A few words on topology

This fits our purposes since we want to recognise our nets as identical even if they are deformed, but not if any chemical bonds are broken. Now, the problem becomes how to describe these topological differences between various nets. 

We first look at how the mathematical discipline of algebraic topology differentiates between the shapes of finite 3D-objects. For example, which objects have the same topology a sphere, and ellipsoid or a torus (doughnut shape)  

Wc will not prove this, but it is important to note that the proof is purely topological, and does not rely on the summing of angles etc. 

For a polyhedron wc have a finite number of vertices, thus the denominator cannot be negative or zero, thus  

This is the same equation as 10.5, giving the five Platonic bodies as the solutions for integer values of n and p. For an infinite number of vertices the denominator should be zero, thus yielding equation 10.4 again. In this formulation it is not evident that these solutions should give 2D-nets- However, an enclosed volume covered by an infinite number of faces also has infinite values of n and p. Since 10.4 gives finite values of n and p these solutions have to correspond to a 2D-net, 

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