120-cell

In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}.

It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron[1] and hecatonicosahedroid.

The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).

[b] As the sixth and largest regular convex 4-polytope,[c] it contains inscribed instances of its four predecessors (recursively).

It is daunting but instructive to study the 120-cell, because it contains examples of every relationship among all the convex regular polytopes found in the first four dimensions.

Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.

[7] Hyper-tetrahedron 5-point Hyper-octahedron 8-point Hyper-cube 16-point 24-point Hyper-icosahedron 120-point Hyper-dodecahedron 600-point Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen.

In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates[9] are the {permutations} and [even permutations] in the left column below: (−1,  √5,  √5,  √5) / 4 (−1,−√5,−√5,  √5) / 4 (−1,−√5,  √5,−√5) / 4 (−1,  √5,−√5,−√5) / 4 ({±1, ±1, ±1, ±1}) / 2 ([0, ±φ−1, ±1, ±φ]) / 2 The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices.

The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point snub 24-cell and the 480-point diminished 120-cell.

[c] The second thing to notice is that each numbered row (each chord) is marked with a triangle △, square ☐, phi symbol 𝜙 or pentagram ✩.

The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope.

The hulls are illustrated as if they were all the same size, but actually they increase in radius as numbered: they are concentric 2-spheres that nest inside each other.

Considering the adjacency matrix of the vertices representing the polyhedral graph of the unit-radius 120-cell, the graph diameter is 15, connecting each vertex to its coordinate-negation at a Euclidean distance of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges.

Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius ⁠φ2/√8⁠ ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices.

These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron.

Each dodecahedron-inscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others face-bonded around it lying only partially within the dodecahedron.

The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.

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).

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 (discrete Hopf fibration).

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.

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.

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.

It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint (Clifford parallel) 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 edges alternating with 6 cell diameter chords, forming an irregular dodecagon in a central plane.

Stereographic projections use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a 3-sphere.

Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size.

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

All of these have a tetrahedral vertex figure {3,3}: The 120-cell is a part of a sequence of 4-polytopes and honeycombs with dodecahedral cells: Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.

[bp] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the tetrahedrally diminished dodecahedron because the 4 vertices removed formed a tetrahedron inscribed in the dodecahedron.

Net
Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic [ o ] rotations. The 120-cell edges of length 𝜁 ≈ 0.270 occur only in the red irregular great hexagon, which also has edges of length √ 2.5 . The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with √ 2.5 edges of inscribed regular 5-cells [ d ] they form 400 irregular great hexagons. [ p ] The 120-cell also contains a compound of several of these great circle polygons in the same central plane, illustrated separately. [ q ] An implication of the compounding is that the edges and characteristic rotations [ t ] of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in the same rotation planes, the hexagonal central planes of the 24-cell. [ u ]
The major [ ak ] chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.
Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an Overall Hull that is a chamfered dodecahedron of Norm= √ 8 .
Hulls 1, 2, & 7 are each pairs of dodecahedrons .
Hull 3 is a pair of icosidodecahedrons .
Hulls 4 & 5 are each pairs of truncated icosahedrons .
Hull 6 is a pair of semi-regular rhombicosidodecahedrons .
Hull 8 is a single non-uniform rhombicosidodecahedron , the central section.
Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.
One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron. [ at ]
Two intertwining rings of the 120-cell.
Two orthogonal rings in a cell-centered projection
In the tetrahedrally diminished dodecahedron , 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.
In triacontagram {30/12}=6{5/2} ,
six of the 120 disjoint regular 5-cells of edge-length √ 2.5 which are inscribed in the 120-cell appear as six pentagrams, the Clifford polygon of the 5-cell . The 30 vertices comprise a Petrie polygon of the 120-cell, [ v ] with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane. [ x ]
The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black), [ ax ] two irregular great hexagons ( red ), [ p ] and four equilateral great triangles (only one is shown, in green ).
In triacontagram {30/9}=3{10/3} we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single Boerdijk–Coxeter helix 30-tetrahedron ring of an inscribed 600-cell.
In triacontagram {30/8}=2{15/4} ,
2 disjoint pentadecagram isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation. [ z ] The pentadecagram edges are #4 chords [ aa ] joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon. [ ab ]
The Petrie polygon of the 120-cell is a skew regular triacontagon {30}. [ ah ] The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel {12} great circle polygons. [ q ]
The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.
Triacontagram {30/5}=5{6} , the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.