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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Section 3.1 Regular Polyhedra

Subsection 3.1.1 Two-Dimensional and Three-Dimensional Objects

Polygons and circles lie in a plane. Figures that lie in a plane and have a height and length are said to be two-dimensional objects. We can measure the area of closed two-dimensional figures. When we limit our geometrical study to objects that lie in one common plane, we are studying two-dimensional geometry. We often use an \(xy\)-coordinate system to represent points on a plane. The \(x\)-axis represents one direction (across) while the \(y\)-axis represents a second direction (up or down). Objects within the plane also include lines (one-dimensional objects) and points (zero-dimensional objects). In fact, the real number line or \(x\)-axis is one of may one-dimenstional lines in the \(xy-plane\text{.}\) When we define parallel and perpendicular lines, we assume that these one-dimensional lines are lying in a common two-dimensional plane.

Checkpoint 3.1.1. Check Your Understanding of Two-Dimensional Objects.

(a)
Lines, segments, and rays live in the plane, but they are considered to be one-dimensional objects. Why?
Answer.
Answers may vary. Linear objects have only dimension length, not width or depth. We cannot measure the area of a linear object.
(b)
Points are considered to be zero-dimensional objects. Why?
Answer.
Points have no length, width, or breadth.
(c)
A circle is considered to be a two-dimensional object, but we typically use one attribute, namely radius, when describing its size. Does a circle have a height and width? If so, how are the height and width of a circle related to its radius?
Answer.
Both the height and width of a circle are equal to the diameter; that is, twice the length of the radius. We can measure the area of a circle.
In this section, we will explore three-dimensional solids. Solids with flat polygonal sides, like cubes and pyramids, are called polyhedra. Other solids, like cylinders and spheres have curved faces.
To fully understand this chapter, you will need to construct some of these figures. Be prepared to print copies of patterns on paper or cardstock so that you can fold and tape the figures together as needed.

Subsection 3.1.2 Defining a Polyhedron

Some three-dimensional solids have polygonal faces.

Definition 3.1.2.

A polyhedron (plural: polyhedra) is a three-dimensional closed figure in which all of the faces are polygons. Both of the solids in Figure 3.1.3 are polyhedra, but the one on the left is said to be concave since we can connect some vertices, like the far left and top vertex with straight segments that pass through the exterior (outside the surface) of the solid. As we work through this chapter, we will assume that a polyhedron is convex so that it has no valleys or depressions. Any line segment connecting two points on the surface of a convex polyhedron will only contain points on the surface or interior of the polyhedron.
A concave polyhedron and a convex polyhedron
Figure 3.1.3. A concave polyhedron and a convex polyhedron

Definition 3.1.4.

A face of a geometrical solid refers to a flat surface that forms part of the boundary of the solid. The line segments where two faces meet are called edges, and the points where edges meet are called the vertices of the polyhedron. The vertices of the polyhedron are also vertices of its polygonal faces.
While it is not incorrect to call a face of a polyhedron a ‘side’, one should be careful to distinguish between a face of the polyhedron and a side of a polygon (which is an edge of the polyhedron). Using the terms face and edge will make our discussions about polyhedra clearer.

Definition 3.1.5.

Note also that a polyhedron has more than one type of angle. In this section, we will be working with angles whose rays are adjacent edges of the polyhedron. Each of these angles is a vertex angle of one of the faces of the polyhedron; hence, we may refer to them as face angles.
 1 
This is not the same as a dihedral angle, measuring the space between two faces.

Checkpoint 3.1.6.

Which of the following solids are polyhedra?
  1. cylinder
  2. cube
  3. triangular prism
  4. sphere
Answer.
A cube and triangular prism are polyhedra. A cylinder and sphere are not polyhedra since they have curved surfaces.

Subsection 3.1.3 Defining Regular Polyhedra

Just as with polygons and tilings, we describe some polyhedra as regular polyhedra. These special solids were identified by Greek mathematicians around 300 B.C., appear in ancient and modern artwork, and are used to model chemical and biological structures today. In this section, we will find that there are only a handful of polyhedra that meet the criteria of being regular.

Definition 3.1.7.

A polyhedron is said to be regular if it satisfies these two conditions:
  • The faces are congruent regular polygons and
  • At every vertex, the number of polygons meeting is the same.

Subsection 3.1.4 What Regular Polyhedra Exist

Exploration 3.1.1. Discovering Regular Polyhedra.

(a) A nonregular polyhedron.
Explain why a square pyramid, Figure 3.4.2, is not a regular polyhedron.
Hint.
Reread the definition for Definition 3.1.7 regular polyhedron.
(b) Angle Measures at a Vertex.
  1. When we constructed tilings of the plane in Section 2.4, we found the sum of the measures of the angles surrounding a vertex was always the same. What was that sum?
  2. Why must the total degree of the face angles surrounding a vertex be less than that amount if we are building a polyhedron instead of a tiling?
Hint.
Look at some polyhedra or attempt to build some. Unless it has curved walls, your classroom is probably a polyhedra.
(c) Regular Polyhedra with Triangular Faces.
For this task, you will need copies of rigid congruent equilateral triangles that can be put together. There are commercially available shapes that can be snapped together, but you can also create your own using tagboard or even straws and string.
Which regular polyhedra can be formed using only equilateral triangles? To answer this question, we will consider the possible number of triangles that could meet at each vertex and attempt to construct the polyhedron, making sure that we keep the vertex arrangement consistent. Later you will be asked to enter the answers to some of these questions into Table 3.1.8.
  1. What happens when you try to construct a polyhedron with only two triangles, and no other shapes, meeting at a vertex?
  2. It is possible to create a regular polyhedron with exactly three triangles meeting at every vertex.
    1. Build this polygon and describe its shape or save a picture of it.
    2. How many triangular faces does this polyhedron have?
    3. How many edges does this polyhedron have?
    4. How many vertices does this polyhedron have?
  3. It is also possible to create a polyhedron with exactly four triangles meeting at every vertex.
    1. Build this polygon and describe its shape or save a picture of it. Check every vertex of the polyhedron to make sure there are exactly four triangles meeting.
    2. How many triangular faces does this polyhedron have?
    3. How many edges does this polyhedron have?
    4. How many vertices does this polyhedron have?
  4. Is it possible to create a regular polyhedron with exactly five triangles meeting at each vertex? If so, build it, record your work, and count the faces, edges, and vertices.
  5. What is the largest number of triangles that can meet at a vertex to form a corner or point of a polyhedron? What happens when you squeeze one more triangle in at a vertex?
  6. Optional Extension: What happens when you squeeze yet another triangle in at each ‘vertex’?
  7. Use your answers above to complete the following table:
Table 3.1.8. Regular Polyhedra with Triangular Faces
Number of faces at each vertex Total number of faces Total number of edges Total number of vertices
3
4
5
(d) A Nonregular Polyhedron made from Regular Triangles.
There is a polyhedron that can be constructed which has exactly six faces, all being equilateral triangles. Build this polyhedron, and then explain why it does not satisfy the definition of a regular polyhedron.
(e) Other Regular Polyhedra.
We now explore whether we can form regular polyhedra using squares, regular pentagons, or other regular polygons.
  1. Regular polyhedra with square faces:
    1. Each angle of a square has what degree measure?
    2. Is it possible to create a regular polyhedron using only squares? If so, determine the number of squares that must meet at each vertex. Then construct or describe the polyhedron. Record a picture.
    3. If the construction is possible, count the faces at each vertex, the total number of faces, the total number of edges, and the total number of vertices. Record the information in Table 3.1.9 below.
    4. Would it be possible to create a regular polyhedron with fewer squares meeting at each vertex? Why or why not?
    5. Would it be possible to create a regular polyhedron with more squares meeting at each vertex? Why or why not?
  2. Regular polyhedra with pentagonal faces:
    1. What is the measure of a vertex angle of a regular pentagon?
    2. Is it possible to create a regular polyhedron using only pentagons? If so, determine the number of regular pentagons that must meet at each vertex and then construct the polyhedron. Carefully describe this polyhedron and/or save a picture.
    3. If the construction was possible, count the faces at each vertex, the total number of faces, the total number of edges, and the total number of vertices. Record the information in Table 3.1.9 below.
    4. Is this the only possible regular polyhedron using pentagons? Explain.
  3. In search of regular polyhedra whose faces have more than five sides:
    1. What is the measure of each vertex angle in a regular hexagon?
    2. Why would surrounding a vertex with three hexagons not result in a possible corner for a polyhedron?
    3. If a regular polygon has more than six sides, then the measure of each of its vertex angles must be between what two numbers?
    4. Write a thorough and convincing argument as to why any regular polyhedron must be bounded by equilateral triangles, squares, or regular pentagons. Conclude that there are exactly five types of regular polyhedra.
  4. Use your answers above to complete the following table:
Table 3.1.9. Regular Polyhedra with Square or Pentagonal Faces
Shape of Face Number of faces at each vertex Total number of faces Total number of edges Total number of vertices
square
pentagon
(f)
Look at the tables Table 3.1.8 and Table 3.1.9. Let \(F\) be the total number of faces, \(E\) be the total number of edges, and \(V\) be the total number of vertices. What do you notice about \(F + V – E\) ?
We have determined that there are exactly five regular polyhedra. They have the names, tetrahedron (four-faced), hexahedron (six-faced), octahedron (eight-faced), dodecahedron (12-faced), and icosahedron (20-faced). Match these names to the polyhedra you created and add these labels to the appropriate lines in your tables.

Subsection 3.1.5 Nets for Polyhedra

While building your polyhedra, you may have wondered whether it was necessary to cut out and glue each polygon separately. Figure 3.1.11 gives a triangle that can be cut out and folded along the interior lines to produce a regular tetrahedron. You may want to leave a little extra paper along some of the external edges so that you have a surface for gluing them together. This pattern is called a net for the polyhedron.

Definition 3.1.10.

A net for a polyhedron is a single-piece pattern for the polyhedron that can be folded and glued to form the solid. A net can be created by cutting the polyhedron along some of its edges and unfolding. There are multiple nets for each polyhedron.
The midpoints of the sides of an equilateral triangle are joined by segments to divide the triangle into four congruent equilateral triangles.
Figure 3.1.11. A net for a regular tetrahedron

Exercises 3.1.6 Exercises

Building your Toolbox

1.
Give a definition of regular polyhedra in your own words. Then, name and describe each of the five regular polyhedra. In particular, you should mention the shape of each face and the number of faces meeting at each vertex.
2.
Identify by name and/or picture three examples of polyhedra that are not regular. Explain why each is not regular.
3.
What must be true about the sum of the angle measures at any vertex of a convex polyhedron?

Skills and Recall

4. Tetrahedral Nets.
Two patterns are given in Figure 3.1.12. Which, if any, will form a regular tetrahedron when folded along the interior lines?
Pattern a gives a string of four equilateral triangles arranged in a line. The fold lines zigzag down the interior of the design. Pattern b gives four equilateral triangles which share a common vertex.
Figure 3.1.12. Possible tetrahedral nets
5. Nets for a Cube.
Sketch a net for a cube and use it to construct a cube.
6. Nets for Polyhedra?
Give an explanation for why each of the patterns in Figure 3.1.13 cannot be a net for a regular polyhedron.
Pattern a has six squares. Four are in a row and there is one below the first and the last. Pattern b has three squares in a row and two equilateral triangles, one each along the top edge of the center square and along the bottom edge of the center square. Pattern c is a strip of six alternating regular triangles. Pattern d has four 1.3x2 rectangles and two 1.3x1.3 squares.
Figure 3.1.13. Not regular polyhedral nets

Extending the Concepts

7. Counting Vertices and Edges.
It is easy to lose track when counting the vertices or edges of a polyhedron with many faces. We can use mathematical reasoning to compute these values for regular polyhedra.
(a)
Every edge gets glued to one edge of another triangle so the assembled octahedron will have half as many edges. How many edges does the octahedron have?
(b)
How many triangles meet at each vertex of an octahedron?
(c)
Divide the total number of vertices of the triangles, 24, by the number of triangles that meet at each vertex of the octahedron. This will give you the number of vertices in the assembled octahedron.
(d)
Use a similar argument to compute the number of edges and vertices in an icosahedron.
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