1. Euler’s characteristic and the sphere
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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.
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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.
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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.
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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.
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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.
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7. Few examples of complexes
A topological object can be represented by many complexes.
Complexes on the sphere.
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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
Write down the formula of Euler Characteristic
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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
Show that the figures have that Euler characteristic
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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.
Show the figures have that euler characteristic
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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.
Talk about the picture and explain that I will show each one individually
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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 – = 2
Write down all euler characteristics with name of each polyhedron
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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: = 2
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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 – = 2
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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 – = 2
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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 – = 2
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17. Theorem 2
The Platonic solids are the only regular polyhedra topologically equivalent to a sphere.
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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
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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.
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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, ...
Write down these formulas. They will be manipulated later to find the separate vertices, edges and faces.
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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.
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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:
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23. Case 1: n=3 (the polygons are triangles)
Since m>2, the only possibilities are m=3, 4, 5.
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24. #1
If , then so, ,
, and So this is going to form the tetrahedron, which had 4 vertices, 6 edges, and 4 faces.
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25. #2
If , then so,
, , and So this is going to form the octahedron, which had 6 vertices, 12 edges, and 8 faces.
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26. #3
If , then so,
, , and So this is going to form the icosahedron, which had 12 vertices, 30 edges, and 20 faces.
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27. Case 2 n=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.
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28. Case 3 n=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.
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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.
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30. References Topology of Surfaces, L. Christine Kinsey Wikipidia
This website has excellent figures also
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