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Euler Characteristic Theorem

- By Megan Ruehl

Doodling

- Take out a piece of paper and a pen or pencil.
- Close your eyes while doodling some lines in a

region on your piece of paper. - Draw a dot at every intersection of each line.

Write it Down

- We will call each dot the vertex
- Now count each line that connects each vertex. We

will call that the edge. - Now count each region of space enclosed by each

line. We will call this space the face.

V - E F ?

- Take the total number vertices and subtract from

it the total amount of edges, then add the amount

of regions or faces that you counted. - What do you get?
- The answer should equal 2!

Definition

- This exercise verifies the Euler Characteristic

Theorem. - In our text book, The Heart of Mathematics, it

states the definition of the Euler Characteristic

Theorem. - For any connected graph in the plane, V - E F

2, where V is the number of vertices E is the

number of edges and F is the number of regions.

Leonhard Euler and Some of His Discoveries

- Born in Switzerland, in the town of Basel, on

April 15, 1707. - Basil was one of the main centers of mathematics

in Europe at the time. - Started school at age 7, while his father hired a

private mathematics tutor for him.

- Eulers talents werent recognized until after he

moved arrived in St. Petersburg on May 24, 1727,

he was 20 years old. - Some areas he worked in included the theory of

production of the human voice, the theory of

sound and music, the mechanics of vision, and his

work on telescopic and microscopic perception.

- Because of Leonhard Eulers work with telescopic

and microscopic perception the construction of

telescopes and microscopes were made possible. - 1741 he moved to Berlin.
- He worked in the Berlin Academy of Sciences and

was appointed as head of the Berlin Observatory.

- Another one of his discoveries was being able to

detect the atmosphere of Venus. - In 1761, when Venus passed over the face of the

sun, he detected the atmosphere of Venus.

Descartes

- Two hundred years before Euler started making

discoveries, a man named René Descartes noticed

something huge. - He observed that in a region with intersecting

lines being the vertices, and with gaps between

them makes regions of edges, vertices, and faces. - He noticed how the vertices minus the number of

edges plus the number of regions always equals 2.

- The only thing Descartes couldnt do was prove

it.

Why the Characteristic Theorem is Eulers

- Being familiar with the philosophies of Descartes

Leonhard Euler noticed this unfinished business. - Euler took René Descartes observations and came

up with a justification that consistently is

true. - Euler proved it as a fact. Which is obviously why

it is called the Euler Characteristic Theorem.

V - E F 2

- This is the equation he came up with.
- Letting V be the number of vertices.
- E the number of edges.
- F the number of faces or regions.
- When plugged into this formula they equal 2.
- We discovered this by doodling.

Five Platonic Solids

- Another way Euler proved this equation to be true

was studying the platonic solids. - Remember the chart we filed out in chapter four

stating each platonic solids vertices, edges,

and faces? - Add another column V - E F.

This graph shows how Euler came to a conclusion

of the formula.

Origami

- Another way we can verify this theorem is to do

the same thing we did with our doodles to the

origamis I passed out. - Lay the origami flat on your desk.
- Draw a dot at each vertex and count them.
- Now count each line connecting each dot.
- Now count the regions within those line being the

faces.

V - E F ?

- Plug in the amount for each to the equation.
- What is the answer you get?
- This example should help you understand the

theory better.

Conclusion

- To this day Euler has discovered things that are

being studied and taught. - Euler was a thinking time bomb until his death in

the year 1783. - Because of the great accomplishments during his

life and all the discoveries he has made that are

still current, Leonhard Euler is known as one of

the founding fathers of modern science.

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