Math

Vertices, edges, faces

A solid body of a few dozen faces, and V − E + F read off it after every edit: triangulate a face, put a vertex in it, split or erase an edge, cut a face out, refine them all - a ledger shows what each does to V, E and F and that χ moves only when a hole is cut. The genus comes from χ with its conditions stated: how many pieces, how many boundary loops, orientable or not - on a Möbius band the formula gives cross-caps instead.

By Mykhailo Makarov

Actual application image

What happens

Count the corners, edges and faces of a solid: V − E + F. On a block it is 2; drill a tunnel through it and the answer is 0; two tunnels, −2. The number does not care how the surface is cut into faces - triangulate a face, put a vertex in it, split an edge, erase one, refine everything, and V, E and F all move while their alternating sum stands still. Only cutting a face OUT changes it, by one. But χ alone is not the shape: the genus follows from it only once the pieces and the boundary loops are counted and the surface is orientable - a torus and a cylinder share χ = 0, and a Möbius band is neither.

Equations used in the module

Equations
χ = V − E + F
orientable, c pieces, b boundary loops:  χ = 2c − 2g − b
not orientable:  χ = 2c − k − b  (k cross-caps)
triangulate an n-gon: ΔE = ΔF = n − 3;  vertex in a face: ΔV = 1, ΔE = n, ΔF = n − 1
split an edge: ΔV = ΔE = 1;  erase an edge: ΔE = ΔF = −1;  cut a face out: ΔF = −1, Δχ = −1

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