posted
I was looking at various multi-sided dice trying to relate the number of sides on equilateral polygons to the minimum number of sides that would be on completly symmetrical die made from repetition of those polygons.
for example, with equilateral triangles, the minimum number of sides on a symmetrical die is 6 (the 5 sided pyramid-shape has a square base and 4 triangular sides) so not symmetrical.
a square can be made into a six-sided die, as we are most familiar with.
a 5 sided equilateral polygon can be made into a D20, if I got it right.
I was trying to figure out the number of sides that a six-sided equilateral polygon (a hexagon) would make if you turned it into a die.
and WHAM!
one of those stupid die collector/gamer sites downloaded SOMETHING to my computer and it started opening and closing stuff and making my pointer into weird shapes, etc.
Virus checking didn't catch it. It must've been transient.
I quickly just hit the power switch 'cuz nothing else worked to stop the merry mayhem.
I have thoroughly checked my machine and there is no adware, spyware, malware, virus or anything. BUT...probably because I turned off my machine while the drive was cycling doing read-writes, I lost the friends and blacklist lists for my mail cleaner program, and my index for my disk search (copernic) was hosed as well.
Sadly, I don't know which site did it because, as is my wont when using Google, I opened several of the more promising sites at once in new tabs.
quote:Originally posted by Bob_Scopatz: . . . for example, with equilateral triangles, the minimum number of sides on a symmetrical die is 6 (the 5 sided pyramid-shape has a square base and 4 triangular sides) so not symmetrical.
posted
Enigmatic! This is all your fault -- he needs to play more D&D. With the full dice set, not that wimpy little adventure you ran at Christmas.
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quote:Originally posted by Bob_Scopatz: I couldn't find a d-4 that was composed of four equal triangles. dkw says that's what a d4 is...she's looking it up for me.
Ic -- completely symmetrical as in all sides are exactly the same shape.
posted
Okay, I think there's a point at which you can't build die out of repeating shapes. And I wonder if the issue is that any shape that doesn't "tile" on a flat surface, completely covering that surface, can't be turned into a die either...
Hmm...
At any rate, I can't find anything for heptagons or higher and I think the problem is the same for both tiling a flat surface and making a three-dimensional "spheroid" out of repeating shapes of n-equilateral sides. There start to be gaps between the items once you get beyond six sides.
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posted
rivka, when I took HS chemistry, tetrahedral molecules didn't exist. We had to make do clunky octahedrals and dodecahedrals. And we carried them around in large leather cases tied to our belts.
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posted
It is so weird but I was working on figuring out what a hexagonal polygon would be today to come up with a three dimensional hex for a spell device. I thought it was a twelve sidded dice but that is a pentagon so what did you come up with?
posted
I found a website from the UK with polyhedron generating software (for $25). It was linked to a site with all the regular polyhedra that someone has built.
I didn't spot anything with high-order figures than pentagons. Although that D32 sure looks like it's made from purely symmetrical hexagons...just not sure whether that is strictly "right" or not.
Anyway, there's nothing out there with repeating figures of 7-or more equal sides.
Something about the length of the sides and the angles that makes it impossible to "tile" after that point. I know I've tiled with hexagons, so I figure that you could make that into a spheroid.
But I'm no mathematician.
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posted
I'm like Icarus, what did you mean by "completly symmetrical die"?
If you mean maximal symmetry, that's the Platonic Solids: tetrahedron (d4), cube (d6), octahedron (d8), dodecahedron(d12), and icosahedron(d20).
The d4, d8, and d20 are all made up of equilateral triangles.
Bob, the d32 you link to is irregular, it's lacking in symmetry. The faces aren't equilateral, and they don't meet smoothly at the vertices.
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posted
Thanks...I just thought it was probably poorly made, but that hexagons would make a D32 if done properly.
My criterion for symmetry was as follows:
I was looking for the minimum number of facets that a three-dimensional object (a polyhedron) would have if it was constructed from equilateral polygons.
So, four identical equilateral triangles can be used to make a 4 sided polyhedron as the minimum number of facets.
six identical squares can be used to make a six-faceted polyhedron.
12 identical equilateral pentagons can be used to make a 12-faceted dodecahedron.
and ... I think... 32 identical equilateral hexagons can be used to make a 32-faceted polyhedron, although I'm not sure on that.
But then, that's it. There is no polyhedron composed entirely of 7-sided equiliateral polygons.
Nor for 8-sided polygons, and so forth.
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quote:Originally posted by Bob_Scopatz: I found a website from the UK with polyhedron generating software (for $25). It was linked to a site with all the regular polyhedra that someone has built.
I didn't spot anything with high-order figures than pentagons. Although that D32 sure looks like it's made from purely symmetrical hexagons...just not sure whether that is strictly "right" or not.
Anyway, there's nothing out there with repeating figures of 7-or more equal sides.
Something about the length of the sides and the angles that makes it impossible to "tile" after that point. I know I've tiled with hexagons, so I figure that you could make that into a spheroid.
But I'm no mathematician.
You can tile a plane with hexagons, sure. But you can't make a completely symmetrical solid with them. Look closely at this d32 you linked to. The sides aren't all of equal length, and neither are the angles.
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quote: The truncated icosahedron is the 32-faced Archimedean solid A_(11) corresponding to the facial arrangement 20{6}+12{5}. It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also the C_(60) structure of pure carbon known as buckyballs a.k.a. fullerenes.
It's 32-faced, but uses hexagons and pentagons. Soccer balls, atomic bombs, and buckyballs, all from the same shape.
edit: Sorry, Bob, I erased a post after this because I thought it could be wrong, and not worth editing. It was about hexagons tiling a plane but unable to make into a regular polyhedron.
quote:Originally posted by Bob_Scopatz: rivka, when I took HS chemistry, tetrahedral molecules didn't exist. We had to make do clunky octahedrals and dodecahedrals. And we carried them around in large leather cases tied to our belts.
So diamonds didn't exist back then when you were young, huh? Wow, Bob, you're even older than I thought!
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posted
Well, you CAN indeed tile a plane with hexagons, so that's not the issue. Whether you could wrap them into a polyhedron without "fudging" is the question.
quote:A polyhedron is said to be regular if its faces and vertex figures are regular (not necessarily convex) polygons (Coxeter 1973, p. 16). Using this definition, there are a total of nine regular polyhedra, five being the convex Platonic solids and four being the concave (stellated) Kepler-Poinsot solids. However, the term "regular polyhedra" is sometimes used to refer exclusively to the Platonic solids (Cromwell 1997, p. 53). The dual polyhedra of the Platonic solids are not new polyhedra, but are themselves Platonic solids.
quote:Originally posted by Bob_Scopatz: rivka, when I took HS chemistry, tetrahedral molecules didn't exist. We had to make do clunky octahedrals and dodecahedrals. And we carried them around in large leather cases tied to our belts.
So diamonds didn't exist back then when you were young, huh? Wow, Bob, you're even older than I thought!
Diamonds take millions of years to form.
This didn't happen until 1962. Until then, a lot of women were walking around with lumps of coal on their fingers, wondering if they'd been duped when their boyfriend promised them, "Trust me. This will be really valuable some day."
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posted
I found this linked to on a dice page you supplied Bob. I thought Euler's equation would constrain the solution for dice, but was too lazy to dig into it. Nice to find a convenient summary for why dice cannot have faces with more than 5 sides.
Later at the link there's a nice table summarizing all possible dice.
quote: Euler's equation, which is good for any convex polyhedron of three or more faces (curved or otherwise), states that V + N = E + 2, where V is the number of vertices in the polyhedron, N the number of faces, and E the number of edges. Since the faces on a proper die are identical, they must all have the same number of sides (and corners), a number we'll call M. We have E = M*N/2, which tells us that N only can be odd if M is even (or E won't be a whole number). Using this substitution, Euler's equation becomes:
(1) V = N(M/2 - 1) + 2
This equation does not cover the situations where M=0 or M=1. These, however, only have one solution each, namely:
(A) N=1, M=0, V=0 (a sphere) (B) N=2, M=1, V=0 (a lens)
In the case M=2, (1) simplifies to V = 2. This has solutions for all N greater than 2, but requires curved surfaces. The shape of these "dice" are prisms that taper in both ends, with N-sided cross sections:
(C) N>=3, M=2, V=2 (edged cigar shapes)
As will be shown later, this is the only way to make dice with an odd number of faces (not counting the sphere). N*M, which with (1) tells us that M =< 6 - 12/N, which means that M <6. The faces must thus be triangles, quadrangles, or pentagons.
posted
I found Euclid's proof that there were only five platonic solids, and I rushed to find this thread, only to find that I had basically been beaten to the punch by rivka and Morbo.
(The "platonic dice" are d4, d6, d8, d12, and d20.) (d10 is not a platonic solid because the faces, which are quadrilaterals, are not regular--all the sides aren't congruent and neither are the angles.) (In the picture of the 32-sided die, several of the faces are clearly pentagons.)
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