Lecture 1: The Euler characteristic
Lecture 1: The Euler characteristic Euler characteristic (simple form): ... Euler characteristic. -1. -2. Solid double torus. The graph: Double torus =.
Euler Characteristic
15 mai 2007 ie. the Euler characteristic is 2 for planar surfaces. ... Double torus (genus 2): v ? e + f = ?2. Euler Characteristic. Rebecca Robinson.
RECOGNIZING SURFACES
boundary components genus
On the dichromatic number of surfaces
the double torus the projective plane
Fundamental Polygons for Coverings of the Double-Torus
of V vertex cycles (vertices on the surface) and the Euler characteristic is ?(S?) = V ? E + 1. Various polygons may yield the same topological surface.
Chapter 10.7: Planar Graphs
Euler's Formula: For a plane graph v ? e + r = 2. The torus has Euler characteristic 0 (it can be tiled with squares
Capturing Eight-Color Double-Torus Maps
For a map on the double torus with eight countries there will be 8 heptagons with 56/2 = 28 edges. Since the Euler characteristic of the two-holed torus is
1 Euler Characteristic
1.2 Euler-Poincar´e Formula sphere. (g=0) torus. (g=1) double torus. (g=2). Clearly not all surfaces look like disks or spheres.
New Classes of Quantum Codes Associated with Surface Maps
3 juil. 2020 a double torus and SEMs on the surface of Euler characteristic -1 and the covering maps. Finally in Section 6
On dual unit balls of Thurston norms - Archive ouverte HAL
13 juil. 2021 where ?(Si) is the Euler characteristic of Si. When a representative S of a ... the torus— on the first homology of the torus.
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
POLYHEDRA
Choose a spanning treeTinG.Euler CharacteristicRebecca Robinson12POLYHEDRA
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=dVEuler CharacteristicRebecca Robinson15
POLYHEDRA
Rearrange:
e=dV/2,f=dV/nBy Euler's formula:
V-dV/2 +dV/n= 2
V(2n+ 2d-nd) = 4n
SincenandVare positive:
2n+ 2d-nd >0
(n-2)(d-2)<4Thus 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= 2Euler CharacteristicRebecca Robinson17
NON-PLANAR SURFACES
Torus (genus 1):v-e+f= 0Euler CharacteristicRebecca Robinson18NON-PLANAR SURFACES
Double torus (genus 2):v-e+f=-2Euler CharacteristicRebecca Robinson19NON-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 invariantEuler CharacteristicRebecca Robinson20
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