Cosmochoron (EntityTopic, 12)

From Hi.gher. Space

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<[#ontology [kind topic] [cats 4D Regular Polytope]]>
{{STS Shape
{{STS Shape
| dim=4
| dim=4
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| elements=120, 720, 1200, 600
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| elements=120 [[Dodecahedron|dodecahedra]], 720 [[pentagon]]s, 1200 [[digon]]s, 600 [[point]]s
| genus=0
| genus=0
| ssc=<nowiki>{{</nowiki>G5<sup>3</sup>}<sup>4</sup>}
| ssc=<nowiki>{{</nowiki>G5<sup>3</sup>}<sup>4</sup>}
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| extra={{STS Uniform polytope
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| ssc2=Ks1
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| extra={{STS Polytope
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| flayout={{FLD|dim=4|left=e3|erev|a5|end2|e3}}
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| petrie=30, 60, 30, 60,<br>60, 60, 60, 60,<br>60, 30, 60, 30, 0
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| dual=[[Hydrochoron]]
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| bowers=Hi
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}}{{STS Uniform polytope
| schlaefli={[[Pentagon|5,]][[Dodecahedron|3,]]3}
| schlaefli={[[Pentagon|5,]][[Dodecahedron|3,]]3}
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| dynkin=x5o3o3o
| vlayout=([[Pentagon|5]][[Dodecahedron|<sup>3</sup>]])<sup>4</sup>
| vlayout=([[Pentagon|5]][[Dodecahedron|<sup>3</sup>]])<sup>4</sup>
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| vfigure=[[テ]]
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| vfigure=[[Tetrahedron]]
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| bowers=Hi
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| kana=ヘカ
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| dual=[[サコ]]
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}}}}
}}}}
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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.
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The '''cosmochoron''', also known as the '''hecatonicosachoron''' and the '''120-cell''', is a 4D polytope bounded by 120 [[dodecahedra]]. It is the highest dimensional analog of the [[pentagon]] and the dodecahedron.
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== Geometry ==
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Each dodecahedron in its [[net]] lies on a ring of ten dodecahedra along a [[great circle]] of the [[glome]]. There are four dodecahedra on such a ring between any dodecahedron and its antipodal dodecahedron.
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=== Equations ===
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*Variables:
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<blockquote>''l'' ⇒ length of the edges of the hecatonicosachoron</blockquote>
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*All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a hecatonicosachoron will satisfy the following equation:
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==Coordinates==
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<blockquote>''Unknown''</blockquote>
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The coordinates of the 120-cell with edge length 2/φ<sup>2</sup> are all permutations of coordinates and changes of sign of:
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*The [[hypervolume]]s of a hecatonicosachoron are given by:
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(2, 2, 0, 0)<br>
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<blockquote>total edge length = 1200''l''<br>
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(√5, 1, 1, 1)<br>
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total surface area = 180''l''<sup>2</sup>sqrt(25+10sqrt(5))<br>
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(φ, φ, φ, 2-φ)<br>
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surcell volume = 300''l''<sup>3</sup>(tan(3π10<sup>-1</sup>))<sup>2</sup>(tan(sin<sup>-1</sup>(2sin(π5<sup>-1</sup>))<sup>-1</sup>))<br>
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(φ+1, φ−1, φ−1, φ−1)<br>
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bulk = ''Unknown''</blockquote>
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*The [[realmic]] [[cross-section]]s (''n'') of a hecatonicosachoron are:
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Together with all even permutations of coordinates and all changes of sign of:
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<blockquote>''Unknown''</blockquote>
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== Projection ==
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(φ+1, 2-φ, 1, 0)<br>
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http://teamikaria.com/dl/Pmyg-el9wlGbaOueUDKBuTbIo5xIx8iFlG8rLaBYljwi-z1f.gif http://teamikaria.com/dl/4TMygnHLhLXtUc0oRXtHFapu60FMLK_ig1kN2fN5GSeSdbFz.gif
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(√5, φ−1, φ, 0)<br>
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(2, 1, φ, φ−1)<br>
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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.
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== Equations ==
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*The [[hypervolume]]s of a cosmochoron with side length ''l'' are given by:
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<blockquote>total edge length = 1200''l''<br>
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total surface area = 180√(25 + 10√5) {{DotHV}}<br>
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surcell volume = 30(15 + 7√5) {{DotHV|3}}<br>
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bulk = {{Over|15|4}}(105 + 47√5) {{DotHV|4}}</blockquote>
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== Relation to other tetrashapes ==
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== Projection ==
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<[#embed [hash XN0PP2G3K4HZT6489B2ZYT8D15]]> <[#embed [hash NK06P7XH2G01YTY3EYK0T5BGDQ]]>
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=== Grand antiprism ===
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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 32 dodecahedra. After this is a fourth layer of 30 dodecahedra lying on the spherical "equator", followed by three more layers mirroring the three layers seen here (not seen here, because they lie on the 4D "back" of the shape).
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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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Note that only the projection is rotating; the 4D shape itself is not.
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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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<[#polytope [id 40]]>
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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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<br clear="all"><br>
 
{{Tetrashapes}}
{{Tetrashapes}}
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[[Category:Regular polychora]]
 

Latest revision as of 15:34, 26 March 2017

The cosmochoron, also known as the hecatonicosachoron and the 120-cell, is a 4D polytope bounded by 120 dodecahedra. It is the highest dimensional analog of the pentagon and the dodecahedron.

Each dodecahedron in its net lies on a ring of ten dodecahedra along a great circle of the glome. There are four dodecahedra on such a ring between any dodecahedron and its antipodal dodecahedron.

Coordinates

The coordinates of the 120-cell with edge length 2/φ2 are all permutations of coordinates and changes of sign of:

(2, 2, 0, 0)
(√5, 1, 1, 1)
(φ, φ, φ, 2-φ)
(φ+1, φ−1, φ−1, φ−1)

Together with all even permutations of coordinates and all changes of sign of:

(φ+1, 2-φ, 1, 0)
(√5, φ−1, φ, 0)
(2, 1, φ, φ−1)

Equations

  • The hypervolumes of a cosmochoron with side length l are given by:
total edge length = 1200l
total surface area = 180√(25 + 10√5) · l2
surcell volume = 30(15 + 7√5) · l3
bulk = 154(105 + 47√5) · l4

Projection

(image) (image)

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 32 dodecahedra. After this is a fourth layer of 30 dodecahedra lying on the spherical "equator", followed by three more layers mirroring the three layers seen here (not seen here, because they lie on the 4D "back" of the shape).

Note that only the projection is rotating; the 4D shape itself is not.

Incidence matrix

Dual: hydrochoron

#TXIDVaEa5aC1aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 5a 55 = pentagon ;
3 C1a 203012 = dodecahedron ;
4 H4.1a 6001200720120 = cosmochoron ;

Usage as facets

This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.


Notable Tetrashapes
Regular: pyrochoronaerochorongeochoronxylochoronhydrochoroncosmochoron
Powertopes: triangular octagoltriatesquare octagoltriatehexagonal octagoltriateoctagonal octagoltriate
Circular: glomecubinderduocylinderspherindersphonecylindronediconeconinder
Torii: tigertorispherespheritorustorinderditorus