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07-05-2008, 12:31 AM
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#843 (permalink) | |
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Bodhar agus leath dall
 
Join Date: Mar 2008
Location: Middle of dog pack
Posts: 16,083
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Quote:
__________________
 It's a joke Nathan!
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07-05-2008, 01:07 AM
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#847 (permalink) | |
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Registered User
Join Date: Jun 2004
Location: Bellows Falls, VT
Posts: 1,277
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Quote:
Polychoron is the word for the four dimensional shapes. Polygon is the word for the two dimensional shapes like squares and triangles. The general word for any dimension is polytope. This object is one of the convex uniform polychora, meaning it's not self-intersecting and the word uniform here means the vertices are all the same. There are 64 such polychora, not including the infinite set analogous to the infinite set of prisms in three dimensions. The clue infinity, 5, 6, 3, 3, 3, 3.... was the numbers of convex regular polytopes in each dimension from two onwards. There is an infinite number of convex regular polygons, including the equilateral triangle, square, regular pentagon, etc. Then in 3d, there are the five Platonic solids. In four dimensions, there are six such shapes. In five dimensions and beyond, there are just three each. The object is one of the six. It's self-dual and has no lower dimensional analogy. It's because most self-dual polytopes are polygons or analogies of the tetrahedron called simplexes. This object is the only Euclidean polytope that is neither. Counting the number of red squares in one of the clues should tell you which one of the six it is, if you find the information on those six. |
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07-05-2008, 01:47 AM
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#852 (permalink) | |
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Registered User
Join Date: Jun 2004
Location: Bellows Falls, VT
Posts: 1,277
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Quote:
The cuboctahedron is a plain old three dimensional creature that I've built before. The object in the pictures I posted does look like a cuboctahedron on the outside from certain points of view in four dimensions, but looks different from other directions. The object is one of the 6 convex regular four dimensional polytopes. Its name is really simple. |
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07-05-2008, 02:00 AM
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#856 (permalink) |
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Registered User
Join Date: Jun 2004
Location: Bellows Falls, VT
Posts: 1,277
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Clue to help finding the name:
The five Platonic solids have names based on the number of faces they have. Tetrahedron -> tetra -> 4 cube -> hexahedron -> hexa -> 6 octahedron -> octa -> 8 dodecahedron -> dodeca -> 12 icosahedron -> icosi -> 20 The -hedron part means face. Polyhedra are three dimensional and have zero dimensional vertices, one dimensional edges and two dimensional faces. Polychora are four dimensional and are made of polyhedra put together into one 4d object. Those 3d parts are called cells. |
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07-05-2008, 02:03 AM
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#857 (permalink) |
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Registered User
Join Date: Jun 2004
Location: Bellows Falls, VT
Posts: 1,277
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24-cell
  Yes, footballnutt got the right object. Those are the names for it plus it's also called the 24-cell because it has 24 octahedral cells. 3d space doesn't have enough room to fit together all of those 24 without distorting them, hence the projections used in the pictures. Looking at a projection of the 4d object into 3d space, like in the first picture is looking at the 3d shadow of the 4d object. It's like looking at the 2d shadow of a 3d object. The other picture with red squares is a projection into 2d space. Those red squares are the 24 points connected by 96 edges, which make up 96 triangular faces that are put together into 24 octahedral cells. The 24-cell is the special polytope that is self-dual while not being a polygon or a simplex. There is no others like that in any other dimension of Euclidean space. A polytope that is dual to another one has vertices that correspond to the faces of the other one and vice versa. For example the cube and octahedron are duals of each other. The 6 points of the octahedron correspond to the 6 faces of the cube and the 8 corners of the cube correspond to the 8 triangles of the octahedron. The tetrahedron is the dual of itself, being the simplex in 3d space, as well as the regular triangle, which is the 2d simplex, plus all other regular polygons. The 24-cell is self-dual, yet isn't a polygon or simplex, making it special. I used the same program used to make some of those pictures to make the picture of the bunch of 24 red octahedra I posted. Fold that in 4d and you get the 24-cell. Here is the tiling of the 24-cell across 4d space I also referred to. Many copies of the 24-cell put together can fill 4d space like many copies of the cube can fill 3d space. 4d space is said to be the richest one because there's so many forms with 64 convex uniform ones outside of infinite series, as opposed to the 18 we have besides prisms in 3d. Including nonconvex uniform ones brings us to 75 (plus one special one) in 3d and well over 1000 in 4d. Then for some reason, spaces of five and higher dimensions have only 3 convex regular polytopes each, rather than 5 and 6 for 3d and 4d space. And that's just flat Euclidean space. There's more in curved space, both positively and negatively curved. Whew I tried to explain all of the clues I used.  Those special properties, especially the self-dual one, was why I picked the 24-cell. I thought the 4d analogy of the cube would be too easy. This was inspired by jenni-m's posting of the dodecahedron picture, which I recognized instantly. For all of your trouble, I give you a link to a calendar printed onto a dodecahedron, one month on each side. Enjoy! 12 sided calendar You are next footballnutt!
Last edited by RedFox; 07-05-2008 at 02:47 AM. |
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