Euler's Formula
(There is another "Euler's Formula" about complex numbers,
this page is about the one used in Geometry and Graphs)
Euler's Formula
For any polyhedron that doesn't intersect itself, the
- Number of Faces
- plus the Number of Vertices (corner points)
- minus the Number of Edges
always equals 2
This can be written: F + V - E = 2
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Try it on the cube:
A cube has 6 Faces, 8 Vertices, and 12 Edges,
so:
6 + 8 - 12 = 2
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To see why this works, imagine taking the cube and adding an edge
(say from corner to corner of one face).
You will have an extra edge, plus an extra face:
7 + 8 - 13 = 2
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Likewise if you included another vertex (say halfway along a line)
you would get an extra edge, too.
6 + 9 - 13 = 2.
"No matter what you do, you always end up with 2"
(But only for this type of Polyhedron ... read on!)
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Example With Platonic Solids
Let's try with the 5 Platonic Solids
* But Not Always ...
Now that you see how this works, I am going to show you how it doesn't work ...!
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What if I joined up two opposite corners of the icosahedron?
It is still an icosahedron (but no longer regular or convex).
In fact it looks a bit like a drum where someone has stitched the top and bottom together.
Now, there would be the same number of edges and faces ... but one less vertex!
So:
F + V - E = 1
Oh No! It doesn't always add to 2!
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The reason it didn't work was that this new shape is basically different.
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All Platonic Solids (and many other solids) are like a Sphere ... you can reshape them so that they become a sphere (move their corner points, then curve their faces a bit). |
But you can't do that with this new shape, because that joined bit in the middle won't let you!
Euler Characteristic
So, F+V-E can equal 2, or 1, and maybe other values, so the more general formula is
F + V - E = χ
Where χ is called the "Euler Characteristic".
Here are a few examples:
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And the Euler Characteristic can also be less than zero.
This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so:
F + V - E = -2
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In fact the Euler Characteristic is a basic idea in Topology (the study of the Nature of Space).
Donut and Coffee Cup
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Lastly, this discussion would be incomplete without showing you that a Donut and a Coffee Cup are really the same!
Well, they can be deformed into one another.
We say the two objects are "homeomorphic" (from Greek homoios = identical and morphe = shape)
Just like the platonic solids are homeomorphic to the sphere.
(Animation courtesy of Wikipedia User:Kieff)
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