Superlesson
Project 12-1
Advanced Algebra
Chapter 12, Discrete Mathematics and Models
Graph theory has both theoretical and practical importance, as the following investigations show.
Part B, Paths and Circuits
1. Study the 13 Archimedean solids at this Web site to answer the following questions. (In the illustration, each solid is paired with its twin directly above or below it.)
a. How are Archimedean solids similar to Platonic solids? Different from Platonic solids?
b. Show that Euler's Formula is valid for a truncated tetrahedron.
c. Complete the table for the given polyhedra.
|
Polyhedron
|
Measures of the Angles Meeting at Each Vertex |
|
D + 360 - S |
V = Number of Vertices |
| Cube |
90°, 90°, 90° |
270° |
90° |
8 |
| Tetrahedron | ||||
| Octahedron | ||||
| Dodecahedron | ||||
| Icosahedron | ||||
| Truncated Tetrahedron |
d. Find a relationship between D and V in the table from question c.
e. Use the relationship to find the number of vertices in a truncated icosahedron.
Part D, Making Connections
2. Use the Distance Web site to answer the following questions.
a. The graph shows eight Texas towns and the roads connecting them. Find the length of each road.
b. To visit all of her customers, a saleswoman completed an Euler path on the graph, beginning in Abilene. Where did the path end? Name the towns the saleswoman passed through and the order in which she passed through them.
c. To visit warehouses in each town on the map, the saleswoman completed a Hamiltonian circuit on the graph, beginning in Abilene. Where did the path end? Name the towns the saleswoman passed through and the order in which she passed through them.
d. Find a minimal spanning tree for the graph. How long is it?
