15 mai 2007 · NON-PLANAR SURFACES Double torus (genus 2): v − e + f = −2 Euler Characteristic Rebecca Robinson 19
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15 mai 2007 · NON-PLANAR SURFACES Double torus (genus 2): v − e + f = −2 Euler Characteristic Rebecca Robinson 19
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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= 2Euler 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= 2Holds 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 Robinson6POLYHEDRA
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 1752Euler 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 Robinson8POLYHEDRA
•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= 20Euler 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