I. MontesI. Montes

Euler’s characteristic and Euler’s characteristic and the spherethe sphere

I. MontesI. Montes

Definition of a cellDefinition of a cell

An An nn-cell is a set whose interior is homeomorphic to the -cell is a set whose interior is homeomorphic to the nn--dimensional disc dimensional disc

with the additional property that its boundary or frontier with the additional property that its boundary or frontier

must be divided into a finite number of lower-dimensional must be divided into a finite number of lower-dimensional cells, called the faces of the cells, called the faces of the nn-cell.-cell.

• A 0-dimensional cell is a point A 0-dimensional cell is a point AA..• A 1-dimensional cell is a line segment A 1-dimensional cell is a line segment a=ABa=AB, and , and A<aA<a, , B<aB<a..• A 2-dimensional cell is a polygon (often a triangle) such as A 2-dimensional cell is a polygon (often a triangle) such as

ABCABC, and then , and then AB, BC, AC . AB, BC, AC . Note that Note that • A 3-dimensional cell is a solid polyhedron (often a A 3-dimensional cell is a solid polyhedron (often a

tetrahedron), with polygons, edges, and vertices as faces.tetrahedron), with polygons, edges, and vertices as faces.

{ :|| || 1}n nD x x R

, so A AB A

I. MontesI. Montes

Facts about n-cellsFacts about n-cells

The faces of an The faces of an nn-cell are lower dimensional -cell are lower dimensional cells: the endpoints of a 1-cell or edge are cells: the endpoints of a 1-cell or edge are 0-cells, the boundary of a 2-cell or polygon 0-cells, the boundary of a 2-cell or polygon consists of edges (1-cells) and vertices (0-consists of edges (1-cells) and vertices (0-cells), etc. These cells will be joined cells), etc. These cells will be joined together to form complexes.together to form complexes.

I. MontesI. Montes

Not examples of cellsNot examples of cells

The figure on the left is not a cell but the one on The figure on the left is not a cell but the one on the right is a cell.the right is a cell.

The figure on the left is not a cell because there The figure on the left is not a cell because there are no vertices. The figure on the right is a cell are no vertices. The figure on the right is a cell because it has three vertices, three edges and because it has three vertices, three edges and one face.one face.

I. MontesI. Montes

Cells form complexesCells form complexes

Cells are glued together to form complexes, by gluing edge Cells are glued together to form complexes, by gluing edge to edge and vertex to vertex and identifying higher-to edge and vertex to vertex and identifying higher-dimensional cells in a similar manner.dimensional cells in a similar manner.

Definition of a complex:Definition of a complex:A complex A complex KK is finite set of cells, is finite set of cells,

such that:such that: if is a cell in if is a cell in KK, then all faces of are elements of , then all faces of are elements of K;K; If and are cells in If and are cells in KK, then, thenThe dimension of The dimension of KK is the dimension of its highest- is the dimension of its highest-

dimension cell.dimension cell.

{ : is a cell}K

( ) ( ) 0Int Int

I. MontesI. Montes

Not examples of complexesNot examples of complexes

Complexes cannot intersectComplexes cannot intersect

A complex is more than a set of points, since it A complex is more than a set of points, since it also comes equipped with the structure given by also comes equipped with the structure given by the allotment of its points into cells of various the allotment of its points into cells of various dimensions. In each case above, notice that the dimensions. In each case above, notice that the intersections are homeomorphic to cells ,but are intersections are homeomorphic to cells ,but are not among the cells of the complex not among the cells of the complex KK..

I. MontesI. Montes

Few examples of complexesFew examples of complexes

A topological object can be represented by A topological object can be represented by many complexes.many complexes.

Complexes on the sphere.Complexes on the sphere.

I. MontesI. Montes

Definition of a Euler CharacteristicDefinition of a Euler Characteristic

Let Let KK be a complex. The Euler characteristic be a complex. The Euler characteristic of of KK is is

For 2-complexes; let For 2-complexes; let ff = #{faces}, = #{faces}, ee = #{edges}, = #{edges}, and and vv = #{vertices}, and then the Euler = #{vertices}, and then the Euler characteristic may be written ascharacteristic may be written as

( )K v e f

( ) #(0 ) #(1 ) #(2 ) #(3 ) ...K cells cells cells cells

I. MontesI. Montes

Example of how to find Euler CharacteristicExample of how to find Euler Characteristic

Consider a polygon with Consider a polygon with nn sides, shown here. The sides, shown here. The complex complex KK has has nn vertices, vertices, nn edges, and one edges, and one face, so face, so

Another examples is Another examples is K'K' given by the standard given by the standard planar diagram of the sphere in the following planar diagram of the sphere in the following figure. figure. K'K' has two vertices (P and Q), one edge, has two vertices (P and Q), one edge, and one face, so and one face, so

P QP Q

( ) 1 1K n n

( ') 2 1 1 2K

a

a

2S

I. MontesI. Montes

Theorem 1Theorem 1Any 2-complex, KAny 2-complex, K'' , such that is topologically , such that is topologically

equivalent to the sphere, has Euler characteristic equivalent to the sphere, has Euler characteristic

The converse of this theorem is not true because there are The converse of this theorem is not true because there are complexes with complexes with

Which are not homeomorphic to the sphere such as:Which are not homeomorphic to the sphere such as:

Two points have no faces, no edges, but two vertices, so Two points have no faces, no edges, but two vertices, so therefore it is not homeomorphic to the sphere.therefore it is not homeomorphic to the sphere.

Also, the following figure is not homeomorphic to the Also, the following figure is not homeomorphic to the sphere, but has a Euler Characteristic of 2.sphere, but has a Euler Characteristic of 2.

| ' |K

( ) 2K

2

( ) 2 0 0 2K

I. MontesI. Montes

Platonic Solids and SpherePlatonic Solids and Sphere

Definition of a regular polyhedronDefinition of a regular polyhedron: A regular : A regular polyhedron is polyhedron whose faces all have the polyhedron is polyhedron whose faces all have the same number of sides, and which also has the same number of sides, and which also has the same number of faces meeting at each vertex.same number of faces meeting at each vertex.

Definition of a platonic solidsDefinition of a platonic solids: the : the Platonic solidsPlatonic solids are are the regular polyhedra which are topologically the regular polyhedra which are topologically equivalent to the sphere. equivalent to the sphere.

Here is a description of the 5 platonic solids.Here is a description of the 5 platonic solids.

I. MontesI. Montes

TetrahedronTetrahedron

Made up of trianglesMade up of triangles Each face has 3 sidesEach face has 3 sides Three faces meet at Three faces meet at

each vertexeach vertex Vertices=4Vertices=4 Edges=6Edges=6 Faces=4Faces=4 Euler characteristic: Euler characteristic:

4 – 6 + 4 = 24 – 6 + 4 = 2

I. MontesI. Montes

CubeCube Properly called a Properly called a

hexahedronhexahedron Is made up of squaresIs made up of squares Each face has 4 sidesEach face has 4 sides 3 faces at each vertex3 faces at each vertex Vertices=8Vertices=8 Edges=12Edges=12 Faces=6Faces=6 Euler characteristic: Euler characteristic:

8 -12 + 6 = 28 -12 + 6 = 2

I. MontesI. Montes

OctahedronOctahedron Made up of trianglesMade up of triangles Each face has three Each face has three

sidessides Four faces at each Four faces at each

vertexvertex Vertices=6Vertices=6 Edges=12Edges=12 Faces=8Faces=8 Euler characteristic: Euler characteristic:

6 – 12 + 8 = 26 – 12 + 8 = 2

I. MontesI. Montes

IcosihedronIcosihedron

Made up of trianglesMade up of triangles Each face has 3 sidesEach face has 3 sides Five faces at each Five faces at each

vertexvertex Vertices=12Vertices=12 Edges=30Edges=30 Faces=20Faces=20 Euler characteristic: Euler characteristic:

12 – 30 + 20 = 212 – 30 + 20 = 2

I. MontesI. Montes

DodecahedronDodecahedron Made up of pentagonsMade up of pentagons Each face has five Each face has five

sidessides Three faces at each Three faces at each

vertexvertex Vertices=20Vertices=20 Edges=30Edges=30 Faces=12Faces=12 Euler characteristic: Euler characteristic:

20 – 30 + 12 = 220 – 30 + 12 = 2

I. MontesI. Montes

Theorem 2Theorem 2

The Platonic solids are the only The Platonic solids are the only regular polyhedra topologically regular polyhedra topologically

equivalent to a sphere.equivalent to a sphere.

I. MontesI. Montes

The ProofThe ProofSo, let So, let KK be a polyhedron whose Euler be a polyhedron whose Euler

characteristic is 2.characteristic is 2.

Let Let ff denote the number of faces in denote the number of faces in KK

Let Let ee denote the number of edges in denote the number of edges in KK

Let Let vv denote the number of vertices in denote the number of vertices in KK

Let Let nn be the number of edges on each face be the number of edges on each face

Let Let mm be the number of faces meeting at each be the number of faces meeting at each vertexvertex

From From Theorem 1Theorem 1, we know that , we know that ( ) 2K v e f

I. MontesI. Montes

Before assembly of the polyhedronLet's consider the polyhedron before it is put together.Let's consider the polyhedron before it is put together.

f'f' will be the number of faces before assembly will be the number of faces before assemblye'e' will be the number of edges before assembly will be the number of edges before assembly

v'v' will be the number of vertices before assembly will be the number of vertices before assemblyHere is the tetrahedron before assembly. Move slider to show two Here is the tetrahedron before assembly. Move slider to show two

triangles being put together.triangles being put together.

I. MontesI. Montes

The number of polygons (faces) is the same before or after The number of polygons (faces) is the same before or after assembly so assembly so

f=f'f=f'

Before attaching, each face has Before attaching, each face has nn edges and edges and nn vertices so vertices so

nf=e'=v'.nf=e'=v'.

The edges are glued together in pairs in The edges are glued together in pairs in KK, so , so

e'=nf=2ee'=nf=2e..

In assembling In assembling KK, , mm faces meet at each vertex of faces meet at each vertex of KK, so , so mm vertices from vertices from mm unglued faces are glued together to make unglued faces are glued together to make

one vertex in one vertex in KK, and , and

v'=mvv'=mv..

Thus, Thus,

v'=mv=nf=2ev'=mv=nf=2e..

So, ...So, ...

I. MontesI. Montes

First of all we start with First of all we start with the euler characteristic the euler characteristic equal to 2equal to 2

Then, so Then, so vv is is replaced and so replaced and so ff is replaced.is replaced.

Then 2 and Then 2 and ee are are factored outfactored out

Lastly, 2 and Lastly, 2 and ee are are moved to the other side moved to the other side of the equation by of the equation by dividingdividing

So, the 2’s cancel and So, the 2’s cancel and you are left with this you are left with this equation.equation.

2ev

m

2 v e f

2 2e ee

m n

1 1 12

2em n

1 1 1 1

2e m n

2ef

n

I. MontesI. Montes

Note that Note that e, n, me, n, m must be integers and that must be integers and that ee>2, >2, nn>2, >2, mm>2, so then >2, so then

Since equations with only integer solutions Since equations with only integer solutions allowed such as the one above are rather allowed such as the one above are rather

difficult to solve, we will analyze each difficult to solve, we will analyze each possible case separately:possible case separately:

1 1 1 1

2e m n

1 1

2e

I. MontesI. Montes

Since Since mm>2, the only possibilities are >2, the only possibilities are mm=3, 4, =3, 4, 5.5.

Case 1: n=3 (the polygons are triangles)

1 1 1 1 1 1 10

2 2 3 6e m m

2 1 1

3 6m

36

2m

I. MontesI. Montes

If , then so, ,If , then so, ,

, and . So this is , and . So this is

going to form the tetrahedron, which going to form the tetrahedron, which

had 4 vertices, 6 edges, and 4 faces.had 4 vertices, 6 edges, and 4 faces.

#1

1 1 1 1 1

3 2 3 6e

24

ev

m 2

4e

fn

6e3m

I. MontesI. Montes

If , then so, If , then so,

, , and . So , , and . So

this is going to form the octahedron, this is going to form the octahedron,

which had 6 vertices, 12 edges, and 8 which had 6 vertices, 12 edges, and 8

faces.faces.

#2

1 1 1 1 1

4 2 3 12e

6v 8f 12e

4m

I. MontesI. Montes

If , then so, If , then so,

, , and . So , , and . So

this is going to form the icosahedron, this is going to form the icosahedron,

which had 12 vertices, 30 edges, and 20 which had 12 vertices, 30 edges, and 20

faces.faces.

#3

1 1 1 1 1

5 2 3 30e

12v 20f 30e

5m

I. MontesI. Montes

Since Since mm>2, the only possibility is>2, the only possibility is

If , then so, If , then so,

, , and . , , and .

This is going to form the cube, which has 8 vertices, 12 This is going to form the cube, which has 8 vertices, 12

edges, and 6 faces.edges, and 6 faces.

Case 2 n=4 (the polygons are squares)1 1 1 1 1 1 1

02 2 4 4e m m

3 1 1

4 4m

44

3m

3m3m

1 1 1 1 1

4 2 3 12e

12e 6f 8v

I. MontesI. Montes

Since Since mm>2, the only possibility is>2, the only possibility is

If , then If , then

, , and . , , and .

This is going to form the dodecahedron, which has 20 This is going to form the dodecahedron, which has 20

vertices, 30 edges, and 12 faces.vertices, 30 edges, and 12 faces.

Case 3 n=5 (the polygons are pentagons)1 1 1 1 1 1 3

02 2 5 10e m m

4 1 3

5 10m

5 10

4 3m

3m3m

30e 12f 20v

I. MontesI. Montes

This cannot happen because This cannot happen because mm>2, so there >2, so there are only the 5 possibilities or solutions.are only the 5 possibilities or solutions.

Case 4 n 6(the polygons are hexagons or bigger)

1 1 1 1 1 1 1 1 10

2 2 6 3e m n m m

1 1

3m

3m

I. MontesI. Montes

ReferencesReferences

Topology of Surfaces, L. Christine KinseyTopology of Surfaces, L. Christine Kinsey WikipidiaWikipidia www.mathsisfun.comwww.mathsisfun.com This website has excellent figures also This website has excellent figures also

http://www.neubert.net/PLASpher.htmlhttp://www.neubert.net/PLASpher.html