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| {{Infobox polychoron |
| | For your own offense, you might have Gunboats which can [http://Www.Britannica.com/search?query=handily+shoot handily shoot] at enemy safeguarding coming from a very long range and Landing Projects which you must satisfy when you train gadgets for example Rifleman, Heavy, Zooka, Warrior and Fish tanks. To your village defenses, you might hold structures like Mortar, Hardware Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Skyrocket Launcher to assist families eradicate enemies.<br><br>The upsides of video gaming applications can include fun, film and even education. The downsides range in addictive game play in which to younger individuals seeing or hearing things they [http://Browse.deviantart.com/?q=continue continue] to be not old enough in order for. With luck, their ideas presented within doing this article can help manage video games incredibly well within your home for many everyone's benefit.<br><br>For those who have just about any inquiries concerning where along with tips on how to make use of [http://circuspartypanama.com how to hack clash of clans], it is possible to e-mail us in the webpage. Temperance is essential in nearly things, and enjoying clash of clans cheats is no special. Playing for hours on finish is simply not good-for-you, bodily or in your mind. There are some games out truth be told there which know this and will include measures to remind someone to take rests. Take the initiative yourself, although! Place an alarm a person don't play for more than an hour right.<br><br>So if you feel like you really targeted your enemy site on in a shooter and still missed, check what weapon you are probably using. Just like in real life, different weapons have different good points and weaknesses. All of the weapon you are operating may not have the short distance required and also the weapon recoil is ordinarily actually putting you a little bit off target.<br><br>Plan not only provides overall tools, there is perhaps even Clash of Clans identify no survey by anyone else. Strict anti ban system allow users to utilize method and play without type of hindrance. If people are interested in best man program, they are just required to visit this wonderful site and obtain typically the hack tool trainer actually. The name of the website is Amazing Cheats. A number of online have different types out of software by which people can get past harsh stages in the task.<br><br>Anyone have are the proud possessor of an ANY lightweight device that runs directly on iOS or android basically a touchscreen tablet computing machine or a smart phone, then you definitely might probably have already been careful of the revolution finding place right now on the world of mobile personal game "The Clash Associated with Clans", and you would be in demand having to do with conflict of families no fee jewels compromise because a lot of gems, elixir and sterling silver are needed seriously so that it will acquire every battle.<br><br>Now you have read this composition, you need to a good easier time locating and therefore loving video games on your life. Notwithstanding your favored platform, from your cellphone to any own computer, playing and enjoying video gaming allows for you to take the advantage of the worries of ones own busy week get specifics. |
| Name=600-cell|
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| Image_File=Schlegel_wireframe_600-cell_vertex-centered.png|
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| Image_Caption=[[Schlegel diagram]], vertex-centered<BR>(vertices and edges)|
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| Type=[[Convex regular 4-polytope]]|
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| Last=[[Rectified 600-cell|34]]|
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| Index=35|
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| Next=[[Truncated 120-cell|36]]|
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| Schläfli={3,3,5}|
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| CD={{CDD|node|5|node|3|node|3|node_1}}|
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| Cell_List=600 ([[Tetrahedron|''3.3.3'']]) [[Image:Tetrahedron.png|20px]]|
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| Face_List=1200 [[triangle|{3}]]|
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| Edge_Count=720|
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| Vertex_Count= 120|
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| Petrie_Polygon=[[regular polygon|30-gon]]|
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| Coxeter_Group=H<sub>4</sub>, [3,3,5], order 14400|
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| Vertex_Figure=[[Image:600-cell verf.png|80px]]<BR>[[icosahedron]]|
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| Dual=[[120-cell]]|
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| Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
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| }}
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| In [[geometry]], the '''600-cell''' (or '''hexacosichoron''') is the [[convex regular 4-polytope]], or [[polychoron]], with [[Schläfli symbol]] {3,3,5}. Its boundary is composed of 600 [[tetrahedron|tetrahedral]] [[cell (mathematics)|cells]] with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons. Each vertex of the 600-cell is a vertex of six such decagons.
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| The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[hypersphere]], only have the values 36° = <math>\pi/5</math>, 60°= <math>\pi/3</math>, 72° = <math>2\pi/5</math>, 90° = <math>\pi/2</math>, 108° = <math>3\pi/5</math>, 120° = <math>2\pi/3</math>, 144° = <math>4\pi/5</math>, and 180° = <math>\pi</math>. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[icosahedron]], at 60° and 120° the 20 vertices of a [[dodecahedron]], at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an [[icosadodecahedron]], and finally at 180° the antipodal vertex of V. ''References:'' S.L. van Oss (1899); F. Buekenhout and M. Parker (1998).
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| The 600-cell is regarded as the 4-dimensional analog of the [[icosahedron]], since it has five [[tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[triangle]]s meeting at every vertex. It is also called a '''tetraplex''' (abbreviated from "tetrahedral complex") and '''[[polytetrahedron]]''', being bounded by tetrahedral [[Cell (geometry)|cells]]. | |
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| Its [[vertex figure]] is an [[icosahedron]], and its [[dual polytope]] is the [[120-cell]].
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| Each cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.
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| ==Coordinates==
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| The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ (where φ = (1+√5)/2 is the [[golden ratio]]), can be given as follows: 16 vertices of the form:<ref>{{mathworld|title=600-cell|urlname=600-Cell}}</ref>
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| :(±½,±½,±½,±½),
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| and 8 vertices obtained from
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| :(0,0,0,±1)
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| by permuting coordinates. The remaining 96 vertices are obtained by taking [[even permutation]]s of
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| :½(±φ,±1,±1/φ,0).
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| Note that the first 16 vertices are the vertices of a [[tesseract]], the second eight are the vertices of a [[16-cell]], and that all 24 vertices together are vertices of a [[24-cell]]. The final 96 vertices are the vertices of a [[snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.
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| When interpreted as [[quaternion]]s, the 120 vertices of the 600-cell form a [[group (mathematics)|group]] under quaternionic multiplication. This group is often called the [[binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[icosahedral group]] ''I''. It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[invariant subgroup]], namely as the subgroup ''2I<sub>L</sub>'' of quaternion left-multiplications and as the subgroup ''2I<sub>R</sub>'' of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of ''2I<sub>L</sub>'' and ''2I<sub>R</sub>''; the pair of opposite elements generate the same element of ''RSG''. The [[Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''-Id''. We have the isomorphism ''RSG ≅ (2I<sub>L</sub> × 2I<sub>R</sub>) / {Id, -Id}''. The order of ''RSG'' equals 120 × 120 / 2 = 7200.
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| The binary icosahedral group is [[isomorphic]] to [[special linear group|SL(2,5)]].
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| The full [[symmetry group]] of the 600-cell is the [[Weyl group]] of [[H4 (mathematics)|H<sub>4</sub>]]. This is a [[group (mathematics)|group]] of order 14400. It consists of 7200 [[Rotation (mathematics)|rotations]] and 7200 rotation-reflections. The rotations form an [[invariant subgroup]] of the full symmetry group. The rotational symmetry group was described by S.L. van Oss (1899); see References.
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| == Visualization ==
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| The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells, and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600-cell is the dual of the 120-cell.
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| == Union of two tori ==
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| The 120-cell can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]]. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to a 10-vertex decagon path in the 600-cell. Start by assembling five tetrahedrons around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each "saucer" with 10 tetrahedrons forming an icosahedron. You can view this as five, vertex stacked, icosahedra, with the five extra annular ring gaps also filled in. The surface is the same as that of ten stacked [[pentagonal antiprism]]s. You now have a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces, 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges. The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above. These paths spiral around the center core path, but mathematicall they are all equivalent. Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges.
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| The 600-cell can be further partitioned into 20 disjoint intertwining rings of 30 cells and ten edges long each, forming a discrete [[Hopf fibration]]. These chains of 30 tetrahedra each form a [[Boerdijk–Coxeter helix]]. Five such [[Helix|helices]] nest and spiral around each of the 10-vertex decagon paths, forming the initial 150 cell torus mentioned above.
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| This decomposition of the 600-cell has [[Coxeter notation|symmetry]] [<span\>[10,2<sup>+</sup>,10]], order 400, the same symmetry as the [[grand antiprism]]. The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of tetrahedra, similar to the belt of an icosahedron with the 5 top and 5 bottom triangles removed (pentagonal antiprism).
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| == Images ==
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| === 2D projections ===
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| {| class="wikitable"
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| |+ [[Orthographic projection]]s by [[Coxeter plane]]s
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| |- align=center
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| !H<sub>4</sub>
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| ! -
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| !F<sub>4</sub>
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| |- align=center
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| |[[File:600-cell graph H4.svg|240px]]<BR>[30]
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| |[[File:600-cell t0 p20.svg|240px]]<BR>[20]
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| |[[File:600-cell t0 F4.svg|240px]]<BR>[12]
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| |- align=center
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| !H<sub>3</sub>
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| !A<sub>2</sub> / B<sub>3</sub> / D<sub>4</sub>
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| !A<sub>3</sub> / B<sub>2</sub>
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| |- align=center
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| |[[File:600-cell t0 H3.svg|240px]]<BR>[10]
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| |[[File:600-cell t0 A2.svg|240px]]<BR>[6]
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| |[[File:600-cell t0.svg|240px]]<BR>[4]
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| |}
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| === 3D projections ===
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| {| class=wikitable
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| !colspan=2|Vertex-first projection
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| |-
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| |[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]]
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| |This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
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| * The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown.
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| * The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow.
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| * The remaining cells are rendered in edge-outline.
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| * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image.
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| |-
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| !colspan=2|Cell-first projection
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| |-
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| |[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]]
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| |This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
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| * The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image.
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| * The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow.
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| * The remaining cells are rendered in edge-outline.
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| * Cells facing away from the 4D viewpoint have been culled for clarity.
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| This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.
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| |-
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| !colspan=2|Stereographic projection (on [[3-sphere]])
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| |-
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| |align=center|[[Image:Stereographic polytope 600cell.png|220px]]
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| |Cell-Centered
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| |-
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| !colspan=2|Simple Rotation
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| |-
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| |align=center|[[Image:600-cell.gif|256px]]
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| |A 3D projection of a 600-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]].
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| |}
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| [[File:Cell600Cmp.ogv]]Frame synchronized animated comparison of the 600 cell using orthogonal isometric (left) and perspective (right) projections.
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| == Related polytopes and honeycombs ==
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| The [[snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell]] and taking the [[convex hull]] of the remaining vertices. This process is a ''diminishing'' of the 600-cell.
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| The [[grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.
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| {{H4_family}}
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| It is similar to three [[regular polychora]]: the [[5-cell]] {3,3,3}, [[16-cell]] {3,3,4} of Euclidean 4-space, and the [[order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have a [[tetrahedron|tetrahedral]] cells.
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| {{Tetrahedral cell tessellations}}
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| This polychora is a part of a sequence of polychora and honeycombs with [[icosahedron]] vertex figures:
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| {{Icosahedral vertex figure tessellations}}
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| ==See also==
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| * [[Uniform polychoron#The H4 family|Uniform polychora family with [5,3,3] symmetry]]
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| * [[Convex regular polychoron]]
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| * [[120-cell]], the dual [[polychoron]] to the 600-cell
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *H. S. M. Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
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| *'''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
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| ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
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| ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
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| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| * [[John Horton Conway|J.H. Conway]] and [[Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
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| * [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
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| *[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf]
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| * Oss, Salomon Levi van: Das regelmässige 600-Zell und seine selbstdeckenden Bewegungen. ''Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 Deel 7 Nummer 1 (Afdeeling Natuurkunde). Amsterdam: 1899.'' Online at URL [http://www.dwc.knaw.nl/english/academy/digital-library/?pagetype=publDetail&pId=PU00011478], reachable from the home page of the KNAW Digital Library at URL [http://www.dwc.knaw.nl/english/academy/digital-library/]. REMARK: Van Oss does not mention the arc distances between vertices of the 600-cell.
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| * F. Buekenhout, M. Parker: The number of nets of the regular convex polytopes in dimension <= 4. ''Discrete Mathematics, Volume 186, Issues 1-3, 15 May 1998, Pages 69-94.'' REMARK: The authors do mention the arc distances between vertices of the 600-cell.
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| == External links ==
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| * {{mathworld | urlname = 600-Cell | title = 600-Cell}}
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| *{{GlossaryForHyperspace | anchor=hexacosichoron | title=Hexacosichoron}}
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| ** {{PolyCell | urlname = section4.html| title = Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 35}}
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| * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o5o - ex}}
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| * [http://www.polytope.de/c600.html Der 600-Zeller (600-cell)] Marco Möller's Regular polytopes in R<sup>4</sup> (German)
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| * [http://eusebeia.dyndns.org/4d/600-cell The 600-Cell] Vertex centered expansion of the 600-cell
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| {{4D regular polytopes}}
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora| 600]]
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