{{Short description|Four-dimensional analog of the dodecahedron}}









!24| ({0, 0, ±2, ±2})!64| ({±φ, ±φ, ±φ, ±φ−2})|!64!64| ({±φ−1, ±φ−1, ±φ−1, ±φ2})|!96!96| ([0, ±φ−2, ±1, ±φ2])!192| ([±φ−1, ±1, ±φ, ±2])|

factoids
Name	120-cell| Image_File=Schlegel wireframe 120-cell.png| Image_Caption=Schlegel diagram(vertices and edges)| Type=Convex regular 4-polytope Snub 24-cell>31| Index=32 Rectified 120-cell>33 | Cell_List=120 {5,3} missing image! - Dodecahedron.png - 20px | Face_List=720 {5} (File:Regular pentagon.svg|20px)| Edge_Count=1200| Vertex_Count= 600| Petrie_Polygon=30-gon| Coxeter_Group=H4, [3,3,5]| Vertex_Figure=(File:120-cell verf.svg|80px)tetrahedron| Dual=600-cell| Property_List=convex, isogonal, isotoxal, isohedral }}File:120-cell net.png|thumb|right|Net ]]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, dodecacontachoronN.W. Johnson: Geometries and Transformations, (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249 and hecatonicosahedroid.Matila Ghyka, The Geometry of Art and Life (1977), p.68The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. Together they form 720 pentagonal faces, 1200 edges, and 600 vertices. It is the 4-dimensional analogue of the regular dodecahedron, since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the dodecaplex has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the 600-cell. Geometry The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 120-cell is the 600-point 4-polytope: sixth and last in the ascending sequence that begins with the 5-point 4-polytope.|name=polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5-cell,{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=Remark 5.7|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional Swiss Army knife: it contains one of everything.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.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detailand perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why Stillwell titled his paper on the 4-polytopes and the history of mathematicsMathematics and Its History, John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}} of more than 3 dimensions The Story of the 120-cell.{{Sfn|Stillwell|2001}}{{Regular convex 4-polytopes}} 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 {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {permutations} and {{bracket|even permutations}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable
24-cell#Squares>24-cell	600-point 120-cell
5}}})|
5}}])		Snub 24-cell#Coordinates>Snub 24-cell
Snub 24-cell#Coordinates>Snub 24-cell

where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the golden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the golden ratio, {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618.

Unit radius coordinates

The unit-radius 120-cell has edge length {{Sfrac|1|φ2{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone|Pileio|Levitt|2010|p=1442|loc=Table 3}} are the {permutations} and {{bracket|even permutations}} in the left column below:{| class="wikitable" style=width:720px!rowspan=3|120!8({±1, 0, 0, 0})16-cell#Coordinates>16-cell24-cell600-cell120-cell!16({±1, ±1, ±1, ±1}) / 2Tesseract#Radial equilateral symmetry>Tesseract!96([0, ±φ−1, ±1, ±φ]) / 2Snub 24-cell!rowspan=7|480!colspan=2|Diminished 120-cell!5-point 5-cell!24-cell!600-cell!32([±φ, ±φ, ±φ, ±φ−2]) / {{radic|8}}(1, 0, 0, 0)(−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4(−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4(−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4(−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4({±{{radic1/2}}, 0, 0})({±1, 0, 0, 0})({±1, ±1, ±1, ±1}) / 2([0, ±φ−1, ±1, ±φ]) / 2!32([±1, ±1, ±1, ±{{radic8}}!32([±φ−1, ±φ−1, ±φ−1, ±φ2]) / {{radic|8}}!96([0, ±φ−1, ±φ, ±{{radic8}}!96([0, ±φ−2, ±1, ±φ2]) / {{radic|8}}!192([±φ−1, ±1, ±φ, ±2]) / {{radic|8}}The unit-radius coordinates of uniform convex 4-polytopes are related by quaternion multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible Quaternion#Multiplication of basis elements{{Sfn>MamoneLevittp=1433ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors left(w,x,y,zright)_1 and left(w,x,y,zright)_2 according tobegin{pmatrix}w_2x_2y_2z_2end{pmatrix}

begin{pmatrix}w_1x_1y_1z_1end{pmatrix}

begin{pmatrix}{w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}{w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}{w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}{w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1}end{pmatrix}}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone|Pileio|Levitt|2010|loc=Table 3|p=1442}}}}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.{{Efn|The 120-cell can be constructed as a compound of {{red|5}} disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or {{red|25}} disjoint 24-cells, or {{red|75}} disjoint 16-cells, or {{red|120}} disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish disjoint multiple instances from merely distinct multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of all the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of completely disjoint inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains {{green|10}} distinct 600-cells, {{green|225}} distinct 24-cells, and {{green|675}} distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}}

Chords

File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between k-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also Clifford parallel) and they intersect in a single point. In double rotations, points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a helical smooth curve called an isocline{{Efn|An isocline is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a straight line, a geodesic. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are linked and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|name=isocline circumference}} To avoid confusion, we always refer to an isocline as such, and reserve the term great circle for an ordinary great circle in the plane.{{Efn|name=isocline circumference}}|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a Clifford displacementClifford displacement{{see also|600-cell#Golden 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|File:Regular_star_figure_6(5,2).svg|thumb|200px|In triacontagram {30/12}=6{5/2}, six of the 120 disjoint regular 5-cells of edge-length {{radic|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,{{Efn|name=two coaxial Petrie 30-gons}} 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{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In 600-cell § Decagons and pentadecagrams, see the illustration of triacontagram {30/6}=6{5}triacontagram {30/6}=6{5}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 {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} 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 {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, pentagonal circuits each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two disjoint 5-cells are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic isoclinic rotation,{{Efn|File:Regular_star_figure_2(15,4).svg|thumb|200px|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.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the same discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}})The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and its inscribed regular 5-cells' opposing 1200 edges.{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|0.𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} 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 {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} 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 which is illustrated separately.{{Efn|name=irregular great dodecagon}}|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete Hopf fibration.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢4 the operation [15]𝑹q3,q3 is the 15 distinct rotational displacements which comprise the class of pentagram isoclinic rotations of an individual 5-cell; in symmetry group 𝛨4 the operation [1200]𝑹q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 #4 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=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).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} 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{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨4|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct isoclinic rotations.{{Efn|name=distinct rotations}}}} 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.{{Efn|name=irregular great hexagon}}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.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical isocline{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its Clifford polygon.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The 120-cell's Petrie polygon is a triacontagon {30} zig-zag skew polygon.{{Efn|File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a skew regular name=15 distinct chord lengths}} 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.{{Efn|name=irregular great dodecagon}}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.{{Efn|name=Petrie polygons of the 120-cell}} 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.{{Efn|name=two coaxial Petrie 30-gons}} 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 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the Petrie polygon can be picked out as a skew polygon of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a Boerdijk–Coxeter helix ring. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered chords of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct Pythagorean distances through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} 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. 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). In this article, we name the 15 unnumbered minor chords by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #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.{{Efn|name=polytopes ordered by size and complexity}}The second thing to notice is that each numbered row is marked with a triangle △, square ☐, or pentagon ✩. The 15 chords lie in central planes of three kinds: great square ☐ planes characteristic of the 16-cell, great hexagon and great triangle △ planes characteristic of the 24-cell, or great decagon and great pentagon ✩ planes characteristic of the 600-cell.{{Efn|The edges and 4𝝅 characteristic rotations of the 16-cell lie in the great square ☐ central planes. Rotations of this type are an expression of the symmetry group B_4. The edges and 5𝝅 characteristic rotations of the 600-cell lie in the great pentagon ✩ (great decagon) central planes. Rotations of this type are an expression of the symmetry group H_4. The edges and characteristic rotations{{Efn|name=isocline circumference}} of the other regular 4-polytopes, the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell,{{Efn|name=120-cell characteristic rotation}} all lie in the great triangle △ (great hexagon) central planes.{{Efn|name=irregular great dodecagon}} Collectively these rotations are expressions of all four symmetry groups A_4, B_4, F_4 and H_4.|name=edge rotation planes}}{| class=wikitable style="white-space:nowrap;text-align:center"!colspan=14|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ−2√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 10 − 160, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2la) are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column a for (30−i)0) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2lb, which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}}!colspan=5|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii 0𝑹, 1𝑹, 2𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈0, 𝐈𝐈1, 𝐈𝐈2, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}}!5-cell!16-cell!8-cell!24-cell!Snub!600-cell!Dimin! style="border-right: none;"|120-cell! style="border-left: none;"|!colspan=5|Vertices58162496120480600{{Efn|name=rays and bases}}!colspan=5|Edges10{{Efn|name=irregular great hexagon}}24329643272012001200{{Efn|name=irregular great hexagon}}!colspan=5|Edge chord#8{{Efn|name=inscribed 5-cells}}#7#5#5#3#3{{Efn180pxIn Triacontagon#Triacontagram 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 600-cell#Boerdijk–Coxeter helix rings>Boerdijk–Coxeter helix 30-tetrahedron ring600-cell#Boerdijk–Coxeter helix rings>Boerdijk–Coxeter helix 30-tetrahedron ring The 120-cell and 600-cell both have 30-gon Petrie polygons.{{EfnSkew polygon#Regular skew polygons in four dimensions>regular skew 30-gon is the Petrie polygon of the 600-cell and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell 600-cell#Boerdijk–Coxeter helix rings (the Petrie polygons of the 600-cell):{{Efn>(File:Regular_star_polygon_30-11.svgthumb600-cell#Boerdijk–Coxeter helix rings>600-cell Petrie polygon is a helical ring which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a Triacontagon#Triacontagram of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn>Sadocpp=577-578name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a 600-cell#Decagons).{{Efn>name=two coaxial Petrie 30-gons}}600-cell#Boerdijk–Coxeter helix rings>600-cell's Petrie helix does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efnname=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew Triacontagon#Triacontagram (versus a skew Triacontagon#Triacontagram>{30/11} polygram).{{Efnname=two coaxial Petrie 30-gons}}#1#1{{Efn|name=120-cell Petrie {30}-gon}}!colspan=5|Isocline chord{{Efn|name=isoclinic rotation}}#8#15#10#10#5#5#4#4{{Efn|name=#4 isocline chord}}!colspan=5|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's Clifford polygon, as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic Clifford displacement.{{Efn|name=isoclinic}}|name=Clifford polygon}}{5/2}{8/3}{{^hexagram>{6/2}}}{6/2}{{^pentadecagram>{15/2}}}{15/2}{{^pentadecagram>{15/4}}}pentadecagram{{Efn>name=120-cell characteristic rotation}}!colspan=2|Chord!Arc!colspan=2|Edge style="background: seashell;"|#1△(File:Regular_polygon_30.svg{30})|120-cell edge{{Efn|name=120-cell Petrie {30}-gon}}{{red1}}1200{{Efn>name=120-cell characteristic rotation}}{{blue|4}}{3,3} style="background: seashell;"||15.5~°0.𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where:{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio {{sfrac|1|φ}}{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽2 ≈ 0.382{{indent|7}}𝜀 = 𝚫2/2 ≈ 0.073and the 120-cell edge-length {{sfrac|1|φ2{{radic|2}}}} is:{{indent|7}}𝛇 = {{radic|𝜀}} ≈ 0.270For example:{{indent|7}}𝛇 = {{radic|0.𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}}|0.270~ style="background: seashell;"|#2☐(File:Regular_star_figure_2(15,1).svg{30/2}=2{15})1/φ{{radic|2}}}}face diagonal{{EfnThe face Pentagon#Regular pentagons (the #2 chord) is in the golden ratio φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn>name=dodecahedral cell metrics}}name=#2 chord}}3600{{blue|12}}2{3,4} style="background: seashell;"||25.2~°0.19~}}|0.437~ style="background: yellow;"|#3✩(File:Regular_star_figure_3(10,1).svg{30/3}=3{10})1/φ}}great decagon phi^{-1}{{green10}}{{Efn>name=inscribed counts}}7207200{{blue|24}}2{3,5} style="background: yellow;"||36°0.𝚫}}|0.618~ style="background: seashell;"|#4△(File:Regular_star_figure_2(15,2).svg{30/4}=2{15/2})name=irregular great dodecagon}}cell diameter{{Efnname=dodecahedral cell metrics}}1200{{blue|4}}{3,3} style="background: seashell;"||44.5~°0.57~}}|0.757~ style="background: paleturquoise;"|#5△(File:Regular_star_figure_5(6,1).svg{30/5}=5{6})|𝝅/3600-cell#Hexagons{{Efn>File:Regular_star_figure_5(6,1).svg180pxTriacontagon#Triacontagram>Triacontagram {30/5}=5{6}Triacontagon#Triacontagram>Triacontagram {30/5}=5{6} Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a 0-gon great circle axis that does not intersect any vertices.{{Efnname=same 200 planes}}|name=great hexagon}}32{{green225}}{{Efn>name=inscribed counts}}96{{green|225}}{{red5}}{{Efn>name=inscribed counts}}12002400{{Efn|name=same 200 planes}}{{blue|32}}4{4,3} style="background: paleturquoise;"||60°1}}|1 style="background: yellow;"|#6✩(File:Regular_star_figure_6(5,1).svg{30/6}=6{5}){{radic|3−φ}}}}600-cell#Decagons and pentadecagrams{{Efn>name=great pentagon}}7207200{{blue|24}}2{3,5} style="background: yellow;"||72°1.𝚫}}|1.175~ style="background: paleturquoise;"|#7☐(File:Regular_star_polygon_30-7.svg{30/7})|𝝅/2600-cell#Squares{{Efn>name=rays and bases}}{{green675}}{{Efn>name=rays and bases}}24{{green|675}}4872180016200{{blue|54}}9{3,4} style="background: paleturquoise;"||90°2}}|1.414~ style="background: seashell;"|#8△(File:Regular_star_figure_2(15,4).svg{30/8}=2{15/4})|5-cell#Boerdijk–Coxeter helix{{Efn>The 5-cell#Boerdijk–Coxeter helix is the pentagon (5-gon), and the Petrie polygon of the 120-cell is the triacontagon (30-gon).{{Efn>name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 30-gon lies completely orthogonal to six 5-cell Petrie 5-gons, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efnname=orthogonal Petrie polygons.}}{{red120}}{{Efn>name=inscribed 5-cells}}107201200{{Efn|name=120-cell characteristic rotation}}{{blue|4}}{3,3} style="background: seashell;"||104.5~°2.5}}|1.581~ style="background: yellow;"|#9✩(File:Regular_star_figure_3(10,3).svg{30/9}=3{10/3})|3𝝅/5golden section phi7207200{{blue|24}}2{3,5} style="background: yellow;"||108°2.𝚽}}|1.618~ style="background: paleturquoise;"|#10△(File:Regular_star_figure_10(3,1).svg{30/10}=10{3})|2𝝅/3great triangle32{{red25}}{{Efn>name=inscribed counts}}9612002400{{blue|32}}4{4,3} style="background: paleturquoise;"||120°3}}|1.732~ style="background: seashell;"|#11△(File:Regular_star_polygon_30-11.svg{30/11})|600-cell#Boerdijk–Coxeter helix rings{{Efn>name={30/11}-gram}}1200{{blue|4}}{3,3} style="background: seashell;"||135.5~°3.43~}}|1.851~ style="background: yellow;"|#12✩(File:Regular_star_figure_6(5,2).svg{30/12}=6{5/2}){{radic|2+φ}}}}great pentagon diag7207200{{blue|24}}2{3,5} style="background: yellow;"|name=dihedral}}3.𝚽}}|1.902~ style="background: seashell;"|#13☐(File:Regular_star_polygon_30-13.svg{30/13})|{30/13}-gram3600{{blue|12}}2{3,4} style="background: seashell;"||154.8~°3.81~}}|1.952~ style="background: seashell;"|#14△(File:Regular_star_figure_2(15,7).svg{30/14}=2{15/7})|{30/14}=2{15/7}1200{{blue|4}}{3,3} style="background: seashell;"||164.5~°3.93~}}|1.982~ style="background: paleturquoise;"|#15△☐✩(File:Regular_star_figure_15(2,1).svg30/15}=15{2})|𝝅diameter{{red75}}{{Efn>name=inscribed counts}}48124860240300{{Efn|name=rays and bases}}{{blue|1}} style="background: paleturquoise;"||180°4}}|2!colspan=5|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}}256425657614400360000{{Efn|name=additional 120-cell chords}}!{{blue|300}}(File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.)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 {{red|red}} 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 {{red|25}} disjoint 24-cells (25 * 24 vertices = 600 vertices).The {{green|green}} 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 {{green|225}} distinct 24-cells which share components.The {{blue|blue}} 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, {{blue|24}} #3 chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral vertex figure 2{3,5}. In total {{blue|300}} major chords{{Efn|name=additional 120-cell 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. 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}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional jigsaw puzzle in which all those polytopes are the parts.{{Sfn|Schleimer|Segerman|2013}} 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.{{Sfn|Stillwell|2001}}The regular 1-polytope occurs in only 15 distinct lengths in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By Alexandrov's uniqueness theorem, convex polyhedra with distinct shapes 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 {{background color|paleturquoise|hypercubic chords}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} 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.{{see also|24-cell#Relationships among interior polytopes}}An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|golden chords}} are square roots of irrational fractions that are functions of {{radic|5}}. 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.{{see also|600-cell#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 regular 5-cells.{{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).{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting (0,0,0,1), and its completely orthogonal 3-facet permuting (0,0,0,-1), comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-demicube, half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to n=4. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} 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).{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting (0,0,0,0,1), and its completely orthogonal 4-facet permuting (0,0,0,0,-1), comprise all 10 vertices of the 5-orthoplex.}} 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.{{Efn|No vertex pair of any of the 120 5-cells (no great digon central plane of a 5-cell) occurs in any of the 675 16-cells (the 675 Cartesian basis sets of 6 orthogonal central planes).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}}{{see also|5-cell#Geodesics and rotations}}

Geodesic rectangles

The 30 distinct chords{{Efn|name=additional 120-cell 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 dodecagon,{{Efn|name=irregular great dodecagon}} {{Backgroundcolor|yellow|✩ planes that intersect {10} vertices}} in a regular decagon, and {{Backgroundcolor|gainsboro|☐ planes that intersect {4} vertices}} in several kinds of rectangle, including a square. 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.{{Efn|In the curved 3-dimensional space of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the apex vertex and its surrounding base polyhedron form a polyhedral pyramid. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} The #1 chord is the "radius" of the 10 section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} 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.{| class=wikitable style="white-space:nowrap;text-align:center"!colspan=13|30 chords (15 180° pairs) make 15 kinds of great circle polygons and polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ−2√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 10 − 160|ps=, but 140 and 160 are congruent opposing sections and 150 opposes itself; there are 29 non-point sections, denoted 10 − 290, in 15 opposing pairs.}}!colspan=4|Short chord!colspan=2|Great circle polygons!Rotation!colspan=4|Long chord style="background: palegreen;"|10#1In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in every central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by two 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central planes,{{Efnall the 120-cell's various isoclinic rotations (not only in its characteristic rotation).>name=12° rotation angle}}1 / phi^2sqrt{2}(File:Irregular great hexagons of the 120-cell.png|100px)400 irregular great hexagons{{Efn|name=irregular great dodecagon}} / 4(600 great rectangles)in 200 △ planes4𝝅{{EfnTriacontagon#Triacontagram>{15/4}{{EfnThe characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efnname=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efnname=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efnname=isocline}} does not intersect any nearer vertices.{{^name=#2 chord}} as do the isocline chords of other characteristic isoclinic rotations whose edge chords are their rotation chords.{{EfnThe #4 chord is {{radicname=#2 chord}} the same relationship the #10 {{radic1}} rotation chord, because the invariant planes of two different characteristic rotations (the 120-cell's irregular great hexagons and the 24-cell's regular hexagons, respectively) both have sets of invariant rotation planes which are distinct subsets of the same 200 △ dodecagon central planes.{{Efnname=regular and irregular great hexagon isoclinic rotations}} The #2 rotation chord is the total vertex displacement as measured in the moving reference frame of one of the invariant planes, the way the #4 isocline chord is the total vertex displacement as measured in the stationary reference frame.name=120-cell rotation angle}}|name=#4 isocline chord}}|phi^{5}sqrt{3} / sqrt{8}290#14 style="background: palegreen;"||15.5~°0.𝜀}}{{Efn|name=fractional square roots}}|0.270~|164.5~°3.93~}}|1.982~ style="background: gainsboro;"|20#2name=#2 chord}}1 / phisqrt{2}(File:25.2° × 154.8° chords great rectangle.png|100px)Great rectanglesin ☐ planes4𝝅{30/13}#13|280#13 style="background: gainsboro;"||25.2~°0.19~}}|0.437~|154.8~°3.81~}}|1.952~ style="background: yellow;"|30#3|pi / 51 / phi(File:Great decagon rectangle.png|100px)720 great decagons / 12(3600 great rectangles)in 720 ✩ planes5𝝅{15/2}#5|4pi / 5sqrt{2+phi}270#12 style="background: yellow;"||36°0.𝚫}}|0.618~name=dihedral}}3.𝚽}}|1.902~ style="background: gainsboro;"|40#4−1|sqrt{1}/sqrt{2}(File:√0.5 × √3.5 great rectangle.png|100px)Great rectanglesin ☐ planes|sqrt{7} / sqrt{2}260#11+1 style="background: gainsboro;"||41.4~°0.5}}|0.707~|138.6~°3.5}}|1.871~ style="background: palegreen;"|50#4Efn|name=regular and irregular great hexagon isoclinic rotations}}sqrt{3} / phisqrt{2}(File:Irregular great dodecagon.png|100px)200 irregular great dodecagons{{Efnblue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radicname=irregular great dodecagon}} Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efnname=isoclinic rotation}} The #4 chord{{Efnregular great hexagons (the 600-cell#Hexagons>24-cell's characteristic rotation), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in irregular great hexagons).{{Efnname=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every regular great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.name=dodecagon rotation}} / 4(600 great rectangles)in 200 △ planes{{Efn|name=#4 isocline chord bridge}}|phi^2 / sqrt{2}250#11 style="background: palegreen;"||44.5~°0.57~}}|0.757~|135.5~°3.43~}}|1.851~ style="background: gainsboro; height:50px"|60#4+1pi / 3.67sim}}(File:49.1° × 130.9° great rectangle.png|100px)Great rectanglesin ☐ planes|240#11−1 style="background: gainsboro;"||49.1~°0.69~}}|0.831~|130.9~°3.31~}}|1.819~ style="background: gainsboro; height:50px"|70#5−1pi / 3.21sim}}(File:56° × 124° great rectangle.png|100px)Great rectanglesin ☐ planes|230#10+1 style="background: gainsboro;"||56°0.88~}}|0.939~|124°3.12~}}|1.766~ style="background: palegreen;"|80#5|pi / 3(File:Great hexagon.png|100px)400 regular 600-cell#Hexagons{{Efn>name=great hexagon}} / 16 (1200 great rectangles)in 200 △ planes4𝝅{{Efncharacteristic rotation of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.>name=isocline circumference}}2{10/3}#4|2pi / 3220#10 style="background: palegreen;"||60°1}}|1|120°3}}|1.732~ style="background: gainsboro; height:50px"|90#5+1pi / 2.72sim}}(File:66.1° × 113.9° great rectangle.png|100px)Great rectangles in ☐ planes|210#10−1 style="background: gainsboro;"||66.1~°1.19~}}|1.091~|113.9~°2.81~}}|1.676~ style="background: gainsboro; height:50px"|100#6−1pi / 2.58sim}}(File:69.8° × 110.2° great rectangle.png|100px)Great rectangles in ☐ planes|200#9+1 style="background: gainsboro;"||69.8~°1.31~}}|1.144~|110.2~°2.69~}}|1.640~ style="background: yellow;"|110#6|2pi/5sqrt{3-phi}(File:Great pentagons rectangle.png|100px)1440 600-cell#Decagons and pentadecagrams{{Efn>name=great pentagon}} / 12(3600 great rectangles)in 720 ✩ planes4𝝅{24/5}#9|3pi / 5phi190#9 style="background: yellow;"||72°1.𝚫}}|1.175~|108°2.𝚽}}|1.618~ style="background: palegreen; height:50px"|120#6+1pi/2.38sim}}sqrt{3} / sqrt{2}(File:Great 5-cell digons rectangle.png|100px)1200 5-cell#Geodesics and rotations{{Efn>The 5-cell#Geodesics and rotations intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic>2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radicname=5-cell rotation}} / 4(600 great rectangles)in 200 △ planes4𝝅{{EfnPentagram>{5/2}#8|sqrt{5} / sqrt{2}180#8 style="background: palegreen;"||75.5~°1.5}}|1.224~|104.5~°2.5}}|1.581~ style="background: gainsboro; height:50px"|130#6+2pi / 2.22sim}}(File:81.1° × 98.9° great rectangle.png|100px)Great rectangles in ☐ planes|170#8−1 style="background: gainsboro;"||81.1~°1.69~}}|1.300~|98.9~°2.31~}}|1.520~ style="background: gainsboro; height:50px"|140#7−1pi / 2.13sim}}(File:84.5° × 95.5° great rectangle.png|100px)Great rectangles in ☐ planes|160#7+1 style="background: gainsboro;"||84.5~°0.81~}}|1.345~|95.5~°2.19~}}|1.480~ style="background: gainsboro;"|150#7|pi / 2(File:Great square rectangle.png|100px)4050 600-cell#Squares{{Efn>name=rays and bases}} / 27in 4050 ☐ planes4𝝅{30/7}#7|pi / 2150#7 style="background: gainsboro;"||90°2}}|1.414~|90°2}}|1.414~Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled double rotation in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its rotation angle and its isocline angle.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an isoclinic rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a simple rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this incorrectly, because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its completely orthogonal plane, but Clifford parallel to it. It rotates with its completely orthogonal plane, but not in it. It is Clifford parallel to its completely orthogonal plane and to the plane it is moving to, and does not intersect them; the plane that it rotates in is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal planes are 90° apart, in the two orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different orientation (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two adjacent great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} of a distinct class,{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The characteristic isoclinic rotation of a 4-polytope is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each Hopf fibration of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an irregular great dodecagon which also has #4 chord edges.{{Efn|name=irregular great dodecagon}} In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons / 4".}} In each class of rotation,{{Efn|Rotations in 4-dimensional Euclidean space are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are invariant, which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually Clifford parallel. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called fibrations, which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular fibers, the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}}

Concentric hulls

File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px|Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm={{radic|8}}.Hulls 1, 2, & 7 are each overlapping pairs of Dodecahedrons.Hull 3 is a pair of Icosidodecahedrons.Hulls 4 & 5 are each pairs of Truncated icosahedrons.Hulls 6 & 8 are pairs of RhombicosidodecahedronRhombicosidodecahedron{{Clear}}

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 {{sfrac|1|φ2{{radic|2}}}} ≈ 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.BOOK, Carlo H. Séquin, Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes, July 2005, 463–472, Mathartfun.com, 9780966520163,weblink March 13, 2023,

Constructions

The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the 600-cell,{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the 24-cell,{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 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.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 rays [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a basis. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a geometric configuration, which in the language of configurations is written as 30096754 to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct 16-cell containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}}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.

Dual 600-cells

(File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.)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 ({{sfrac|φ2|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ2{{radic|2}} ≈ 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 {{sfrac|1|φ2{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ2}} ≈ 1.080.File:Dodecahedron_vertices.svg|thumb|150px|right|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.{{Efn|In the dodecahedral cell of the unit-radius 120-cell, the length of the edge (the #1 chordchordReciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ2|{{radic|8}}}} ≈ 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 {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ2}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ2, which is smaller than {{Radic|8}} 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{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} 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.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} 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).{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a compound of five octahedra, which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a compound of ten tetrahedra. Further, we can choose one tetrahedron from each stella octangula, so as to derive a compound of five tetrahedra, which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be chiral."}} 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.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 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.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-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.{{Efn|name=truncated apex}}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.{{Efn|As we saw in the 600-cell, these 12 tetrahedra belong (in pairs) to the 6 icosahedral clusters of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} 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.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} 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) : V1O(0010) : V2O(0001) : V3T'=sqrt{2}{V1oplus V2oplus 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]:rrightarrow r'=prq and [p,q]^*:rrightarrow r''=pbar 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}^4oplus p^i T
• the 600-cell I=T+S=sum_{i=0}^4oplus p^i T
• the 120-cell J=sum_{i,j=0}^4oplus p^ibar p^{dagger j}T'
• the alternate snub 24-cell S'=sum_{i=1}^4oplus p^ibar 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.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}}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.{| class=wikitable!H4||{{CDD|node_1|5|node|3|node|3|node}}! k-face||fk||f0 || f1||f2||f3||k-fig!Notes align=rightnode_xnodenodenode}} ( ) !f0 600 4 6 4 regular tetrahedron >| H4/A3 = 14400/24 = 600 align=rightnode_1node_xnodenode}} { }!f1 2 1200 3 3 equilateral triangle >| H4/A2A1 = 14400/6/2 = 1200 align=rightnode_1nodenode_xnode}} {5}!f2 5 5 720 2 { } H4/H2A1 = 14400/10/2 = 720 align=rightnode_1nodenodenode_x}} {5,3}!f3 20 30 12 120 ( ) H4/H3 = 14400/120 = 120

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.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}}

Layered stereographic projection

The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer|Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "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, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on 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.{| class="wikitable"! Layer #! Number of Cells! Description! Colatitude! Region 1 1 cell| North Pole 0° Northern Hemisphere 2 12 cells| First layer of meridional cells / "Arctic Circle" 36° 3 20 cells| Non-meridian / interstitial 60° 4 12 cells| Second layer of meridional cells / "Tropic of Cancer" 72° 5 30 cells| Non-meridian / interstitial 90° Equator 6 12 cells| Third layer of meridional cells / "Tropic of Capricorn" 108° Southern Hemisphere 7 20 cells| Non-meridian / interstitial 120° 8 12 cells| Fourth layer of meridional cells / "Antarctic Circle" 144° 9 1 cell| South Pole 180°! Total! 120 cells! colspan="3" |The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell.

Intertwining rings

missing image!
- 120-cell rings.jpg -
Two intertwining rings of the 120-cell.

(File:120-cell_two_orthogonal_rings.png|thumb|300px|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.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer|Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} 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.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B2|pp=23-29}} 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 (Clifford parallel) 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 edges alternating with 6 cell diameter chords, forming an irregular dodecagon in a central plane.{{Efn|(File:Great dodecagon of the 120-cell.png|thumb|200px|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),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).)The 120-cell has an irregular dodecagon {12} great circle polygon of 6 edges (#1 chords marked {{Color|red|𝜁}}) alternating with 6 dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} The irregular great dodecagon contains two irregular great hexagons ({{color|red|red}}) inscribed in alternate positions.{{Efn|name=irregular great hexagon}} Two regular great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The twelve regular hexagon edges (#5 chords), the six cell-diameter edges of the dodecagon (#4 chords), and the six 120-cell edges of the dodecagon (#1 chords), are all chords of the same great circle, but the other 24 zig-zag edges (#1 chords, not shown) that bridge the six #4 edges of the dodecagon do not lie in this great circle plane. 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 dodecagon planes. The 120-cell contains 200 such {12} central planes (100 completely orthogonal pairs), the same 200 central planes each containing a hexagon that are found in each of the 10 inscribed 600-cells.{{Efn|The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} All ten 600-cells occupy the same set of 200 irregular great dodecagon central planes.{{Efn|name=irregular great dodecagon}} 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 hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint great 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).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells.|name=same 200 planes}}|name=irregular great dodecagon}} Both these great circle paths have dual great circle paths in the 600-cell. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a decagon.{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path 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), forming a hexagon.Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges illustrated above.{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each irregular great dodecagon, in alternate positions.{{Efn|name=irregular great dodecagon}}

Perspective projections {|class"wikitable"

!colspan=2|Projections to 3D of a 4D 120-cell performing a simple rotation(File:120-cell.gif|256px)(File:120-cell-inner.gif|256px)|From outside the 3-sphere in 4-space.600-cell#Boundary envelopes>3D surface of the 3-sphere.As in all the illustrations in this article, only the edges of the 120-cell appear in these renderings. All the other chords are not shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.These projections use perspective projection, from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint.A comparison of perspective projections of the 3D dodecahedron to 2D (below left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. Schlegel diagrams use perspective to show depth in the dimension which has been flattened, choosing a view point above a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. 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. Other perspective projection methods exist, such as the rotating animations above, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must).{| class="wikitable" style="width:540px;"|+Comparison with regular dodecahedron!width=80|Projection!Dodecahedron!120-cell!missing image!
- Dodecahedron schlegel.svg">220px12 pentagon faces in the planeFile:Schlegel wireframe 120-cell.png -