Difference between revisions of "Euler Characteristic"

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(New page: {{k12}} {{worksheet}} In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces f, edges e, and vertices v of a polyhedron: {{boxed|<m...)
 
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{{worksheet}}
 
{{worksheet}}
  
In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces f, edges e, and vertices v of a polyhedron:
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In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron:
  
{{boxed|<math>v-e + f = 2</math>}}
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He found that V - E + F = 2
  
As a first step to understanding this equation, we will calculate v, e, and f for the Platonic solids and check that
+
Let's check this formula on some of the shapes below.
<math>v-e + f = 2</math> in these cases.
 
  
 
===Tetrahedron===
 
===Tetrahedron===
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 +
[[Image:Tetrahedron-labeled.svg|500px]]
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 +
How many vertices ("corners") V do you see? _______
 +
 +
How may edges E do you see?  ______
 +
 +
How many faces ("sides") F do you see?  ______
 +
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Now find V - E + F =  ___ - ___ + ___ =
 +
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Did the answer come out to 2?  ______
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===Octahedron===
 +
 +
[[Image:Octahedron-labeled.svg|500px]]
 +
 +
How many vertices ("corners") V do you see? _______
 +
 +
How may edges E do you see?  ______
 +
 +
How many faces ("sides") F do you see? This may be easier to count in the figure on the right.  ______
 +
 +
Now find V - E + F =  ___ - ___ + ___ =
 +
 +
Did the answer come out to 2?  ______

Revision as of 10:44, 6 May 2009

K-12: Materials at high school level.



In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron:

He found that V - E + F = 2

Let's check this formula on some of the shapes below.

Tetrahedron

Tetrahedron-labeled.svg

How many vertices ("corners") V do you see? _______

How may edges E do you see? ______

How many faces ("sides") F do you see? ______

Now find V - E + F = ___ - ___ + ___ =

Did the answer come out to 2? ______


Octahedron

Octahedron-labeled.svg

How many vertices ("corners") V do you see? _______

How may edges E do you see? ______

How many faces ("sides") F do you see? This may be easier to count in the figure on the right. ______

Now find V - E + F = ___ - ___ + ___ =

Did the answer come out to 2? ______