1. Topology and The Euler Number
Richard Earl
Mathematical Institute

2. Early Geometry
Much of the geometry taught in schools was first discovered by the Ancient Greeks
Some names may be familiar from their discoveries and theorems:
Pythagoras
Euclid
Archimedes
Apollonius

3. Platonic Solids Vertices, Edges, Faces In particular they found
the five regular polyhedra
known as Platonic solids.
However, as they were
mainly interested in their
geometry (lengths, areas,
volumes) they missed a
simple pattern relating the solids’
Vertices, Edges, Faces

4. Some Examples Image Name Tetra- hedron Cuboid Pyramid Dodeca-hedron
Carbon 60 V 4 8 5 20 60 E 6 12 30 90 F 32
V-E+F 2

5. Leonard Euler (1707-1783) The relation is named after the Swiss
V – E + F = 2
is named after the Swiss
mathematician, Leonard Euler,
who discovered it in 1758.

6. Topology
Having nothing to do with lengths or areas, this relationship isn’t a result from Geometry.
Rather it was one of the first ever results from
Topology
The London Tube
Map is an example of
a topological map. It
shows connections,
but not distances.

7. Topology (Ctd.) Topology is sometimes called Rubber-sheet Geometry.
Unlike in geometry, topological properties remain true even when objects are stretched and deformed, rather than just rotated and translated.
The Tube map doesn’t reflect the distances involved in the real Underground. But certain facts remain true: e.g. which stations are connected to which others, and how many stops there are between stations.

8. Proving the Formula
Remove a face of the polyhedron and flatten the remaining faces.
Cut the remaining faces into triangles.
Introducing a triangle
increases F by 1
increases E by 1
keeps V the same
And so has no effect on
V – E + F

9. Proof (Ctd.) Begin removing the surrounding triangles to: Or
decrease E by 1
decrease F by 1
keep V the same Or
decrease E by 2
decrease V by 1
Both of which keep V–E+F the same.

10. Proof (Ctd.) When left with one triangle we have
V – E + F = 3 – = 1.
As we began by removing a face then originally
V – E + F = 2.
QED(?)

11. Counter-examples (?) Shape Sphere Cylinder Torus (as above) V 160 E 2
160 E 2
320 F 1 3
V – E + F

12. Counter-examples (Ctd.)
These are only counter-examples because we didn’t state our relation carefully enough.
A proper face can’t be spherical.
And the single “face” around the cylinder isn’t allowed.
Faces have to be disc or polygon shaped.
The torus fails because of the hole through it.

13. Tetrahedron, pyramid, cuboid,
Back to Topology
The reason we got V – E + F = 2 for the
Tetrahedron, pyramid, cuboid,
dodecahedron, C60
Is because they each have the underlying shape of a sphere.
Each of them could be deformed from a sphere by flattening out faces.
From the point of view of topology they’re all the same.

14. Genus V – E + F = 2 – 2g V – E + F is called the Euler Number
Any polyhedron, with a single hole through it, will give
V – E + F = 0
Because its underlying shape is a torus.
The number of holes in a shape is called its genus, and has the symbol g.
For such shapes,
V – E + F = 2 – 2g
V – E + F is called the Euler Number

15. Nets To make polyhedra we can glue together the edges of a net.
For example the net below would make a dodecahedron.

16. Gluing Together Surfaces
Similarly if we glue together the sides of a square we first make a cylinder.
Then gluing the top and bottom produces a torus.

17. More Surfaces
If instead we glue the ends of the cylinder in reverse fashion, we get a Klein Bottle.
It is named after the German mathematician Felix Klein ( )

18. The Klein Bottle
The Klein Bottle has no inside or outside – it is called non-orientable.
This is similar to the Mobius Strip having one side.
It’s impossible to draw the Klein Bottle in three dimensions without it intersecting itself.

19. Distinguishing Surfaces
The torus and Klein Bottle were both made
from a single square (F=1)
with the four edges glued in pairs to make two edges (E=2)
with the four corners glued together to make the same point (V=1)
The arrows on the squares indicate how to glue the sides.

20. Distinguishing Surfaces (Ctd.)
So Euler’s formula for both the torus and Klein Bottle gives
V – E + F = 1 – = 0
Yet the torus and Klein Bottle have different shapes as one is orientable (two-sided) and one is non-orientable (one-sided).

21. Classification Theorem
But it turns out the Euler number is enough to distinguish between the two-sided surfaces.
Theorem: A surface which is
Closed (no boundary)
Orientable (two-sided)
has the same underlying shape as a torus with
g holes (for some g = 0,1,2,3,…) and has Euler
number 2 – 2g.

22. Examples Below are tori (plural of torus) with 0, 1, 2, 3 holes.
Their Euler numbers are 2, 0, -2, -4.

23. One Last Problem
Question: Below is an octagon, the edges of which we will glue together according to labelled arrows.
When we are done what type of surface will we have?

24. Solution There is one face – the octagon itself.
There are four edges a, b, c, d, as the eight original edges are glued in pairs.
When we start gluing, we also glue some of the points together:
front of a = front of b =
front of c = front of d =
back of b = back of c =
back of a = back of d
So there is only one vertex

25. Solution (Ctd.) As V = 1, E = 4, F = 1, then
Euler Number = 1 – = -2
which is the Euler number of a torus with two holes.
So after gluing,
becomes