyramid. You should have built a triangular dipyramid in Task 3.1.1.d.
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An n-gonal tripyramid has two vertices where n triangles meet, but at each of the n vertices where the pyramids are “glued together”, there are four isosceles triangles touching. A net for a pentagonal dipyramid is given in Figure 3.2.11.
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Determine the number of vertices, faces, and edges for each of the following types of dipyramids. Do not count the bases of the pyramids as a face.
  1. A triangular dipyramid
  2. A pentagonal dipyramid
  3. A hexagonal dipyramid
  4. An n-gonal dipyramid
A net for a pentagonal dipyramid consisting of ten isosceles triangles.
Figure 3.2.11. Net for a pentagonal dipyramid
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Writing Prompts

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6.
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Write a letter to a friend describing one thing you found to be surprising or interesting in this section or Section 3.1. What questions do you have that would be interesting to explore?
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7.
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When three-dimensional objects are described verbally or illustrated in a two-dimensional space, they can be difficult to visualize. Discuss the challenges you have encountered with visualizing these solids and the techniques you have used to overcome these challenges.
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