Euler’s characteristic and the sphere

1. Euler’s characteristic and the sphere I. Montes

2. Definition of a cell An n-cell is a set whose interior is homeomorphic to the n-dimensional disc with the additional property that its boundary or frontier must be divided into a finite number of lower-dimensional cells, called the faces of the n-cell. • A 0-dimensional cell is a point A. • A 1-dimensional cell is a line segment a=AB, and A<a, B<a. • A 2-dimensional cell is a polygon (often a triangle) such as ABC, and then AB, BC, AC . Note that • A 3-dimensional cell is a solid polyhedron (often a tetrahedron), with polygons, edges, and vertices as faces. I. Montes

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

4. Not examples of cells The figure on the left is not a cell but the one on the right is a cell. The figure on the left is not a cell because there are no vertices. The figure on the right is a cell because it has three vertices, three edges and one face. I. Montes

5. Cells form complexes Cells are glued together to form complexes, by gluing edge to edge and vertex to vertex and identifying higher-dimensional cells in a similar manner. Definition of a complex: A complex K is finite set of cells, such that: • if is a cell in K, then all faces of are elements of K; • If and are cells in K, then The dimension of K is the dimension of its highest-dimension cell. I. Montes

6. Not examples of complexes Complexes cannot intersect A complex is more than a set of points, since it also comes equipped with the structure given by the allotment of its points into cells of various dimensions. In each case above, notice that the intersections are homeomorphic to cells ,but are not among the cells of the complex K. I. Montes

7. Few examples of complexes A topological object can be represented by many complexes. Complexes on the sphere. I. Montes

8. Definition of a Euler Characteristic Let K be a complex. The Euler characteristic of K is For 2-complexes; let f = #{faces}, e = #{edges}, and v = #{vertices}, and then the Euler characteristic may be written as I. Montes

9. Example of how to find Euler Characteristic Consider a polygon with n sides, shown here. The complex K has n vertices, n edges, and one face, so Another examples is K' given by the standard planar diagram of the sphere in the following figure. K' has two vertices (P and Q), one edge, and one face, so P Q I. Montes

10. Theorem 1 Any 2-complex, K' , such that is topologically equivalent to the sphere, has Euler characteristic The converse of this theorem is not true because there are complexes with Which are not homeomorphic to the sphere such as: Two points have no faces, no edges, but two vertices, so therefore it is not homeomorphic to the sphere. Also, the following figure is not homeomorphic to the sphere, but has a Euler Characteristic of 2. I. Montes

11. Platonic Solids and Sphere Definition of a regular polyhedron: A regular polyhedron is polyhedron whose faces all have the same number of sides, and which also has the same number of faces meeting at each vertex. Definition of a platonic solids: the Platonic solids are the regular polyhedra which are topologically equivalent to the sphere. Here is a description of the 5 platonic solids. I. Montes

12. Tetrahedron • Made up of triangles • Each face has 3 sides • Three faces meet at each vertex • Vertices=4 • Edges=6 • Faces=4 • Euler characteristic: 4 – 6 + 4 = 2 I. Montes

13. Cube • Properly called a hexahedron • Is made up of squares • Each face has 4 sides • 3 faces at each vertex • Vertices=8 • Edges=12 • Faces=6 • Euler characteristic: 8 -12 + 6 = 2 I. Montes

14. Octahedron • Made up of triangles • Each face has three sides • Four faces at each vertex • Vertices=6 • Edges=12 • Faces=8 • Euler characteristic: 6 – 12 + 8 = 2 I. Montes

15. Icosihedron • Made up of triangles • Each face has 3 sides • Five faces at each vertex • Vertices=12 • Edges=30 • Faces=20 • Euler characteristic: 12 – 30 + 20 = 2 I. Montes

16. Dodecahedron • Made up of pentagons • Each face has five sides • Three faces at each vertex • Vertices=20 • Edges=30 • Faces=12 • Euler characteristic: 20 – 30 + 12 = 2 I. Montes

17. Theorem 2 The Platonic solids are the only regular polyhedra topologically equivalent to a sphere. I. Montes

18. The Proof So, let K be a polyhedron whose Euler characteristic is 2. • Let f denote the number of faces in K • Let e denote the number of edges in K • Let v denote the number of vertices in K • Let n be the number of edges on each face • Let m be the number of faces meeting at each vertex • From Theorem 1, we know that I. Montes

19. Before assembly of the polyhedron Let's consider the polyhedron before it is put together. f' will be the number of faces before assembly e' will be the number of edges before assembly v' will be the number of vertices before assembly Here is the tetrahedron before assembly. Move slider to show two triangles being put together. I. Montes

20. The number of polygons (faces) is the same before or after assembly so f=f' Before attaching, each face has n edges and n vertices so nf=e'=v'. The edges are glued together in pairs in K, so e'=nf=2e. In assembling K, m faces meet at each vertex of K, so m vertices from m unglued faces are glued together to make one vertex in K, and v'=mv. Thus, v'=mv=nf=2e. So, ... I. Montes

21. First of all we start with the euler characteristic equal to 2 • Then, so v is replaced and so f is replaced. • Then 2 and e are factored out • Lastly, 2 and e are moved to the other side of the equation by dividing • So, the 2’s cancel and you are left with this equation. I. Montes

22. Note that e, n, m must be integers and that e>2, n>2, m>2, so then Since equations with only integer solutions allowed such as the one above are rather difficult to solve, we will analyze each possible case separately: I. Montes

23. Case 1: n=3 (the polygons are triangles) Since m>2, the only possibilities are m=3, 4, 5. I. Montes

24. #1 If , then so, , , and . So this is going to form the tetrahedron, which had 4 vertices, 6 edges, and 4 faces. I. Montes

25. #2 If , then so, , , and . So this is going to form the octahedron, which had 6 vertices, 12 edges, and 8 faces. I. Montes

26. #3 If , then so, , , and . So this is going to form the icosahedron, which had 12 vertices, 30 edges, and 20 faces. I. Montes

27. Case 2n=4 (the polygons are squares) Since m>2, the only possibility is If , then so, , , and . This is going to form the cube, which has 8 vertices, 12 edges, and 6 faces. I. Montes

28. Case 3n=5 (the polygons are pentagons) Since m>2, the only possibility is If , then , , and . This is going to form the dodecahedron, which has 20 vertices, 30 edges, and 12 faces. I. Montes

29. Case 4 n 6(the polygons are hexagons or bigger) This cannot happen because m>2, so there are only the 5 possibilities or solutions. I. Montes

30. References • Topology of Surfaces, L. Christine Kinsey • Wikipidia • www.mathsisfun.com • This website has excellent figures also http://www.neubert.net/PLASpher.html I. Montes