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In 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.

Contents

1 Elements

1.1 As a configuration

2 Cartesian coordinates 3 Visualization

3.1 Layered stereographic projection 3.2 Intertwining rings 3.3 Other great circle constructs

4 Projections

4.1 Orthogonal projections 4.2 Perspective projections

5 Related polyhedra and honeycombs 6 See also 7 Notes 8 References 9 External links

Elements[edit]

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]

As a configuration[edit] 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]

[

600

4

6

4

2

1200

3

3

5

5

720

2

20

30

12

120

]

displaystyle begin bmatrix begin matrix 600&4&6&4\2&1200&3&3\5&5&720&2\20&30&12&120end matrix end bmatrix

Cartesian coordinates[edit] 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)

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. Visualization[edit] 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. Layered stereographic projection[edit] 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 Hemisphere

2 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 Hemisphere

7 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. Intertwining rings[edit]

Two intertwining rings of the 120-cell.

Two orthogonal rings in a cell-centered projection

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. Other great circle constructs[edit] 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). Projections[edit] Orthogonal projections[edit] 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

Play media

Animated 4D rotation

Perspective projections[edit] 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

Schlegel diagram

12 pentagon faces in the plane

120 dodecahedral cells in 3-space

Stereographic projection

With transparent faces

Perspective projection

Cell-first 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 120-cell) culled to minimize clutter 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 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 120-cell performing a simple rotation.

A 3D projection of a 120-cell performing a simple rotation (from the inside).

Animated 4D rotation

Related polyhedra and honeycombs[edit] The 120-cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:

H4 family polytopes

120-cell rectified 120-cell truncated 120-cell cantellated 120-cell runcinated 120-cell cantitruncated 120-cell runcitruncated 120-cell omnitruncated 120-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 rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated 600-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

Vertex figure

3,3

3,4

3,5

3,6

3,7

3,8

3,∞

See also[edit]

Uniform 4-polytope family with [5,3,3] symmetry 57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra. 600-cell - the dual 4-polytope to the 120-cell

Notes[edit]

^ Matila Ghyka, The Geometry of Art and Life (1977), p.68 ^ Coxeter, Regular polytopes, p.293 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 ^ Weisstein, Eric W. "120-cell". MathWorld. 

References[edit]

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]

(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 Davis, Michael W. (1985), "A hyperbolic 4-manifold", Proceedings of the American Mathematical Society, 93 (2): 325–328, doi:10.2307/2044771, ISSN 0002-9939, MR 0770546  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]

External links[edit]

Weisstein, Eric W. "120-Cell". MathWorld.  Olshevsky, George. "Hecatonicosachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007. 

Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 32, George Olshevsky.

Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5x - hi".  Der 120-Zeller (120-cell) Marco Möller's Regular polytopes in R4 (German) 120-cell explorer – 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 YouTube animation of the construction of the 120-cell Gian Marco Todesco.

H4 family polytopes

120-cell rectified 120-cell truncated 120-cell cantellated 120-cell runcinated 120-cell cantitruncated 120-cell runcitruncated 120-cell omnitruncated 120-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 rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated 600-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

v t e

Regular 4-polytopes

Convex

5-cell 8-cell 16-cell 24-cell 120-cell 600-cell

3,3,3 pentachoron 4-simplex

4,3,3 tesseract 4-cube

3,3,4 hexadecachoron 4-orthoplex

3,4,3 icositetrachoron octaplex

5,3,3 hecatonicosachoron dodecaplex

3,3,5 hexacosichoron tetraplex

Star

icosahedral 120-cell small stellated 120-cell great 120-cell grand 120-cell great stellated 120-cell grand stellated 120-cell great grand 120-cell great icosahedral 120-cell grand 600-cell great grand stellated 120-cell

3,5,5/2 icosaplex

5/2,5,3 stellated dodecaplex

5,5/2,5 great dodecaplex

5,3,5/2 grand dodecaplex

5/2,3,5 great stellated dodecaplex

5/2,5,5/2 grand stellated dodecaplex

5,5/2,3 great grand dodecaplex

3,5/2,5 great icosaplex

3,3,5/2 grand tetraplex

5/2,3,3 great grand stellated dodecaplex

v t e

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 polyt

.

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