Answer the following questions to determine why Euler’s
formula continues to hold for the polyhedron formed by
cutting corners off of an octahedron, :
a. For an octahedron V = , ____,F = ____, and E =
_____.
b. When you slice off one corner of the octahedron,
you (gain or lose) ____ vertices, (gain or lose) ____
faces, and (gain or lose) ____ edges.
c. Therefore, the total change in V is ____; the total
change in F is ____; and the total change in V + F is
____.
d. The total change in E is ____.
e. What does the comparison of the total change in V + F
to the total change in E tell you?
In case you are not already familiar with the notations,
V=number of vertices (corners) of the octahedron.
F=number of faces
E=number of edges
The following link shows what happens when a corner of the octahedron is cut away. I'll let you complete the answers to the question.
http://img204.imageshack.us/img204/5302/euler.jpg
a. For an octahedron V = 6, F = 8, and E = 12.
b. When you slice off one corner of the octahedron, you lose one vertex, gain one face, and gain three edges.
c. Therefore, the total change in V is -1; the total change in F is +1; and the total change in V + F is (-1) + (+1) = 0.
d. The total change in E is +3.
e. The comparison of the total change in V + F to the total change in E tells us that the change in the number of vertices plus the change in the number of faces is equal to the change in the number of edges for this operation. In other words, the change in V + F = change in E. This is consistent with Euler's formula, V + F - E = 2, which continues to hold for the polyhedron formed by cutting corners off of the octahedron.