Subsection3.1.1Two-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.
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?
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.
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 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.
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.
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.
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?
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 3.1.9.
What happens when you try to construct a polyhedron with only two triangles, and no other shapes, meeting at a vertex?
It is also possible to create a polyhedron with exactly four triangles meeting at every vertex.
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.
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.
This question is slightly more complex. Imagine starting with a central vertex and arranging five triangles around it. How do the edges of these triangles connect to form a larger, symmetrical shape?
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?
Each triangle has a 60 degree corner. How many of these corners can fit together at a single point before the total angle is too large to bend and form a corner?
(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.
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.
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 3.1.10 below.
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.
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 3.1.10 below.
The sum of the interior angles of a pentagon is 540 degrees. Use this sum to find the measure of one vertex angle. Then determine how many of these can meet at a single point without the faces lying flat.
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.
Look at the tables Table 3.1.9 and Table 3.1.10. 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.
While building your polyhedra, you may have wondered whether it was necessary to cut out and glue each polygon separately. Figure 3.1.12 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.
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.
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.
Recall that a regular tetrahedron has four triangular faces. Each triangular face connects to three other faces. Visualize or sketch how each net (Figure a and Figure b) would fold up in 3D.
Recall that a cube has six square faces. In a notebook, try sketching six connected squares that can easily fold into a 3D shape. Each face should have four edges that connect to other faces.
A valid net for a regular polyhedron must fold into a completely closed 3D shape without any overlaps or gaps. Pay close attention to how individual faces are connected and whether edges meet when folded.
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.
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?
Recall from earlier in this section that an octahedron has eight faces. How can the number of faces of an octahedron be used to calculate its number of edges?
Recall from the previous problem that an octahedron has eight triangular faces and 12 edges. First, determine the total number of vertices. Then, consider how many faces must meet at each of those vertices.
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.
