 120cell

120cell
Schlegel diagram
(vertices and edges)Type Convex regular 4polytope Schläfli symbol {5,3,3} CoxeterDynkin diagram Cells 120 {5,3} Faces 720 {5} Edges 1200 Vertices 600 Vertex figure
tetrahedronPetrie polygon 30gon Coxeter group H_{4}, [3,3,5] Dual 600cell Properties convex, isogonal, isotoxal, isohedral Uniform index 32 In geometry, the 120cell (or hecatonicosachoron) is the convex regular 4polytope with Schläfli symbol {5,3,3}.
The boundary of the 120cell is composed of 120 dodecahedral cells with 4 meeting at each vertex.
It can be thought of as the 4dimensional analog of the dodecahedron and has been called a dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, and polydodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedrons, with 3 around each edge.
The Davis 120cell, introduced by Davis (1985), is a compact 4dimensional hyperbolic manifold obtained by identifying opposite faces of the 120cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4dimensional hyperbolic space.
Contents
Elements
 There are 120 cells, 720 pentagonal faces, 1200 edges, and 600 vertices.
 There are 4 dodecahedra, 6 pentagons, and 4 edges meeting at every vertex.
 There are 3 dodecahedra and 3 pentagons meeting every edge.
 The dual polytope of the 120cell is the 600cell.
 The vertex figure of the 120cell is a tetrahedron.
Cartesian coordinates
The 600 vertices of the 120cell include all permutations of:^{[1]}
 (0, 0, ±2, ±2)
 (±1, ±1, ±1, ±√5)
 (±φ^{2}, ±φ, ±φ, ±φ)
 (±φ^{1}, ±φ^{1}, ±φ^{1}, ±φ^{2})
and all even permutations of
 (0, ±φ^{2}, ±1, ±φ^{2})
 (0, ±φ^{1}, ±φ, ±√5)
 (±φ^{1}, ±1, ±φ, ±2)
where φ (also called τ) is the golden ratio, (1+√5)/2.
Visualization
The 120cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.
Layered stereographic projection
The cell locations lend themselves to a hyperspherical description. Pick an arbitrary cell and label it the "North Pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the 5th "South Pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4*12).
Starting at the North Pole, we can build up the 120cell in 9 latitudinal layers, with allusions to terrestrial 2sphere topography in the table below. With the exception of the poles, each layer represents a separate 2sphere, with the equator being a great 2sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles.
Layer # Number of Cells Description Colatitude Region 1 1 cell North Pole 0° Northern Hemisphere 2 12 cells First layer of meridian cells 36° 3 20 cells Nonmeridian / interstitial / "Tropic of Cancer" 60° 4 12 cells Second layer of meridian cells 72° 5 30 cells Nonmeridian / interstitial 90° Equator 6 12 cells Third layer of meridian cells 108° Southern Hemisphere 7 20 cells Nonmeridian / interstitial / "Tropic of Capricorn" 120° 8 12 cells Fourth layer of meridian cells 144° 9 1 cell South Pole 180° Total 120 cells Layers 3 and 7's cells are located directly over the pole cell's vertices. Layer 5's cells are located over the pole cell's edges.
Intertwining rings
The 120cell can be partitioned into 12 disjoint 10cell great circle rings, forming a discrete/quantized Hopf fibration. Starting with one 10cell ring, one can place another ring along side it that spirals around the original ring one complete revolution in ten cells. Five such 10cell rings can be placed adjacent to the original 10cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical torsion. They are all equivalent. The spiraling is a result of the 3sphere curvature. The inner ring and the five outer rings now form a six ring, 60cell solid torus. One can continue adding 10cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120cell, like the 3sphere, is the union of these two (Clifford) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle. Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint great circles.
Other great circle constructs
There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 cells and 6 edges. Both the above great circle paths have dual great circle paths in the 600cell. The 10 cell face to face path above maps to a 10 vertices path solely traversing along edges in the 600cell, forming a decagon. The alternating cell/edge path above maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex in the 600cell.
Projections
Orthogonal projections
Orthogonal projections of the 120cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction.
Orthographic projections by Coxeter planes H_{4}  F_{4}
[30]
[20]
[12]H_{3} A_{2} / B_{3} / D_{4} A_{3} / B_{2}
[10]
[6]
[4]3dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image.
3D orthographic projections
3D isometric projection
Animated 4D rotationPerspective projections
These projections use perspective projection, from a specific view point in 4dimensions, and projecting the model as a 3D shadow. Therefore faces and cells that look larger are merely closer to the 4D viewpoint. Schlegel diagrams use perspective to show 4 dimensional figures, choosing a point above a specific cell, thus making the cell as the envelope of the 3D model, and other cells are smaller seen inside it. Stereographic projection use the same approach, but are shown with curved edges, representing the polytope a tiling of a 3sphere.
A comparison of perspective projections from 3D to 2D is shown in anology.
Comparison with regular dodecahedron Projection Dodecahedron Dodecaplex Schlegel diagram
12 pentagon faces in the plane
120 dodecahedral cells in 3spaceStereographic projection
With transparent facesPerspective projection Cellfirst perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied:  Nearest dodecahedron to the 4D viewpoint rendered in yellow
 The 12 dodecahedra immediately adjoining it rendered in cyan;
 The remaining dodecahedra rendered in green;
 Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120cell) culled to minimize clutter in the final image.
Vertexfirst perspective projection at 5 times the distance from center to a vertex, with these enhancements:  Four cells surrounding nearest vertex shown in 4 colors
 Nearest vertex shown in white (center of image where 4 cells meet)
 Remaining cells shown in transparent green
 Cells facing away from 4D viewpoint culled for clarity
A 3D projection of a 120cell performing a simple rotation. A 3D projection of a 120cell performing a simple rotation (from the inside). Animated 4D rotation See also
 Uniform polychora family with [5,3,3] symmetry
 57cell – an abstract regular polychoron constructed from 57 hemidodecahedra.
Notes
 ^ Weisstein, Eric W., "120cell" from MathWorld.
References
 H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and SemiRegular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 J.H. Conway and M.J.T. Guy: FourDimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
 Davis, Michael W. (1985), "A hyperbolic 4manifold", Proceedings of the American Mathematical Society 93 (2): 325–328, doi:10.2307/2044771, ISSN 00029939, MR770546
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 Fourdimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [2]
External links
 Weisstein, Eric W., "120Cell" from MathWorld.
 Olshevsky, George, Hecatonicosachoron at Glossary for Hyperspace.
 Richard Klitzing, 4D uniform polytopes (polychora), o3o3o5x  hi
 Der 600Zeller (600cell) Marco Möller's Regular polytopes in R^{4} (German)
 120cell explorer – A free interactive program that allows you to learn about a number of the 120cell symmetries. The 120cell is projected to 3 dimensions and then rendered using OpenGL.
 Construction of the HyperDodecahedron
 YouTube animation of the construction of the 120cell Gian Marco Todesco.
H_{4} family polytopes by name, CoxeterDynkin diagram, and Schläfli symbol 120cell rectified
120celltruncated
120cellcantellated
120cellruncinated
120cellbitruncated
120cellcantitruncated
120cellruncitruncated
120cellomnitruncated
120cell{5,3,3} t_{1}{5,3,3} t_{0,1}{5,3,3} t_{0,2}{5,3,3} t_{0,3}{5,3,3} t_{1,2}{5,3,3} t_{0,1,2}{5,3,3} t_{0,1,3}{5,3,3} t_{0,1,2,3}{5,3,3} 600cell rectified
600celltruncated
600cellcantellated
600cellruncinated
600cellbitruncated
600cellcantitruncated
600cellruncitruncated
600cellomnitruncated
600cell{3,3,5} t_{1}{3,3,5} t_{0,1}{3,3,5} t_{0,2}{3,3,5} t_{0,3}{3,3,5} t_{1,2}{3,3,5} t_{0,1,2}{3,3,5} t_{0,1,3}{3,3,5} t_{0,1,2,3}{3,3,5} Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: Fourdimensional geometry
 Polychora
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