Compositional data

Template:Infobox polychoron 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 tetrahedral 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.

The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = $\pi /5$, 60°= $\pi /3$, 72° = $2\pi /5$, 90° = $\pi /2$, 108° = $3\pi /5$, 120° = $2\pi /3$, 144° = $4\pi /5$, and 180° = $\pi$. 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).

The 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex (abbreviated from "tetrahedral complex") and polytetrahedron, being bounded by tetrahedral cells.

Its vertex figure is an icosahedron, and its dual polytope is the 120-cell.

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.

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:

(±½,±½,±½,±½),

and 8 vertices obtained from

(0,0,0,±1)

by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of

½(±φ,±1,±1/φ,0).

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.

When interpreted as quaternions, the 120 vertices of the 600-cell form a 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_L of quaternion left-multiplications and as the subgroup 2I_R of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2I_L and 2I_R; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion -Id. We have the isomorphism RSG ≅ (2I_L × 2I_R) / {Id, -Id}. The order of RSG equals 120 × 120 / 2 = 7200.

The binary icosahedral group is isomorphic to SL(2,5).

The full symmetry group of the 600-cell is the Weyl group of H₄. This is a group of order 14400. It consists of 7200 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.

Visualization

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.

Union of two tori

The 120-cell can be decomposed into 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 antiprisms. 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.

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 helices nest and spiral around each of the 10-vertex decagon paths, forming the initial 150 cell torus mentioned above.

This decomposition of the 600-cell has symmetry [<span\>[10,2⁺,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).

Images

2D projections

Orthographic projections by Coxeter planes

H4	-	F4
[30]	[20]	[12]
H3	A2 / B3 / D4	A3 / B2
[10]	[6]	[4]

3D projections

Vertex-first projection
	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: The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown. The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow. The remaining cells are rendered in edge-outline. 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.
Cell-first projection
	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: The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image. The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow. The remaining cells are rendered in edge-outline. Cells facing away from the 4D viewpoint have been culled for clarity. This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.
Stereographic projection (on 3-sphere)
	Cell-Centered
Simple Rotation
	A 3D projection of a 600-cell performing a simple rotation.

File:Cell600Cmp.ogvFrame synchronized animated comparison of the 600 cell using orthogonal isometric (left) and perspective (right) projections.

Related polytopes and honeycombs

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.

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.

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 tetrahedral cells. On another contemplation, when you pass it on to a direct relative, then your recipient would fill in, in your nonattendance and can be given the proceeds. At least 1 term paper is required per class; and sometimes, the deadlines for these papers coincide with each another. But life insurance has come a long way since the early days and these days there you can cover yourself with cheap term life insurance. Its 2007 research report placed Australia at the sixteenth position in terms of life insurance density and penetration. It ensures the debt payments of the policy holder incase of his unfortunate death. Section 125 deductions are exempt from income tax and not deemed as income for tax computation and FICA (social security and Medicare) purposes. Which one will save you the most money if you only keep it about ten years.

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See also

• Uniform polychora family with [5,3,3] symmetry
• Convex regular polychoron
• 120-cell, the dual polychoron to the 600-cell

Notes

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References

• H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
• 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 [1]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [2]
• 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 [3], reachable from the home page of the KNAW Digital Library at URL [4]. REMARK: Van Oss does not mention the arc distances between vertices of the 600-cell.
• 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.

External links

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• Template:GlossaryForHyperspace
  • Template:PolyCell
• Template:KlitzingPolytopes
• Der 600-Zeller (600-cell) Marco Möller's Regular polytopes in R⁴ (German)
• The 600-Cell Vertex centered expansion of the 600-cell