120-cell - talata.istvan.ymmf.hu
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120-cell
Schlegel diagram(vertices and edges)
Type Convex regular4-polytope
Schläflisymbol
{5,3,3}
Coxeterdiagram
Cells 120 {5,3}
Faces 720 {5}
Edges 1200
Vertices 600
Vertex figure
tetrahedron
Petriepolygon
30-gon
Coxetergroup
H4, [3,3,5]
Dual 600-cell
Properties convex, isogonal,isotoxal, isohedral
Uniformindex
32
120-cellIn geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is also called a C120,
dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron,
hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.[1]
The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be
thought of as the 4-dimensional analog of the dodecahedron. 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 dodecahedra, with 3 around
each edge.
The Davis 120-cell, introduced by Davis (1985), is a compact 4-dimensional hyperbolic manifold obtained by
identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb {5,3,3,5} of
4-dimensional hyperbolic space.
ElementsAs a configuration
Cartesian coordinates
VisualizationLayered stereographic projectionIntertwining ringsOther great circle constructs
ProjectionsOrthogonal projectionsPerspective projections
Related polyhedra and honeycombs
See also
Notes
References
External links
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 120-cell is the 600-cell.The vertex figure of the 120-cell is a tetrahedron.The dihedral angle (angle between facet hyperplanes) of the 120-cell is 144°[2]
The elements of a regular polytopes can be expressed in a configuration matrix. Rows and columns reference
vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent
the number of row elements are incident to the column element. The configurations for dual polytopes can be seen
by rotating the matrix elements by 180 degrees.[3][4]
The 600 vertices of the 120-cell include all permutations of:[5]
(0, 0, ±2, ±2)(±1, ±1, ±1, ±√5)(±ϕ−2, ±ϕ, ±ϕ, ±ϕ)(±ϕ−1, ±ϕ−1, ±ϕ−1, ±ϕ2)
Net
Contents
Elements
As a configuration
Cartesian coordinates
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and all even permutations of
(0, ±ϕ−2, ±1, ±ϕ2)(0, ±ϕ−1, ±ϕ, ±√5)(±ϕ−1, ±1, ±ϕ, ±2)
where ϕ (also called τ) is the golden ratio, (1+√ 5)/2.
The 120-cell 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 24-cell). 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.
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 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the
exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. 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 Hemisphere2 12 cells First layer of meridian cells / "Arctic Circle" 36°
3 20 cells Non-meridian / interstitial 60°
4 12 cells Second layer of meridian cells / "Tropic of Cancer" 72°
5 30 cells Non-meridian / interstitial 90° Equator
6 12 cells Third layer of meridian cells / "Tropic of Capricorn" 108°
Southern Hemisphere7 20 cells Non-meridian / interstitial 120°
8 12 cells Fourth layer of meridian cells / "Antarctic Circle" 144°
9 1 cell South Pole 180°
Total 120 cells
Layers' 2, 4, 6 and 8 cells are located over the pole cell's faces. 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.
The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized
Hopf fibration. Starting with one 10-cell ring, one can place another ring alongside it that spirals
around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed
adjacent to the original 10-cell 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
3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus.
One can continue adding 10-cell 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 120-cell, like the 3-sphere, 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.
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 600-cell. The 10 cell face to face path above maps to a 10 vertices path solely traversing
along edges in the 600-cell, 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 (six triangular bipyramids) in the 600-cell. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub
24-cell (or icosahedral pyramids in the 600-cell).
Visualization
Layered stereographic projection
Intertwining rings
Two intertwining rings of the 120-cell.
Other great circle constructs
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Orthogonal projections of the 120-cell can be done in 2D by defining two orthonormal basis vectors
for a specific view direction.
The H3 decagonal projection shows the plane of the van Oss polygon.
Orthographic projections by Coxeter planes
H4 - F4
[30] [20] [12]
H3 A2 / B3 / D4 A3 / B2
[10] [6] [4]
3-dimensional 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 rotation
Two orthogonal rings in a cell-centered projection
Projections
Orthogonal projections
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These projections use perspective projection, from a specific view point in four dimensions, 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 four-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 3-sphere.
A comparison of perspective projections from 3D to 2D is shown in analogy.
Comparison with regular dodecahedron
Projection Dodecahedron Dodecaplex
Schlegeldiagram
12 pentagon faces in the plane 120 dodecahedral cells in 3-space
Stereographicprojection
With transparent faces
Perspective projections
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Perspective projection
Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancementsapplied:
Nearest dodecahedron to the 4D viewpoint rendered in yellowThe 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 120-cell) culled to minimizeclutter in the final image.
Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements:
Four cells surrounding nearest vertex shown in 4 colorsNearest vertex shown in white (center of image where 4 cells meet)Remaining cells shown in transparent greenCells facing away from 4D viewpoint culled for clarity
A 3D projection of a 120-cell performing a simple rotation.
A 3D projection of a 120-cell performing a simple rotation (from the inside).
Animated 4D rotation
The 120-cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:
Related polyhedra and honeycombs
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H4 family polytopes
120-cell rectified120-cell
truncated120-cell
cantellated120-cell
runcinated120-cell
cantitruncated120-cell
runcitruncated120-cell
omnitruncated120-cell
{5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3{5,3,3} tr{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
600-cell rectified600-cell
truncated600-cell
cantellated600-cell
bitruncated600-cell
cantitruncated600-cell
runcitruncated600-cell
omnitruncated600-cell
{3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}
It is similar to three regular 4-polytopes: the 5-cell {3,3,3}, tesseract {4,3,3}, of Euclidean 4-space, and hexagonal tiling honeycomb of hyperbolic space. All of
these have a tetrahedral vertex figure.
{p,3,3} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ...{∞,3,3}
Image
Cells{p,3}
{3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3}
This honeycomb is a part of a sequence of 4-polytopes and honeycombs with dodecahedral cells:
{5,3,p} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {5,3,3} {5,3,4} {5,3,5} {5,3,6} {5,3,7} {5,3,8} ... {5,3,∞}
Image
Vertexfigure
{3,3} {3,4} {3,5} {3,6} {3,7} {3,8} {3,∞}
Uniform 4-polytope family with [5,3,3] symmetry57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra.600-cell - the dual 4-polytope to the 120-cell
Matila Ghyka, The Geometry of Art and Life (1977), p.681.
Coxeter, Regular polytopes, p.2932.
Coxeter, Regular Polytopes, sec 1.8 Configurations3.
See also
Notes
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Coxeter, Complex Regular Polytopes, p.1174.
Weisstein, Eric W. "120-cell" (http://mathworld.wolfram.com/120-Cell.html). MathWorld.5.
H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] (http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html)
(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 39,1965Davis, Michael W. (1985), "A hyperbolic 4-manifold", Proceedings of the American Mathematical Society, 93 (2): 325–328, doi:10.2307/2044771(https://doi.org/10.2307%2F2044771), ISSN 0002-9939 (https://www.worldcat.org/issn/0002-9939), MR 0770546 (https://www.ams.org/mathscinet-getitem?mr=0770546)N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966Four-dimensional Archimedean Polytopes (http://www.polytope.de) (German), Marco Möller, 2004 PhD dissertation [2] (http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf)
Weisstein, Eric W. "120-Cell" (http://mathworld.wolfram.com/120-Cell.html). MathWorld.Olshevsky, George. "Hecatonicosachoron" (https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#hecatonicosachoron).Glossary for Hyperspace. Archived from the original (http://members.aol.com/Polycell/glossary.html#hecatonicosachoron) on 4 February 2007.
Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 32 (https://web.archive.org/web/20070204075028/members.aol.com/Polycell/section4.html), George Olshevsky.
Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5x - hi" (https://bendwavy.org/klitzing/dimensions/polychora.htm).Der 120-Zeller (120-cell) (http://www.polytope.de/c120.html) Marco Möller's Regular polytopes in R4 (German)120-cell explorer (http://www.gravitation3d.com/120cell/) – A free interactive program that allows you to learn about a number of the 120-cell symmetries.The 120-cell is projected to 3 dimensions and then rendered using OpenGL.Construction of the Hyper-Dodecahedron (http://www.theory.org/geotopo/120-cell/)YouTube animation of the construction of the 120-cell (https://www.youtube.com/watch?v=MFXRRW9goTs/) Gian Marco Todesco.
H4 family polytopes
120-cell rectified120-cell
truncated120-cell
cantellated120-cell
runcinated120-cell
cantitruncated120-cell
runcitruncated120-cell
omnitruncated120-cell
{5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3{5,3,3} tr{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
600-cell rectified600-cell
truncated600-cell
cantellated600-cell
bitruncated600-cell
cantitruncated600-cell
runcitruncated600-cell
omnitruncated600-cell
{3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}
References
External links
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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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