Transcript Euler Characteristic Theorem - ceadserv1

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.
 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.”
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 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.
Number Number Number
Of
Of
Of
Vertices Edges
Faces
V-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.