Verify Euler formula for n sided prism
Answers
Answer:
Step-by-step explanation:
o smaller faces.
Dividing faces
Figure 10: Dividing faces.
We repeat this with our chosen face until the face has been broken up into triangles.
In the end we are left with triangular faces.
Figure 11: In the end we are left with triangular faces.
If there is a further face with more than three sides, we use Step 1 on that face until it too has been broken up into triangular faces. In this way, we can break every face up into triangular faces, and we get a new network, all of whose faces are triangular. We illustrate this process by showing how we would transform the network we made from a cube.
Transforming the network of the cube.
Figure 12: This is what happens to the cube's network as we repeatedly perform Step 1.
We go back to Step 1, and look at the network we get after performing Step 1 just once. Now, by drawing a diagonal we added one edge. Our original face has become two faces, so we have added one to the number of faces. We haven't changed the number of vertices. The network now has V vertices, E + 1 edges and F + 1 faces. So how has
V - E + F changed after we performed Step 1 once? Using what we know about the changes in V, E and F we can see that V - E + F has become V - (E + 1) + (F + 1). Now we have
V - (E + 1) + (F + 1) = V - E - 1 + F + 1 = V - E + F.
So V - E + F has not changed after Step 1! Because each use of Step 1 leaves V - E + F unchanged, it is still unchanged when we reach our new network made up entirely of triangles! The effect on V - E + F as we transform the network made from the cube is shown in the table below.
Round V E F V - E + F
(a) 8 12 6 2
(b) 8 13 7 2
(c) 8 14 8 2
(d) 8 15 9 2
(e) 8 16 10 2
(f) 8 17 11 2
We now introduce Steps 2 and 3. They will remove faces from around the outside of the network, reducing the number of faces step by step. Once we begin to do this the network probably won't represent a polyhedron anymore, but the important property of the network is retained.
Step 2 We check whether the network has a face which shares only one edge with the exterior face. If it does, we remove this face by removing the one shared edge. The area which had been covered by our chosen face becomes part of the exterior face, and the network has a new boundary. This is illustrated by the diagram below for the network made from the cube.
Removing faces
Figure 13: Removing faces with one external edge.
Now, we will take V, E and F to be the numbers of vertices, edges and faces the network made up of triangular faces had before we performed Step 2. We now look at how the number V - E + F has changed after we perform Step 2 once. We have removed one edge, so our new network has E - 1 edges. We have not touched the vertices at
all, so we still have V vertices. The face we used for Step 2 was merged with the exterior face, so we now have F - 1 faces. So V - E + F has become V - (E - 1) + (F - 1) and
V - (E - 1) + (F - 1) = V - E + 1 + F - 1 = V - E + F.
So once again V - E + F has not changed.
Step 3 We check whether our network has a face which shares two edges with the exterior face. If it does, we remove this face by removing both these shared edges and their shared vertex, so that again the area belonging to our chosen face becomes part of the exterior face. This is illustrated below in the case of the cube network
Figure 14: Removing faces with two external edges.
As we did before we now take V, E and F to be the numbers of vertices, edges and faces of the network we're starting with. Now how has the number V - E + F been affected by Step 3? We have removed one vertex — the one between the two edges — so there are now V - 1 vertices. We have removed two edges, so there are now
E - 2 edges. Finally, our chosen face has merged with the exterior face, so we now have F - 1 faces. So V - E + F has become (V - 1) - (E - 2) - (F - 1) and
(V - 1) - (E -2) + (F - 1) = V - 1 - E + 2 + F - 1 = V - E + F.
So once more V - E + F has not