Difference between revisions of "Euler Characteristic"
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[[Image:Dodecahedron.png|200px]] [[Image:Dodecahedron-tombstone.jpg|200px]] | [[Image:Dodecahedron.png|200px]] [[Image:Dodecahedron-tombstone.jpg|200px]] | ||
| − | Above you see a drawing of a dodecahedron and a tombstone in | + | 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). <br> So we have that V - E + F = 20 - | + | There are 20 vertices (V = 20), 30 edges (E = 30) and 12 faces (F = 12). <br> |
| + | So we have that V - E + F = ___ - ___ + ___ = ___ | ||
| + | |||
| + | ==Icosahedron== | ||
| + | |||
| + | An icosahedron is made up of triangles. There are 20 triangles in one icosahedron. | ||
| + | |||
| + | [[Image:Chem star.png|200px]] [[Image:Icosaedro.jpg|200px]] | ||
| + | |||
| + | 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). <br> | ||
| + | So we have that V - E + F = ___ - ___ + ___ = ___ | ||
Latest revision as of 12:06, 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
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).
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).
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).
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.
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.
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 = ___ - ___ + ___ = ___
