Exam 3 Review

Vocabulary

Polyhedron

Faces, vertices, and edges of a polyhedron

Net of a polyhedron

Regular polyhedron

Semi–regular polyhedron

Euler’s formula for a polyhedron

Defect at a vertex of a polyhedron

Great circle

Line on a sphere

Excess for a triangle on a sphere

Cell division

Euler number of a surface

Torus

Klein bottle

Non–orientable surfaces

Formula that will be provided:

Area of a triangle on a sphere: Area = (Excess ´ p ¸ 180) ´ R²

You should be able to:

1. Name the five regular polyhedra and describe each completely.

Name the five regular polyhedra and give the number of faces, edges and vertices, the type of faces, and how many faces meet at each vertex.

2. Explain why there can only be five regular polyhedra.

Explain why the five regular exist and why there can be no more than those.

3. Determine the number of edges in a polyhedron given the number and type of faces.

One of the Archimedean solids has 32 triangular and 6 square faces. How many edges does the solid have? Explain your method.

4. Determine the number of vertices in a polyhedron given the number and type of faces and the number of faces meeting at each vertex.

Using the solid from the previous problem, determine the number of vertices if 4 triangles and 1 square meet at each vertex, and explain your method.

5. Show that Euler’s formula is satisfied by a given polyhedron.

Using the information from the previous two problems, show that this solid satisfies Euler’s formula.

6. Determine the number of vertices or edges of a given polyhedron from Euler’s formula and other information.

The deltoidal icositetrahedron has 26 vertices and 48 edges. Compute the number of faces.

7. Find the defect at each vertex of a given polyhedron.

A biaugmented truncated cube has 8 vertices at which 3 squares and 1 triangle meet, 16 vertices at which 1 square, 2 triangles and 1 octagon meet, and 8 vertices at which 2 octagons and 1 triangle meet. All of the faces are regular polygons. Find the defect at each of the three types of vertices of this solid.

8. Show that Descartes’ Theorem holds for a given polyhedron.

Show that Descartes’ Theorem holds for the biaugmented truncated cube.

9. Compare the differences between geometry on a sphere and geometry on the plane.

Write out explanations of five ways that geometry on a sphere differs from geometry on the plane.

10. Find the excess for a triangle on a sphere.

Suppose a triangle on a sphere has angles of . Find the excess for the triangle.

11. Solve an area problem for a triangle on a sphere.

If the sphere in the previous problem has a radius of 8 cm, find the area of the triangle given in that problem.

12. Describe the regular tilings of a sphere.

Describe all the regular tilings of a sphere and explain why those are the only ones.

13. Find the number of vertices, faces, and edges of a cell division on a sphere.

A cell division on a sphere has 4 triangles and 3 quadrilaterals. Three faces meet at each of 4 vertices while 4 faces meet at the rest. Find the number of faces, edges, and vertices.

14. Use a cell division on a sphere to find the Euler number of a sphere.

Use the cell division on a sphere from the previous problem to calculate the Euler number of a sphere.

15. Determine equivalent tic–tac–toe positions on a torus.

Given the tic-tac-toe position below, place X’s and O’s on the second board so the two positions are equivalent.

		O
X	O
		X

X

O

16. Find a winning move for a tic–tac–toe position on a torus.

For the following tic-tac-toe board on a torus, find a winning move for X.

		O
X	O
		X

17. Find the number of vertices, faces, and edges of a cell division on a torus.

Identify and count all the vertices, edges, and faces of the cell division of the torus shown below.

18. Use a cell division on a torus to find the Euler number of a torus.

Use the cell division of the torus above to find the Euler number of a torus.

19. Determine equivalent tic–tac–toe positions on a Klein bottle.

Using the first tic-tac-toe board from #15, complete the board below with an equivalent position.

	X
O

20. Find a winning move for a tic–tac–toe position on a Klein bottle.

For the following tic-tac-toe board on a Klein bottle, find a winning move for X.

		O
X	O
		X

21. Find the number of vertices, faces, and edges of a cell division on a Klein bottle.

For the cell division of a Klein bottle shown below, identify and count all the vertices, edges, and faces.

22. Use a cell division on a Klein bottle to find the Euler number of a Klein bottle.

Use the cell division of a Klein bottle in the previous problem to find the Euler number of a Klein bottle.