Cosmochoron (EntityTopic, 12)
From Hi.gher. Space
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| + | The '''hecatonicosachoron''', or '''600-cell''', is a 4D polytope bounded by 600 tetrahedra, meeting 20 to a vertex, 5 to an edge. It is the 4D equivalent of an icosahedron. Its vertex figure is an icosahedron. | ||
== Geometry == | == Geometry == | ||
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Here are two perspective projections of the hecatonicosachoron into 3D. The highlighted center cell is closest to the 4D viewpoint. It is surrounded by 12 [[dodecahedra]] in a second layer. The right-hand image also shows the third layer of 47 dodecahedra. There are three more layers which are mirror images of the first three and cannot be seen as they are on the 4D "back" of the shape. Note that only the projection is rotating; the 4D shape itself is not. | Here are two perspective projections of the hecatonicosachoron into 3D. The highlighted center cell is closest to the 4D viewpoint. It is surrounded by 12 [[dodecahedra]] in a second layer. The right-hand image also shows the third layer of 47 dodecahedra. There are three more layers which are mirror images of the first three and cannot be seen as they are on the 4D "back" of the shape. Note that only the projection is rotating; the 4D shape itself is not. | ||
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| + | == Relation to other tetrashapes == | ||
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| + | === Grand antiprism === | ||
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| + | Given two adjacent vertices, the cells that meet at each vertex define two icosahedra that overlap in precisely 5 tetrahedra. The 5 tetrahedra surround the edge that connects these two vertices. One may identify another set of 5 tetrahedra in the second icosahedron such that they do not share any ridges with these 5 tetrahedra, but do share a vertex. This second set of 5 tetrahedra is the intersection of the second icosahedron with a third. This gives a third vertex connected to the second, such that the second and third vertices lie on the antipodes of an icosahedron. By repeatedly applying this procedure, one obtains a ring of 10 vertices around the 600-cell, lying on one of its great circles. Another ring of vertices may be obtained in the same way such that the two rings lie on two mutually orthogonal planes. If these two rings of vertices are removed from the 600-cell, the convex hull of the remaining vertices is the [[grand antiprism]]. | ||
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| + | One method of obtaining the second ring, given the first, makes use of the fact that the two rings corresponds with the two bounding 3-manifolds of the [[duocylinder]]. The set of vertices directly connected to (but not including) the first ring of vertices lie on a 2-manifold analogous to the ridge of the duocylinder. The set of vertices directly connected to ''this'' set (but excluding the vertices in the first ring) lie on another similar 2-manifold, farther away from the first ring, and closer to the second ring. By iterating this procedure, the manifolds thus obtained will eventually converge onto the second ring. (In fact, only 3 iterations are necessary: the first iteration yields the vertices of one ring of pentagonal antiprisms in the grand antiprism, the second iteration yields the vertices of the other ring of pentagonal antiprisms, and the third iteration yields the second ring.) | ||
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| + | === Snub 24-cell === | ||
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| + | Another uniform polychoron may be obtained from the 600-cell by a different diminishing of it: inscribe a [[24-cell]] in a 600-cell such that the 24 vertices of the latter coincide with 24 vertices in the 600-cell, then remove these 24 vertices from the 600-cell and recompute the convex hull of the remaining vertices. The result is the [[snub 24-cell]]. | ||
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Revision as of 20:10, 2 September 2008
The hecatonicosachoron, or 600-cell, is a 4D polytope bounded by 600 tetrahedra, meeting 20 to a vertex, 5 to an edge. It is the 4D equivalent of an icosahedron. Its vertex figure is an icosahedron.
Geometry
Equations
- Variables:
l ⇒ length of the edges of the hecatonicosachoron
- All points (x, y, z, w) that lie on the surcell of a hecatonicosachoron will satisfy the following equation:
Unknown
- The hypervolumes of a hecatonicosachoron are given by:
total edge length = 1200l
total surface area = 180l2sqrt(25+10sqrt(5))
surcell volume = 300l3(tan(3π10-1))2(tan(sin-1(2sin(π5-1))-1))
bulk = Unknown
- The realmic cross-sections (n) of a hecatonicosachoron are:
Unknown
Projection
http://teamikaria.com/dl/Pmyg-el9wlGbaOueUDKBuTbIo5xIx8iFlG8rLaBYljwi-z1f.gif http://teamikaria.com/dl/4TMygnHLhLXtUc0oRXtHFapu60FMLK_ig1kN2fN5GSeSdbFz.gif
Here are two perspective projections of the hecatonicosachoron into 3D. The highlighted center cell is closest to the 4D viewpoint. It is surrounded by 12 dodecahedra in a second layer. The right-hand image also shows the third layer of 47 dodecahedra. There are three more layers which are mirror images of the first three and cannot be seen as they are on the 4D "back" of the shape. Note that only the projection is rotating; the 4D shape itself is not.
Relation to other tetrashapes
Grand antiprism
Given two adjacent vertices, the cells that meet at each vertex define two icosahedra that overlap in precisely 5 tetrahedra. The 5 tetrahedra surround the edge that connects these two vertices. One may identify another set of 5 tetrahedra in the second icosahedron such that they do not share any ridges with these 5 tetrahedra, but do share a vertex. This second set of 5 tetrahedra is the intersection of the second icosahedron with a third. This gives a third vertex connected to the second, such that the second and third vertices lie on the antipodes of an icosahedron. By repeatedly applying this procedure, one obtains a ring of 10 vertices around the 600-cell, lying on one of its great circles. Another ring of vertices may be obtained in the same way such that the two rings lie on two mutually orthogonal planes. If these two rings of vertices are removed from the 600-cell, the convex hull of the remaining vertices is the grand antiprism.
One method of obtaining the second ring, given the first, makes use of the fact that the two rings corresponds with the two bounding 3-manifolds of the duocylinder. The set of vertices directly connected to (but not including) the first ring of vertices lie on a 2-manifold analogous to the ridge of the duocylinder. The set of vertices directly connected to this set (but excluding the vertices in the first ring) lie on another similar 2-manifold, farther away from the first ring, and closer to the second ring. By iterating this procedure, the manifolds thus obtained will eventually converge onto the second ring. (In fact, only 3 iterations are necessary: the first iteration yields the vertices of one ring of pentagonal antiprisms in the grand antiprism, the second iteration yields the vertices of the other ring of pentagonal antiprisms, and the third iteration yields the second ring.)
Snub 24-cell
Another uniform polychoron may be obtained from the 600-cell by a different diminishing of it: inscribe a 24-cell in a 600-cell such that the 24 vertices of the latter coincide with 24 vertices in the 600-cell, then remove these 24 vertices from the 600-cell and recompute the convex hull of the remaining vertices. The result is the snub 24-cell.
| Notable Tetrashapes | |
| Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
| Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
| Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
| Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
