The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above). As the sixth and largest regular convex 4-polytope, it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the
5-cell, which is not found in any of the others. The 120-cell 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. That is why
Stillwell titled his paper on the 4-polytopes and the history of mathematics of more than 3 dimensions
The Story of the 120-cell.
Cartesian coordinates 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.
√8 radius coordinates The 120-cell with long radius = 2 ≈ 2.828 has edge length 4−2φ = 3− ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {
permutations} and of the following: where φ (also called 𝝉) is the
golden ratio, ≈ 1.618.
Unit radius coordinates The unit-radius 120-cell has edge length ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates are the {
permutations} and in the left column below: 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 (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.
Chords . The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|, six of the 120 disjoint regular 5-cells of edge-length 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, with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell are 6 great pentagons{{Efn|In
600-cell § Decagons and pentadecagrams, see the illustration of
triacontagram {30/6}=6{5}.}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length . No regular 4-polytopes except the 5-cell and the 120-cell contain chords (the #8 chord). The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells. Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic
isoclinic rotation,{{Efn|,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. The pentadecagram edges are #4 chords joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.The characteristic isoclinic rotation of the 120-cell takes place in the invariant planes of its 1200 edges and
its inscribed regular 5-cells' opposing 1200 edges.{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 (#1 chords), and 3 inscribed regular 5-cell edges of length (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon. The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an
irregular great dodecagon, a compound
great circle polygon of the 120-cell.|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete
Hopf fibration. In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle with 15 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord). This means that the two planes are separated by two equal 12° angles, and they are occupied by adjacent
Clifford parallel great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}2{15/4} projection visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}6{5/2} projection.|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits. They also give the 120-cell a characteristic great circle polygon: an
irregular great hexagon in which three 120-cell edges alternate with three 5-cell edges. The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the
5-cell and the
8-cell tesseract, they form zig-zag
Petrie polygons instead. The
120-cell's Petrie polygon is a
triacontagon {30} zig-zag
skew polygon.{{Efn| regular
triacontagon {30}. 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.The 120-cell contains 80 distinct
30-gon Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons. The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound
triacontagram {30/9}3{10/3} of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon. The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has at least 15 distinct chord lengths, ranging from its edge length to its diameter. Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15
major chords are so numbered because the #
n chord connects two vertices which are
n edge lengths apart on a Petrie polygon of the 120-cell. The 15 major chords lie on great circles in central planes that contain regular and irregular polygons of {4}, {10}, or {12} vertices. There are
30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). The 15
minor chords lie on rectangular {4} great circles and do not occur anywhere except inside the 120-cell. In this article, we name the 15 unnumbered minor chords by their arc-angles, e.g. 41.4~° which, with length , falls between the #3~4 chords.|name=additional 120-cell chords}} 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. The second thing to notice is that each numbered row (each chord) is marked with a triangle △, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐
characteristic of the 16-cell, great hexagons and great triangles △
characteristic of the 24-cell, great decagons and great pentagons 𝜙
characteristic of the 600-cell, and skew pentagrams ✩
characteristic of the 5-cell which circle through a set of central planes and form face polygons but not great polygons. The annotated chord table is a complete
bill of materials for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the
#3 chord row, the 600-cell's 72 great decagons contain 720
#3 chords in all. The '''''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains distinct 24-cells which share components. The '''''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the
#3 chord row,
#3 chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral
vertex figure 2{3,5}. In total major chords of 15 distinct lengths meet at each vertex of the 120-cell.
Relationships among interior polytopes The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes,
except for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which
does occur. Various regular
skew polygons {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15} and {30}.|name=elements}} It is a four-dimensional
jigsaw puzzle in which all those polytopes are the parts. Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes. The regular 1-polytope occurs in only
15 distinct lengths in any of the component polytopes of the 120-cell. By
Alexandrov's uniqueness theorem, convex polyhedra with shapes distinct from each other also have distinct
metric spaces of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four , , and are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built from them. An additional 4 of the 15 chords are required to build the 600-cell. The four are square roots of irrational fractions that are functions of . The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. All 15 chords, and 15 other distinct chordal distances enumerated below, occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{Efn|Dodecahedra emerge as
visible features in the 120-cell, but they also occur in the 600-cell as
interior polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the
regular 5-cell (the
simplex regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the
3-simplex and
5-orthoplex. An
n-simplex is bounded by n+1 vertices and n+1 (n-1)-simplex facets, and has n+1 long diameters (its edges) of length \sqrt{n+1}/\sqrt{n} radii. An
n-orthoplex is bounded by 2n vertices and 2^n (n-1)-simplex facets, and has n long diameters (its orthogonal axes) of length 2 radii. An
n-cube is bounded by 2^n vertices and 2n (n-1)-cube facets, and has 2^{n-1} long diameters of length \sqrt{n} radii.{{Efn|The n-simplex's facets are larger than the n-orthoplex's facets. For n=4, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of \sqrt{5} to \sqrt{4} to \sqrt{2}.|name=root 5/root 4/root 2}} The \sqrt{3} long diameters of the 3-cube are shorter than the \sqrt{4} axes of the 3-orthoplex. The
coordinates of the 4-orthoplex are the permutations of (0,0,0,\pm 1), and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of (0,0,0,1). The \sqrt{4} long diameters of the 4-cube are the same length as the \sqrt{4} axes of the 4-orthoplex. The
coordinates of the 5-orthoplex are the permutations of (0,0,0,0,\pm 1), and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of (0,0,0,0,1). The \sqrt{5} long diameters of the 5-cube are longer than the \sqrt{4} axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells. Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.
Compound of five 600-cells The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways. As a consequence of being a compound of five disjoint 600-cells, the 120-cell has 200 irregular great dodecagon {12} central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 {12} central planes originate as the compounds of the hexagonal central planes of the 25 disjoint inscribed 24-cells and the digon central planes of the 120 disjoint inscribed regular 5-cells; they contain all the 24-cell and 5-cell edges, and also the 120-cell edges. Thus the edges and characteristic rotations of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in these same 200 rotation planes. Each of the ten 600-cells occupies the entire set of 200 planes. The 120-cell's irregular
dodecagon {12} great circle polygon has 6 short edges (#1
chords marked ) alternating with 6 longer dodecahedron cell-diameters ( chords). Inscribed in the irregular great dodecagon are two irregular great hexagons () in alternate positions. Two
regular great hexagons with edges of a third size (, the #5 chord) are also inscribed in the dodecagon. The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of 200 dodecagon planes. They occur as 100 completely orthogonal pairs, and they are the
same 200 central planes each containing a
hexagon that are found in
each of the 10 inscribed 600-cells. There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares regular hexagons with 5 other 600-cells. Each disjoint pair of 600-cells occupies the opposing pair of disjoint regular hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25). Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells.
Geodesic rectangles The 30 distinct chords found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Backgroundcolor|palegreen|△ planes that intersect {12} vertices}} in an
irregular great dodecagon, {{Backgroundcolor|yellow|𝜙 planes that intersect {10} vertices}} in a regular decagon, and ☐ planes that intersect {4} vertices in several kinds of , including a . Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing
polyhedral sections of the 120-cell, beginning with a vertex, the 00 section. The correspondence is that each 120-cell vertex is surrounded by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius. The #1 chord is the "radius" of the 10 section, the tetrahedral vertex figure of the 120-cell. The #14 chord is the "radius" of its congruent opposing 290 section. The #7 chord is the "radius" of the central section of the 120-cell, in which two opposing 150 sections are coincident.
Concentric hulls of Norm=. 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. These hulls illustrate sections 1 - 8 of the 120-cell beginning with a cell (hull 1).{{Sfn|Coxeter|1973|p=299|loc=Table V (iv) Sections of {5,3,3} beginning with a cell (right half of table)}} A
section is a flat 3-dimensional hyperplane slice through the
3-sphere: a 2-sphere (ordinary sphere). It is dimensionally analogous to a flat 2-dimensional plane slice through a 2-sphere: a 1-sphere (ordinary circle). 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. Every cell of the 120-cell is the smallest hull in its own set of 8 concentric hulls. There are 120 distinct nesting sets of 8 hulls. The cell-first projection of the 120-cell actually has 15 sections beginning with a cell, numbered 1 - 15 with number 8 in the center. After increasing in size from 1 to 8, the hulls get smaller again. Sections 1 and 15 are both a hull 1, the smallest hull, a dodecahedral cell of the 120-cell. Section 8 is the central section, the largest hull, with the same radius as the 120-cell. Except for the central section 8, the sections occur in parallel pairs, on either side of the central section. Hull 8 is dimensionally analogous to the equator, while hulls 1 - 7 are dimensionally analogous to lines of latitude. There are 120 of each kind of hull 1 - 7 in the 120-cell, but only 60 of the central hull 8. A visualization of these 15 simplified sections is available in WP Commons with subgroup sections (when the inscribed solid has more than one permutation in its orbit) is here. The vertex-first projection (below) of the {5,3,3} 120-cell has 31 sections.
Polyhedral graph 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. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are
3-colorable. The graph is
Eulerian having degree 4 in every vertex. Its edge set can be decomposed into two
Hamiltonian cycles.
Constructions The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the
600-cell, just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the
24-cell, the 24-cell can be deconstructed into three distinct instances of its predecessor the
tesseract (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the
16-cell. The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell. The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse
golden ratio), but the 120-cell's edge length is ~0.270 times its radius. The 120-cell is also the convex hull of the regular compound of 120 disjoint regular 5-cells. This can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point 600-cell, expand each vertex into a regular 5-cell. For each of the 120 vertices, add 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point 600-cells: a 120-cell.
Dual 600-cells Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ( ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ2 ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The
unit radius 120-cell (with edge-length ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius ≈ 1.080. Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given
above of long radius = 2 ≈ 2.828 and edge-length = 3− ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ2, which is smaller than in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ2 ≈ 7.405 given by Coxeter can be constructed in this manner just outside a 600-cell of long radius φ4 and edge-length φ3. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps.
Cell rotations of inscribed duals Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them. Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously). This shows that the 120-cell contains, among its many interior features, 120
compounds of ten tetrahedra, each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells. All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell. Therefore, the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells.
Augmentation Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing
4-pyramids of some kind on the cells of the 600-cell. 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. Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. 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. The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.
Weyl orbits Another construction method uses
quaternions and the
Icosahedral symmetry of
Weyl group orbits O(\Lambda)=W(H_4)=I of order 120. The following describe T and T'
24-cells as quaternion orbit weights of D4 under the Weyl group W(D4): O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2} O(1000) : V1 O(0010) : V2 O(0001) : V3 T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix}; With quaternions (p,q) where \bar p is the conjugate of p and [p,q]:r\rightarrow r'=prq and [p,q]^*:r\rightarrow r''=p\bar rq, then the
Coxeter group W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace is the symmetry group of the
600-cell and the 120-cell of order 14400. Given p \in T such that \bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p and p^\dagger as an exchange of -1/\varphi \leftrightarrow \varphi within p, we can construct: • the
snub 24-cell S=\sum_{i=1}^4\oplus p^i T • the
600-cell I=T+S=\sum_{i=0}^4\oplus p^i T • the 120-cell J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T' • the alternate snub 24-cell S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T' • the
dual snub 24-cell = T \oplus T' \oplus S'.
As a configuration This
configuration matrix represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. \begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix} Here is the configuration expanded with
k-face elements and
k-figures. The diagonal element counts are the ratio of the full
Coxeter group order, 14400, divided by the order of the subgroup with mirror removal. == Visualization ==