Math 350
Section 8.2 Classwork Solutions
Classwork 1: (i) Rotate the polycube one more 1/4 turn. Record on square dot paper what the
observer sees. (ii) Then place the polycube in the original position (orientation) at left above,
and rotate it 1/4 turn in the opposite direction. Is the result the same as you obtained in (i)?
(i)
(ii) This is the same as (i).
Classwork 2: Sketch on square dot paper the original polycube at left above. Then sketch the
three images obtained by rotating the polycube about a horizontal (left-to-right) axis.
Answer:
Classwork 3: There is also an axis of space rotation symmetry through each pair
of opposite vertices of the cube.
(i) What part of a full rotation occurs? One-third of a rotation, 120 degrees.
One way to see this is that there are 3 faces that meet at a vertex,
and the rotation must take one face to the next one at that vertex.
(ii) What is the order of this space rotation symmetry?
It is order 3.
(iii) How many such axes of space rotation symmetry are there? Why?
There are as many axes as there are pairs of opposite vertices for the axes to pass
through. Since a cube has 8 vertices, there are 8/2 = 4 pairs of opposite vertices,
and so 4 such axes of space rotation symmetry.
Classwork 4: There is an axis of space rotation symmetry through a vertex and the
center of the opposite face. Use your model of a tetrahedron; place your thumb at
the center of a face and your index finger on the opposite vertex. Rotate the model.
(i) What part of a rotation around this axis is necessary to make the
tetrahedron coincide with the original configuration? One-third of a full
rotation, since there are three faces meet at the vertex.
(ii) What is the order of this space rotation symmetry? It is order 3.
(iii) How many such axes of space rotation symmetry are there? There are four such
axes, one through each vertex of the tetrahedron. So a tetrahedron has four axes of
order 3 rotation symmetry.
Classwork 5: There is an axis of space rotation symmetry through the midpoints
of opposite edges. Use your model of a tetrahedron; place your thumb at the
midpoint of an edge and your index finger at the midpoint of the opposite edge.
Rotate the model.
(i) What part of a rotation around this axis is necessary to make the
tetrahedron coincide with the original configuration? One-half of a full rotation, since
it takes such a rotation to take one half of the edge to the other half of the edge.
(ii) What is the order of this space rotation symmetry? It is order 2.
(iii) How many such axes of space rotation symmetry are there? There are as many such
axes as there are pairs of opposite edges. There are 6 edges in a tetrahedron, so
there are 6/2 = 3 such axes of order 2 rotation symmetry.
Classwork 6: Consider the right pentacube on the previous page. Draw (on square dot paper)
this pentacube and its image if it is reflected through a mirror plane that is horizontal and above
the pentacube.
Classwork 7: A tetrahedron has mirror symmetries. Each plane of mirror
symmetry contains an edge of the terahedron and passes through the midpoint of
the opposite side, as shown at right. How many such mirror planes of reflection
symmetry does a tetrahedron have?
Answer: There are six such mirror planes of reflection symmetry, one for each of the 6 edges of
the tetrahedron, which the plane can contain.
Classwork 8: Let D be a right pyramid whose base is a regular n-gon but which is not a regular
polyhedron. How many planes (total) of reflection symmetry does D have? Describe their
locations.
Answer: Recall from Chapter 5 that a regular n-gon has n lines of reflection symmetry. For each
of these in one base of the prism there is a plane of mirror reflection
symmetry of the prism containing the line and perpendicular to the base of
the prism (and passing through the apex of the prism).
For example, for each mirror line of reflection symmetry of the base
of the square pyramid there is a mirror plane of reflection symmetry passing
through that line and the apex of the pyramid.