Editing 120-cell (section)

===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 [[Rotations in 4-dimensional Euclidean space#Double rotations|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 [[Link (knot theory)|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|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic 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}} 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.|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 [[SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains a compound of several of these great circle polygons in the same central plane, illustrated separately.{{Efn|name=irregular great dodecagon}} An implication of the compounding is that the edges and characteristic rotations{{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}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in the same rotation planes, the hexagonal central planes of the 24-cell.{{Efn|name=edge rotation planes}}]]
{{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 [[Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> 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 [[5-cell#Boerdijk–Coxeter helix|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|600-cell § Decagons and pentadecagrams]], see the illustration of [[Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{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, [[5-cell#Geodesics and rotations|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 [[SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>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 [[5-cell#Geodesics and rotations|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|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 [[#Chords|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 𝛢<sub>4</sub> the operation [15]𝑹<sub>q3,q3</sub> is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentadecagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨<sub>4</sub> the operation [1200]𝑹<sub>q3,q13</sub> 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 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 𝛨<sub>4</sub>|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' 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#Geodesics and rotations|5-cell]] and the [[8-cell|8-cell tesseract]], they form zig-zag [[Petrie polygon]]s 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 [[Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[triacontagon]] {30} zig-zag [[Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[skew polygon|skew]] regular [[triacontagon]] {30}.{{Efn|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 [[Triacontagon#Triacontagram|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 [[600-cell#Boerdijk–Coxeter helix rings|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|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 distance]]s 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 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord connects two vertices which are ''n'' edge lengths apart on a Petrie polygon of the 120-cell. The 15 major chords lie on great circles in central planes that contain regular and irregular polygons of {4}, {10}, or {12} vertices. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). The 15 ''minor chords'' do not lie on great circles (except their own irregular {2} digon great circles) and do not occur anywhere except inside the 120-cell. In this article, we name the 15 unnumbered minor chords by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3~4 chords.|name=additional 120-cell chords}}

The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=polytopes ordered by size and complexity}}

The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ or decagrams [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which are Petrie polygons that circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}}

{| class=wikitable style="white-space:nowrap;text-align:center"
!colspan=15|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<sup>−2</sup>√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1<sub>0</sub> − 16<sub>0</sub>, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') 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'')<sub>0</sub>) 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 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}}
|-
!colspan=6|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 <sub>0</sub>𝑹, <sub>1</sub>𝑹, <sub>2</sub>𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈<sub>0</sub>, 𝐈𝐈<sub>1</sub>, 𝐈𝐈<sub>2</sub>, ... 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 24-cell|Snub]]
![[600-cell]]
![[#Tetrahedrally diminished 120-cell|Dimin]]
! style="border-right: none;"|120-cell
! style="border-left: none;"|
|-
!colspan=6|Vertices
| style="background: seashell;"|5
| style="background: paleturquoise;"|8
| style="background: paleturquoise;"|16
| style="background: paleturquoise;"|24
| style="background: yellow;"|96
| style="background: yellow;"|120
| style="background: seashell;"|480
| style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}}
|rowspan=6 style="background: seashell; border: none;"|
|-
!colspan=6|Edges
| style="background: seashell;"|10{{Efn|name=irregular great hexagon}}
| style="background: paleturquoise;"|24
| style="background: paleturquoise;"|32
| style="background: paleturquoise;"|96
| style="background: yellow;"|432
| style="background: yellow;"|720
| style="background: seashell;"|1200
| style="background: seashell;"|1200{{Efn|name=irregular great hexagon}}
|-
!colspan=6|Edge chord
| style="background: seashell;"|#8{{Efn|name=inscribed 5-cells}}
| style="background: paleturquoise;"|#7
| style="background: paleturquoise;"|#5
| style="background: paleturquoise;"|#5
| style="background: yellow;"|#3
| style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[Skew 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|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie  is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-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|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={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|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[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{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=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|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}}
| style="background: seashell;"|#1
| style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}}
|-
!colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[SO(4)#Double rotations|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, [[16-cell#Rotations|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 entanglement|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 great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}}
| style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]]
| style="background: paleturquoise;"|[[16-cell#Helical construction|#15]]
| style="background: paleturquoise;"|#10
| style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]]
| style="background: yellow;"|#5
| style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]]
| style="background: seashell;"|#4
| style="background: seashell;"|#4{{Efn|The 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{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=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{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}}
|-
!colspan=6|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}}
| style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]]
| style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]]
| style="background: paleturquoise;"|
| style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]]
| style="background: yellow;"|
| style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]]
| style="background: seashell;"|
| style="background: seashell;"|[[pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}}
|-
!colspan=3|Chord
!Arc
!colspan=2|Edge
| style="background: seashell;"|
| style="background: paleturquoise;"|
| style="background: paleturquoise;"|
| style="background: paleturquoise;"|
| style="background: yellow;"|
| style="background: yellow;"|
| style="background: seashell;"|
| style="background: seashell;"|
|- style="background: seashell;"|
|rowspan=2|#1<br>△
|rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]]
|rowspan=2|30
|
|colspan=2|120-cell edge{{Efn|name=120-cell Petrie {30}-gon}}
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}}
|rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3}
|- style="background: seashell;"|
|15.5~°
|{{radic|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 <small>{{sfrac|1|φ}}</small>
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽<sup>2</sup> = <small>{{sfrac|1|φ<sup>2</sup>}}</small> ≈ 0.382
{{indent|7}}𝜀 = 𝚫<sup>2</sup>/2 = <small>{{sfrac|1|2φ<sup>4</sup>}}</small> ≈ 0.073<br>
and the 120-cell edge-length is:
{{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|φ<sup>2</sup>{{radic|2}}}} ≈ 0.270<br>
For example:
{{indent|7}}𝛇 = {{radic|𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}}
|0.270~
|- style="background: seashell;"|
|rowspan=2|#2<br><big>☐</big>
|rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]]
|rowspan=2|15
|
|colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1<sub>0</sub>. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[Pentagon#Regular pentagons|pentagon diagonal]] (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=face pentagon chord}}|name=#2 chord}}
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>3600<br>
|rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4}
|- style="background: seashell;"|
|25.2~°
|{{radic|0.19~}}
|0.437~
|- style="background: yellow;"|
|rowspan=2|#3<br><big>𝜙</big>
|rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]]
|rowspan=2|10
|𝝅/5
|colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math>
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720
|rowspan=2|
|rowspan=2|<br>7200
|rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5}
|- style="background: yellow;"|
|36°
|{{radic|0.𝚫}}
|0.618~
|- style="background: seashell;"|
|rowspan=2|#4<br>△
|rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]]
|rowspan=2|{{sfrac|15|2}}
|{{Efn|name=irregular great dodecagon}}
|colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}}
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>1200
|rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3}
|- style="background: seashell;"|
|44.5~°
|{{radic|0.57~}}
|0.757~
|- style="background: paleturquoise;"|
|rowspan=2|#5<br>△
|rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]]
|rowspan=2|6
|𝝅/3
|colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] 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.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different central planes.{{Efn|name=same 200 planes}}|name=great hexagon}}
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>32
|rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96
|rowspan=2|{{green|<big>'''225'''</big>}}<br><br>
|rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200
|rowspan=2|
|rowspan=2|<br>2400{{Efn|name=same 200 planes}}
|rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3}
|- style="background: paleturquoise;"|
|60°
|{{radic|1}}
|1
|- style="background: yellow;"|
|rowspan=2|#6<br><big>𝜙</big>
|rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]]
|rowspan=2|5
|2𝝅/5
|colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}}
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>720
|rowspan=2|
|rowspan=2|<br>7200
|rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5}
|- style="background: yellow;"|
|72°
|{{radic|1.𝚫}}
|1.175~
|- style="background: paleturquoise;"|
|rowspan=2|#7<br><big>☐</big>
|rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]]
|rowspan=2|{{sfrac|30|7}}
|𝝅/2
|colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}}
|rowspan=2|
|rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24
|rowspan=2|{{green|<big>'''675'''</big>}}<br>48
|rowspan=2|<br>72
|rowspan=2|
|rowspan=2|<br>1800
|rowspan=2|<br>
|rowspan=2|<br>9000
|rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4}
|- style="background: paleturquoise;"|
|90°
|{{radic|2}}
|1.414~
|- style="background: #FFCCCC;"|
|rowspan=2|#8<br><big>✩</big>
|rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]]
|rowspan=2|{{sfrac|15|4}}
|
|colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}}
|rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>720
|rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}}
|rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3}
|- style="background: #FFCCCC;"|
|104.5~°
|{{radic|2.5}}
|1.581~
|- style="background: yellow;"|
|rowspan=2|#9<br><big>𝜙</big>
|rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]]
|rowspan=2|{{sfrac|10|3}}
|3𝝅/5
|colspan=2|[[golden section]] <math>\phi</math>
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>720
|rowspan=2|
|rowspan=2|<br>7200
|rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5}
|- style="background: yellow;"|
|108°
|{{radic|2.𝚽}}
|1.618~
|- style="background: paleturquoise;"|
|rowspan=2|#10<br>△
|rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]]
|rowspan=2|3
|2𝝅/3
|colspan=2|[[24-cell#Triangles|great triangle]]
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>32
|rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96
|rowspan=2|
|rowspan=2|<br>1200
|rowspan=2|
|rowspan=2|<br>2400
|rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3}
|- style="background: paleturquoise;"|
|120°
|{{radic|3}}
|1.732~
|- style="background: seashell;"|
|rowspan=2|#11<br><big>✩</big>
|rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]]
|rowspan=2|{{sfrac|30|11}}
|
|colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}}
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>1200
|rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3}
|- style="background: seashell;"|
|135.5~°
|{{radic|3.43~}}
|1.851~
|- style="background: yellow;"|
|rowspan=2|#12<br><big>𝜙</big>
|rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]]
|rowspan=2|{{sfrac|5|2}}
|4𝝅/5
|colspan=2|great [[Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}}
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>720
|rowspan=2|
|rowspan=2|<br>7200
|rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5}
|- style="background: yellow;"|
|144°{{Efn|name=dihedral}}
|{{radic|3.𝚽}}
|1.902~
|- style="background: seashell;"|
|rowspan=2|#13<br><big>✩</big>
|rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]]
|rowspan=2|{{sfrac|30|13}}
|
|colspan=2|[[Triacontagon#Triacontagram|{30/13}-gram]]
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>3600<br>
|rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4}
|- style="background: seashell;"|
|154.8~°
|{{radic|3.81~}}
|1.952~
|- style="background: seashell;"|
|rowspan=2|#14<br>△
|rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]]
|rowspan=2|{{sfrac|15|7}}
|
|colspan=2|[[Triacontagon#Triacontagram|{30/14}=2{15/7}]]
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|
|rowspan=2|<br>1200<br>
|rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3}
|- style="background: seashell;"|
|164.5~°
|{{radic|3.93~}}
|1.982~
|- style="background: paleturquoise;"|
|rowspan=2|#15<br><small>△☐𝜙</small>
|rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]]
|rowspan=2|2
|𝝅
|colspan=2|[[diameter]]
|rowspan=2|
|rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4
|rowspan=2|<br>8
|rowspan=2|<br>12
|rowspan=2|<br>48
|rowspan=2|<br>60
|rowspan=2|<br>240
|rowspan=2|<br>300{{Efn|name=rays and bases}}
|rowspan=2|{{blue|<big>'''1'''</big>}}<br><br>
|- style="background: paleturquoise;"|
|180°
|{{radic|4}}
|2
|-
!colspan=6|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}}}}
| style="background: seashell;"|25
| style="background: paleturquoise;"|64
| style="background: paleturquoise;"|256
| style="background: paleturquoise;"|576
| style="background: yellow;"|
| style="background: yellow;"|14400
| style="background: seashell;"|
| style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}}
!<big>{{blue|'''300'''}}</big>
|}

[[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 <big>{{red|'''25'''}}</big> 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 <big>{{green|'''225'''}}</big> 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, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell.