120-cell structure – Trinh Do

 

  • In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}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 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 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.

    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 120-cell is the 600-cell.
    • The vertex figure of the 120-cell is a tetrahedron.

    Cartesian coordinates

    The 600 vertices of the 120-cell include all permutations of:

    (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

    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

    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 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

    Two intertwining rings of the 120-cell.

    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

    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 in the 600-cell.

    Projections

    Orthogonal projections

    Orthogonal projections of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction.

    Orthographic projections by Coxeter planes
    H4 F4
    120-cell graph H4.svg
    [30]
    120-cell t0 p20.svg
    [20]
    120-cell t0 F4.svg
    [12]
    H3 A2 / B3 / D4 A3 / B2
    120-cell t0 H3.svg
    [10]
    120-cell t0 A2.svg
    [6]
    120-cell t0 A3.svg
    [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
    120Cell 3D.png
    3D isometric projection
    File:Cell120.ogv 

    Animated 4D rotation

    Perspective projections

    These projections use perspective projection, from a specific view point in 4-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 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 3-sphere.

    A comparison of perspective projections from 3D to 2D is shown in anology.

    Comparison with regular dodecahedron
    Projection Dodecahedron Dodecaplex
    Schlegel diagram Dodecahedron schlegel diagram.png
    12 pentagon faces in the plane
    Schlegel wireframe 120-cell.png
    120 dodecahedral cells in 3-space
    Stereographic projection Dodecahedron stereographic projection.png Stereographic polytope 120cell faces.png
    With transparent faces
    Perspective projection
    120-cell perspective-cell-first-02.png 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.
    120-cell perspective-vertex-first-02.png 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
    120-cell.gif A 3D projection of a 120-cell performing a simple rotation.
    120-cell-inner.gif A 3D projection of a 120-cell performing a simple rotation (from the inside).
    File:Cell120Persp.ogv 
    Animated 4D rotation

    Related polyhedra and honeycombs

    It is similar to three regular polychora: 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 atetrahedral vertex figure.

    {p,3,3}
    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 Stereographic polytope 5cell.png Stereographic polytope 8cell.png Stereographic polytope 120cell faces.png H3 633 FC boundary.png
    Coxeter diagrams
    subgroups
    1 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    4 CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
    6 CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
    12 CDel nodes 11.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.pngCDel 8.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.pngCDel infin.pngCDel node.png
    24 CDel nodes 11.pngCDel 2.pngCDel nodes 11.png CDel branch 11.pngCDel splitcross.pngCDel branch 11.png Cdel tet4 1111.png Cdel tetinfin 1111.png
    Cells
    {p,3}
    CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.png
    Tetrahedron.png
    {3,3}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    Hexahedron.png
    {4,3}
    CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
    CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
    CDel nodes 11.pngCDel 2.pngCDel node 1.png
    Dodecahedron.png
    {5,3}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
    Uniform tiling 63-t0.png
    {6,3}
    CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
    CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
    CDel branch 11.pngCDel split2.pngCDel node 1.png
    H2 tiling 237-1.png
    {7,3}
    CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
    H2 tiling 238-1.png
    {8,3}
    CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
    CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
    CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
    H2 tiling 23i-1.png
    {∞,3}
    CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
    CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
    CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png

    This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

    {5,3,p}
    Space S3 H3
    Form Finite Compact Paracompact Noncompact
    Name {5,3,3}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    {5,3,4}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
    CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
    {5,3,5}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
    {5,3,6}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
    CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
    {5,3,7}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
    {5,3,8}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
    CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
    … {5,3,∞}
    CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
    CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
    Image Schlegel wireframe 120-cell.png H3 534 CC center.png H3 535 CC center.png H3 536 CC center.png H3 53i UHS plane at infinity.png
    Vertex
    figure
    CDel node 1.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
    Tetrahedron.png
    {3,3}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    Octahedron.png
    {3,4}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
    Icosahedron.png
    {3,5}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
    Uniform tiling 63-t2.png
    {3,6}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
    H2 tiling 237-4.png
    {3,7}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
    H2 tiling 238-4.png
    {3,8}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
    H2 tiling 23i-4.png
    {3,∞}
    CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png

Source: Wikipedia

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