Euler Characteristic

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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

A tetrahedon is a simple shape that is made up of 4 triangles. Below you see a picture with labels on the vertices (V) and edges (E).

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

A tetrahedon is a simple shape that is made up of 8 triangles. Below you see two pictures, the one on the left is given with labels on the vertices (V) and edges (E).

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? ______

Cube

A cube is a simple shape that is made up of 6 squares. Below you see a picture with labels on the vertices (V) and edges (E).

Cube-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? ______


Dodecahedron

A dodecahedron is made up of pentagons (5-gons). There are 12 pentagons in one dodecahedron.

Dodecahedron.png Dodecahedron-tombstone.jpg

Above you see a drawing of a dodecahedron and a tombstone in the form of a dodecahedron.

There are 20 vertices (V = 20), 30 edges (E = 30) and 12 faces (F = 12).
So we have that V - E + F = ___ - ___ + ___ = ___

Icosahedron

An icosahedron is made up of triangles. There are 20 triangles in one icosahedron.

Chem star.png Icosaedro.jpg

Above you see a drawing of a icosahedron and a game piece (like dice) in the form of a icosahedron.

There are 12 vertices (V = 12), 30 edges (E = 30) and 20 faces (F = 20).
So we have that V - E + F = ___ - ___ + ___ = ___