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.
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.
Choose one of the following vertex arrangements, \(3.6.6\text{,}\)\(4.6.6\text{,}\)\(3.4.4.4\text{,}\)\(3.4.3.4\text{,}\)\(3.10.10\text{,}\) or \(5.6.6\text{.}\)
As a bonus, repeat this task for other vertex arrangements. If you are up for a challenge, you may choose \(3.4.5.4\text{,}\)\(3.3.3.3.4\text{,}\) or \(3.3.3.3.5\text{.}\)
(a)Semiregular Polyhedra with Vertex Arrangement \(4.4.n\).
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.
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.
(b)Semiregular Polyhedra with Vertex Arrangement \(3.3.3.n\).
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.
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
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.
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.
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.
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?
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.
