[PDF] [PDF] Euler Characteristic - User Web Pages

[PDF] Euler Characteristic - User Web Pages

15 mai 2007 · NON-PLANAR SURFACES Double torus (genus 2): v − e + f = −2 Euler Characteristic Rebecca Robinson 19 

[PDF] Euler Characteristic - Csse Monash Edu Au

15 mai 2007 · NON-PLANAR SURFACES Double torus (genus 2): v − e + f = −2 Euler Characteristic Rebecca Robinson 19 

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Euler characteristic is a very important topological property which started out as nothing Definition: A graph, G, consists of two sets: a nonempty finite set V of vertices and Now let's see how to calculate the Euler characteristic of a torus

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Theorem 7 1 Any two maps on the same surface have the same value of V -E+F Euler characteristic V -E +F by considering a larger map obtained by drawing An n-fold torus is a surface obtained from a sphere by adding n handles, or 

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Euler Characteristic

Rebecca Robinson

May 15, 2007

Euler CharacteristicRebecca Robinson1

PLANAR GRAPHS

1 Planar graphs

v-e+f= 2v= 6,e= 7,f= 3v= 4,e= 6,f= 4v= 5,e= 4,f= 1 v-e+f= 2v-e+f= 2

Euler CharacteristicRebecca Robinson2

PLANAR GRAPHS

Euler characteristic:χ=v-e+f

If a finite, connected, planar graph is drawn in the plane without any edge intersections, and: •vis the number of vertices, •eis the number of edges, and •fis the number of faces then:

χ=v-e+f= 2

ie. the Euler characteristic is 2 for planar surfaces.

Euler CharacteristicRebecca Robinson3

PLANAR GRAPHS

Proof.

Start with smallest possible graph:

v= 1,e= 0,f= 1 v-e+f= 2

Holds for base case

Euler CharacteristicRebecca Robinson4

PLANAR GRAPHS

Increase size of graph:

•either add a new edge and a new vertex, keeping the number of faces the same:Euler CharacteristicRebecca Robinson5

PLANAR GRAPHS

•or add a new edge but no new vertex, thus completing a new cycleand increasing the number of faces:Euler CharacteristicRebecca Robinson6

POLYHEDRA

2 Polyhedra

•Euler first noticed this property applied to polyhedra •He first mentions the formulav-e+f= 2in a letter to Goldbach in 1750 •Proved the result for convex polyhedra in 1752

Euler CharacteristicRebecca Robinson7

POLYHEDRA

•Holds for polyhedra where the vertices, edges and faces correspond to the vertices, edges and faces of a connected, planar graphEuler CharacteristicRebecca Robinson8

POLYHEDRA

•In 1813 Lhuilier drew attention to polyhedra which did not fitthis formula v-e+f= 0v= 16,e= 24,f= 12 v-e+f= 4v= 20,e= 40,f= 20

Euler CharacteristicRebecca Robinson9

POLYHEDRA

Euler's theorem.(Von Staudt, 1847) LetPbe a polyhedron which satisfies: (a) Any two vertices ofPcan be connected by a chain of edges. (b) Any loop onPwhich is made up of straight line segments (not necessarily edges) separatesPinto two pieces.

Thenv-e+f= 2forP.

Euler CharacteristicRebecca Robinson10

POLYHEDRA

Von Staudt's proof:

For a connected, planar graphG, define thedual graphG?as follows: •add a vertex for each face ofG; and

•add an edge for each edge inGthat separates two neighbouring faces.Euler CharacteristicRebecca Robinson11

POLYHEDRA

Choose a spanning treeTinG.Euler CharacteristicRebecca Robinson12

POLYHEDRA

Now look at the edges in the dual graphG?ofT?scomplement (G-T).

The resulting graphT?is a spanning tree ofG?.

Euler CharacteristicRebecca Robinson13

POLYHEDRA

•Number of vertices in any tree=number of edges+1. |V(T)| - |E(T)|= 1and|V(T?)| - |E(T?)|= 1 |V(T)| -[|E(T)|+|E(T?)|] +|V(T?)|= 2 |V(T)|=|V(G)|, sinceTis a spanning tree ofG |V(T?)|=|F(G)|, sinceT?is a spanning tree ofG's dual |E(T)|+|E(T?)|=|E(G)| •ThereforeV-E+F= 2.

Euler CharacteristicRebecca Robinson14

POLYHEDRA

•Platonic solid:a convex, regular polyhedron, i.e. one whose faces are identical and which has the same number of faces around each vertex. •Euler characteristic can be used to show there are exactly five Platonic solids.

Proof.

Letnbe the number of edges and vertices on each face. Letdbe the degree of each vertex. nF= 2E=dV

Euler CharacteristicRebecca Robinson15

POLYHEDRA

Rearrange:

e=dV/2,f=dV/n

By Euler's formula:

V-dV/2 +dV/n= 2

V(2n+ 2d-nd) = 4n

SincenandVare positive:

2n+ 2d-nd >0

(n-2)(d-2)<4

Thus there are five possibilities for(d,n):

(5,3)(icosahedron).

Euler CharacteristicRebecca Robinson16

NON-PLANAR SURFACES

3 Non-planar surfaces

•χ=v-e+f= 2applies for graphs drawn on the plane - what about other surfaces? •Genusof a graph: a number representing the maximum number of cuttings that can be made along a surface without disconnecting it - the number ofhandlesof the surface. •In general:χ= 2-2g, wheregis the genus of the surface •Plane has genus 0, so2-2g= 2

Euler CharacteristicRebecca Robinson17

NON-PLANAR SURFACES

Torus (genus 1):v-e+f= 0Euler CharacteristicRebecca Robinson18

NON-PLANAR SURFACES

Double torus (genus 2):v-e+f=-2Euler CharacteristicRebecca Robinson19

NON-PLANAR SURFACES

•Topological equivalence:two surfaces are topologically equivalent (or homeomorphic) if one can be `deformed' into the other without cutting or gluing. •Examples: the sphere is topologically equivalent to any convex polyhedron; a torus is topologically equivalent to a `coffee cup' shape. •Topologically equivalent surfaces have the same Euler number: the Euler characteristic is called atopological invariant

Euler CharacteristicRebecca Robinson20

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