Impossible Worlds
Topology "=" Shapes
Combinatorics "=" Counting
Locally looks 2-dimensional
Locally looks n-dimensional
We can cut the sphere into pieces:
What can we say about the number of vertices, edges and faces in a decomposition of the sphere?
Consider a connected graph drawn on the sphere, with v vertices, e edges, and f faces. Then :
v-e+f=2
To show the designers couldn't have made a sphere, it's enough to prove their maps couldn't have v-e+f=2.
1. Every vertex has four arrows
2. Every edge has two arrows
Similar argument show that for a video game graph we have: 4f=2e
So 2v=e=2f
So v-e+f=v-2v+v=0
Violates Euler's Theorem!