Christmas Cheer
2006-12-19
The only decoration we have in the office is this:
Don't blame me, I was bored.
For bonus points, tell me its name, and/or how many sides it has...
Addenda:
It's an icosahedron, having twenty faces each of which is an equilateral triangle. Here's one to play with:
Feel free to click and drag...
As for how they knew, they might be familiar with these:
The maths way goes a little like this:
Assume that the view of this side is representative of the whole solid (which it is, I'm not that mean).
Assume it's a regular [Platonic] solid (all angles equal, all edges the same length), because all the observable faces appear to be.
There are three regular Platonic solids that have triangular faces, having 4 faces (Tetrahedron), 8 faces (Octahedron) and 20 faces (Icosahedron). Approximately 8 faces are visible and by assumption 1 there are more on the other side, so it must be 20.
The proof that there are exactly three Platonic solids with triangular faces (in addition to one with squares and another with pentagons) is left as an exercise to the reader, since it's quite elegant ;-)
The Java applet is one that came with an old version of the JDK: it's probably still in the latest version. The object file I wrote; it turns out that the coordinates for the points are the cyclic permutations of {±1,±τ,0}.
Don't blame me, I was bored.
For bonus points, tell me its name, and/or how many sides it has...
Addenda:
It's an icosahedron, having twenty faces each of which is an equilateral triangle. Here's one to play with:
Feel free to click and drag...
As for how they knew, they might be familiar with these:
The maths way goes a little like this:
Assume that the view of this side is representative of the whole solid (which it is, I'm not that mean).
Assume it's a regular [Platonic] solid (all angles equal, all edges the same length), because all the observable faces appear to be.
There are three regular Platonic solids that have triangular faces, having 4 faces (Tetrahedron), 8 faces (Octahedron) and 20 faces (Icosahedron). Approximately 8 faces are visible and by assumption 1 there are more on the other side, so it must be 20.
The proof that there are exactly three Platonic solids with triangular faces (in addition to one with squares and another with pentagons) is left as an exercise to the reader, since it's quite elegant ;-)
The Java applet is one that came with an old version of the JDK: it's probably still in the latest version. The object file I wrote; it turns out that the coordinates for the points are the cyclic permutations of {±1,±τ,0}.
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