Dice update
Jun. 12th, 2007 01:27 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jack1. You could eliminate the rolling problem on a d7 by using a sphere and painting seven equal areas on the surface (in an appropriately semi-symmetric non-striped way). The only flaw being it not ending on one certain face.
2. You could do that, but have a heptahedral die with little airbags that expands into a sphere for rolling, and then you collapse remotely causing it to fall on one face. But that's over-engineered.
3. ptc suggests the question with the d0 is you're looking at external faces, which should be the only ones that count. If you have a gigantic cube surrounding the earth, that could be a d0. Though rolling it is dangerous.
4. What is the generalisation of a dn for n not positive? Consider Euler's formula. Define D=2+E-V, where E=number of edges and V=number of vertices. For simply connected solids, eg. polyhedrons, D=F. Aha, we're onto a concept we can generalise. For D is also defined for other shapes†. A torus divided into n has D=n+2 which isn't very helpful‡. However, two spheres with n total regions has D=n-2.
So if n is two, (eg. Each sphere consists of a single vertex and single face, and no edges -- I think this isn't pathological by euler characteristic standards), D=0 as desired.
Except this isn't very vivid. I can show you two marbles and say "Lo! These two marbles are a no-sided die" and probably be a hit at Buddhist drug raves, but it doesn't actually illuminate the problem very helpfully, or suggest interesting further maths/engineering to do :)
† Use the Euler Characteristic, explanations tomorrow. In short, V-E+F=Chi, where Chi is defined topologically. Polyhedrons are all topologically equivalent to a sphere, and Chi is two. Donuts and holed solids have Chi=2-2g where g is the genus is number of holes. Two spheres have Chi=4, as you can see by adding up V, E, F and getting 2 for each.
‡ Except maybe to act as a warning bell that our generalisation may be starting to come apart at the seams already. In what sense is a torus havign seven regions equivalent to a d9?
2. You could do that, but have a heptahedral die with little airbags that expands into a sphere for rolling, and then you collapse remotely causing it to fall on one face. But that's over-engineered.
3. ptc suggests the question with the d0 is you're looking at external faces, which should be the only ones that count. If you have a gigantic cube surrounding the earth, that could be a d0. Though rolling it is dangerous.
4. What is the generalisation of a dn for n not positive? Consider Euler's formula. Define D=2+E-V, where E=number of edges and V=number of vertices. For simply connected solids, eg. polyhedrons, D=F. Aha, we're onto a concept we can generalise. For D is also defined for other shapes†. A torus divided into n has D=n+2 which isn't very helpful‡. However, two spheres with n total regions has D=n-2.
So if n is two, (eg. Each sphere consists of a single vertex and single face, and no edges -- I think this isn't pathological by euler characteristic standards), D=0 as desired.
Except this isn't very vivid. I can show you two marbles and say "Lo! These two marbles are a no-sided die" and probably be a hit at Buddhist drug raves, but it doesn't actually illuminate the problem very helpfully, or suggest interesting further maths/engineering to do :)
† Use the Euler Characteristic, explanations tomorrow. In short, V-E+F=Chi, where Chi is defined topologically. Polyhedrons are all topologically equivalent to a sphere, and Chi is two. Donuts and holed solids have Chi=2-2g where g is the genus is number of holes. Two spheres have Chi=4, as you can see by adding up V, E, F and getting 2 for each.
‡ Except maybe to act as a warning bell that our generalisation may be starting to come apart at the seams already. In what sense is a torus havign seven regions equivalent to a d9?
