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.

• 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.

Write it Down

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.

Euler’s talents weren’t 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 Euler’s 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.”

DescartesTwo 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 couldn’t do was prove it.

Why the Characteristic Theorem is Euler’s

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.

NumberOf

Vertices

Number Of

Edges

Number Of

FacesV - E + F

Tetrahedron 4 6 4 2

Cube 8 12 6 2

Octahedron 6 12 8 2

Dodecahedron 20 30 12 2

Icosahedron 12 30 20 2

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 origami’s 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.