MATH 5210/7210 Homework
Homework 10, due Tuesday, February 26
1. In class we have discussed the following space filling collections of polyhedra. Make models that illustrate one of these 3D tessellations. You're encouraged to work with a group, so everyone can make a piece of the puzzle.
cubes
tetrahedra + octahedra
triangular prisms
hexagonal prisms
truncated octahedra
truncated cubes + octahedra
cuboctahedra + octahedra
truncated tetrahedra + tetrahedra
rhombic dodecahedra
2. For each of the 3D tessellations in problem 1, compute the dihedral angles of the polyhedral "bricks." (Use radian angle measure.) Then make the following computations for each of the tessellations:
(a) Check that the sum of the dihedral angles around every edge is 2π radians. You have to visualize how many polyhedral bricks of each type share an edge. For example, in the tetrahedra + octahedra space filling, there are 2 tetrahedra and 2 octahedra sharing each edge, and they are arranged in the order tetra-octa-tetra-octa around the edge.
(b) Check that the sum of the solid angles around a vertex is 4π radians. You have to visualize how many polyhedral bricks of each type share a vertex in the space filling you're looking at. For example, in the tetrahedra + octahedra space filling, there are 8 tetrahedra and 6 octahedra sharing each vertex.
To compute the solid angles of each brick, first compute the dihedral angles. (We have already computed many of them.) Then use Girard's formula to compute the solid angles. Here's how it works for a tetrahedron. The dihedral angle between two faces is the angle between the altitude of one face and the inradius of the other face. The altitude of an equilateral triangle with edge length 1 is √3/2, and the inradius is √3/6, so the dihedral angle is arccos[(√3/6)/(√3/2)] = arccos 1/3 = 1.230959417 radians. Girard's formula for spherical polygons implies that the solid angle at the apex (top vertex) of a convex infinite q-sided pyramid is the sum of the dihedral angles minus (q-2)π. For a vertex of a regular tetrahedron we get that the solid angle equals 1.230959417 + 1.230959417 + 1.230959417 - (3-2)π = 0.551285598 steradians. [Warning: Girard's formula does not work for degrees; it only works for radians.]
Homework 11, due Tuesday, March 25
1. Find pictures of objects displaying radial symmetry, and classify their symmetry types as Cn or Dn (n = 1, 2, 3, . . . ). Try to find as many different types as you can. Good examples of radial symmetry are car wheels and flowers. Note that you can paste digital pictures into GSP and then label the lines of symmetry and rotation angles. For extra credit, take pictures yourself with a digital camera.
2. Compute the multiplication table of the square, as we did in class for the equilateral triangle. Explain your notation for the symmetry transformations, and explain how your table is organized. Be careful of the fact that the multiplication is not commutative!
Homework 12, due Tuesday, April 1
1. Prove the statements 3(a), 3(b), and 3(c) in the discussion of the classification of finite plane symmetry groups, using the defintions of rotation and reflection. These statements say roughly (a) the product of two rotations is a rotation, (b) the product of two reflections is a rotation, and (c) the product of a rotation and a reflection is a reflection.
The data needed to define a rotation R of the plane is a center O and and angle α. Then R(O) = O, and if P ≠ O, then R(P) = P', where P' is the point such that d(P',O) = d(P,O) and the angle POP' is α.
The data needed to define reflection F is a line M, the mirror (or line of symmetry) of the reflection. Then if P lies on M, F(P) = P, and if P does not lie on M, F(P) = P', where P' is the point such that M is the perpendicular bisector of the segment PP'.
2. Describe all the symmetries of a cube, as we did in class for a regular tetrahedron. (You do not have to determine the multiplication table. We'll get into that next week.)
Here are some helpful hints: All of the orientation-preserving symmetries of a cube are rotations. (A rotation is determined by an axis and an angle.) The number of orientation-reversing symmetries of a cube equals the number of orientation-preserving symmetries. Some of the orientation-reversing symmetries are reflections (determined by a plane mirror) and some are "turn-flips" (determined by an axis, an angle, and a mirror, with the mirror perpendicular to the axis).
Homework 13, due Tuesday, April 8 - Multiplication of symmetries of a cube
I've divided the class into six groups, similar to the groups that worked together on homework 8. If you want to switch groups, just email me. I want to keep all the groups 4 or 5 people, and I will let you switch or trade places if this size restriction is maintained.
Each group is assigned to work on one of three types of multiplication of symmetries: (rotation)(rotation), (reflection)(reflection), or (rotation)(reflection). (We will save turn-flips until we have understood these other types.)
The goal in each case is to describe all the possible combinations of symmetries that can occur. For (rotation)(rotation), how can the two axes be related? For (reflection)(reflection) how can the two mirrors be related? For (rotation)(reflection) how can the axis and mirror be related? Once you understand the different types that can occur, then you need to look at the possible angles of rotation (except for the (reflection)(reflection) case).
Someone in each group should make two identical cubes, with the vertices labelled, so that you can keep track of the symmetries just like we did with the triangle cut-outs in our previous class activity. This should be done before class Thursday, so the groups can work with the models.
We will have discussion at the beginning and end of class Thursday, and then each group will write up a report for Tuesday.
Group A: Molly, Karina, Christina, Nicholas - (rotation)(rotation)
Group B: Mary, Christy, Erin, Julie - (reflection)(reflection)
Group C: Amanda B., Andrea, Pepper, Amanda H., Lauren - (reflection)(reflection)
Group D: Avleen, Charlie, Katie, Denna - (rotation)(reflection) and (reflection)(rotation)
Group E: Sherry, Sharon, Jisun, Ron, Felicia - (rotation)(reflection) and (reflection)(rotation)
Group F: Colleen, Ryan, Dana, Sarah - (rotation)(rotation)