Answer the following questions to determine why Euler’s
formula continues to hold for the polyhedron formed by
cutting corners off of an octahedron, as in Figure 11.35(b):
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?
a. For an octahedron V = 6, F = 8, and E = 12.
To determine the values of V, F, and E for an octahedron, we need to consider its properties. An octahedron has 8 faces, 6 vertices, and 12 edges. Therefore, V = 6, F = 8, and E = 12.
b. When you slice off one corner of the octahedron, you gain 1 vertex, lose 4 faces, and gain 4 edges.
To understand the changes that occur when one corner of the octahedron is sliced off, we need to examine the results. Slicing off a corner creates a new vertex where the corner used to be, resulting in a gain of 1 vertex. It also removes the 4 faces that were adjacent to the corner, resulting in a loss of 4 faces. Finally, it creates 4 new edges connecting the newly created vertex to the neighboring vertices, resulting in a gain of 4 edges.
c. Therefore, the total change in V is 1; the total change in F is -4; and the total change in V + F is -3.
To determine the total changes in V, F, and V + F, we can sum up the changes from part (b). The total change in V is 1 because we gained 1 vertex. The total change in F is -4 because we lost 4 faces. The total change in V + F is -3 because we add the total changes in V and F, which gives us 1 + (-4) = -3.
d. The total change in E is 0.
To determine the total change in E, we need to consider the changes in part (b). Although we gained 4 edges by slicing off one corner, we also lost 4 edges when the 4 adjacent faces were removed. Therefore, the total change in E is 4 - 4 = 0. We neither gained nor lost any edges.
e. The comparison of the total change in V + F to the total change in E tells us that the total change in V + F equals the total change in E.
The total change in V + F (-3) equals the total change in E (0). This comparison tells us that the changes in the sum of vertices and faces are equivalent to the changes in the edges. In other words, for this particular transformation of the octahedron, the change in the number of vertices and faces is balanced by the change in the number of edges. This observation aligns with Euler's formula, which states that for any polyhedron, V + F - E = 2. In this case, the change in V + F equals the change in E, reinforcing the validity of Euler's formula for this specific transformation.