Editing 120-cell (section)

=== 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|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and {{Backgroundcolor|seashell|<big>☐</big> 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 [[#Concentric hulls|polyhedral sections]] of the 120-cell, beginning with a vertex, the 0<sub>0</sub> 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 1<sub>0</sub> section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the "radius" of its congruent opposing 29<sub>0</sub> section. The #7 chord is the "radius" of the central section of the 120-cell, in which two opposing 15<sub>0</sub> sections are coincident.

{| class=wikitable style="white-space:nowrap;text-align:center"
!colspan=10|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<sup>−2</sup>√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1<sub>0</sub> − 16<sub>0</sub>|ps=, but 14<sub>0</sub> and 16<sub>0</sub> are congruent opposing sections and 15<sub>0</sub> opposes itself; there are 29 non-point sections, denoted 1<sub>0</sub> − 29<sub>0</sub>, in 15 opposing pairs.}}
|-
!colspan=4|Short chord
!colspan=2|Great circle polygons
!colspan=4|Long chord
|- style="background: palegreen;"|
|rowspan=2|1<sub>0</sub><br><br>#1
|{{Efn|In 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'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}}
|colspan=2|<math>1 / \phi^2\sqrt{2}</math>
|rowspan=2|[[File:Irregular great hexagons of the 120-cell.png|100px]]
|rowspan=2|400 irregular great hexagons{{Efn|name=irregular great dodecagon}}<br>
(600 great rectangles)<br>
in 200 △ planes
|
|colspan=2|
|rowspan=2|29<sub>0</sub><br><br>#14
|- style="background: palegreen;"|
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|0.270~
|164.5~°
|{{radic|3.93~}}
|1.982~
|- style="background: seashell;"|
|rowspan=2|2<sub>0</sub><br><br>#2
|{{Efn|name=#2 chord}}
|colspan=2|<math>1 / \phi\sqrt{2}</math>
|rowspan=2|[[File:25.2° × 154.8° chords great rectangle.png|100px]]
|rowspan=2|Great rectangles<br>in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|28<sub>0</sub><br><br>#13
|- style="background: seashell;"|
|25.2~°
|{{radic|0.19~}}
|0.437~
|154.8~°
|{{radic|3.81~}}
|1.952~
|- style="background: yellow;"|
|rowspan=2|3<sub>0</sub><br><br>#3
|<math>\pi / 5</math>
|colspan=2|<math>1 / \phi</math>
|rowspan=2|[[File:Great decagon rectangle.png|100px]]
|rowspan=2|720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes
|<math>4\pi / 5</math>
|colspan=2|<math>\sqrt{2+\phi}</math>
|rowspan=2|27<sub>0</sub><br><br>#12
|- style="background: yellow;"|
|36°
|{{radic|0.𝚫}}
|0.618~
|144°{{Efn|name=dihedral}}
|{{radic|3.𝚽}}
|1.902~
|- style="background: gainsboro;"|
|rowspan=2|4<sub>0</sub><br><br>#3~4
|
|colspan=2|<math>\sqrt{1}/\sqrt{2}</math>
|rowspan=2|[[File:√0.5 × √3.5 great rectangle.png|100px]]
|rowspan=2|Great rectangles<br>in <big>☐</big> planes
|
|colspan=2|<math>\sqrt{7} / \sqrt{2}</math>
|rowspan=2|26<sub>0</sub><br><br>#11~12
|- style="background: gainsboro;"|
|41.4~°
|{{radic|0.5}}
|0.707~
|138.6~°
|{{radic|3.5}}
|1.871~
|- style="background: palegreen;"|
|rowspan=2|5<sub>0</sub><br><br>#4
|
|colspan=2|<math>\sqrt{3} / \phi\sqrt{2}</math>
|rowspan=2|[[File:Irregular great dodecagon.png|100px]]
|rowspan=2|200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} 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 {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown).{{Efn|name=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.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' 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).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=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=#4 isocline chord bridge}}|name=dodecagon rotation}}<br>(600 great rectangles)<br>in 200 △ planes
|
|colspan=2|<math>\phi^2 / \sqrt{2}</math>
|rowspan=2|25<sub>0</sub><br><br>#11
|- style="background: palegreen;"|
|44.5~°
|{{radic|0.57~}}
|0.757~
|135.5~°
|{{radic|3.43~}}
|1.851~
|- style="background: gainsboro; height:50px"|
|rowspan=2|6<sub>0</sub><br><br>#4~5
|
|colspan=2|
|rowspan=2|[[File:49.1° × 130.9° great rectangle.png|100px]]
|rowspan=2|Great rectangles<br>in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|24<sub>0</sub><br><br>#10~11~11
|- style="background: gainsboro;"|
|49.1~°
|{{radic|0.69~}}
|0.831~
|130.9~°
|{{radic|3.31~}}
|1.819~
|- style="background: gainsboro; height:50px"|
|rowspan=2|7<sub>0</sub><br><br>#4~5~5
|
|colspan=2|
|rowspan=2|[[File:56° × 124° great rectangle.png|100px]]
|rowspan=2|Great rectangles<br>in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|23<sub>0</sub><br><br>#10~11
|- style="background: gainsboro;"|
|56°
|{{radic|0.88~}}
|0.939~
|124°
|{{radic|3.12~}}
|1.766~
|- style="background: palegreen;"|
|rowspan=2|8<sub>0</sub><br><br>#5
|<math>\pi / 3</math>
|colspan=2|
|rowspan=2|[[File:Great hexagon.png|100px]]
|rowspan=2|400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}}<br> (1200 great rectangles)<br>in 200 △ planes
|<math>2\pi / 3</math>
|colspan=2|
|rowspan=2|22<sub>0</sub><br><br>#10
|- style="background: palegreen;"|
|60°
|{{radic|1}}
|1
|120°
|{{radic|3}}
|1.732~
|- style="background: gainsboro; height:50px"|
|rowspan=2|9<sub>0</sub><br><br>#5~6
|
|colspan=2|
|rowspan=2|[[File:66.1° × 113.9° great rectangle.png|100px]]
|rowspan=2|Great rectangles<br> in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|21<sub>0</sub><br><br>#9~10~10
|- style="background: gainsboro;"|
|66.1~°
|{{radic|1.19~}}
|1.091~
|113.9~°
|{{radic|2.81~}}
|1.676~
|- style="background: gainsboro; height:50px"|
|rowspan=2|10<sub>0</sub><br><br>#5~6~6
|
|colspan=2|
|rowspan=2|[[File:69.8° × 110.2° great rectangle.png|100px]]
|rowspan=2|Great rectangles<br> in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|20<sub>0</sub><br><br>#9~10
|- style="background: gainsboro;"|
|69.8~°
|{{radic|1.31~}}
|1.144~
|110.2~°
|{{radic|2.69~}}
|1.640~
|- style="background: yellow;"|
|rowspan=2|11<sub>0</sub><br><br>#6
|<math>2\pi/5</math>
|colspan=2|<math>\sqrt{3-\phi}</math>
|rowspan=2|[[File:Great pentagons rectangle.png|100px]]
|rowspan=2|1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}}<br>(3600 great rectangles)<br>
in 720 <big>𝜙</big> planes
|<math>3\pi / 5</math>
|colspan=2|<math>\phi</math>
|rowspan=2|19<sub>0</sub><br><br>#9
|- style="background: yellow;"|
|72°
|{{radic|1.𝚫}}
|1.175~
|108°
|{{radic|2.𝚽}}
|1.618~
|- style="background: palegreen; height:50px"|
|rowspan=2|12<sub>0</sub><br><br>#6~6~7
|
|colspan=2|<math>\sqrt{3} / \sqrt{2}</math>
|rowspan=2|[[File:Great 5-cell digons rectangle.png|100px]]
|rowspan=2|1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] 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 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12<sub>0</sub> sections and 18<sub>0</sub> sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}}<br>(600 great rectangles)<br>
in 200 △ planes
|
|colspan=2|<math>\sqrt{5} / \sqrt{2}</math>
|rowspan=2|18<sub>0</sub><br><br>#8
|- style="background: palegreen;"|
|75.5~°
|{{radic|1.5}}
|1.224~
|104.5~°
|{{radic|2.5}}
|1.581~
|- style="background: gainsboro; height:50px"|
|rowspan=2|13<sub>0</sub><br><br>#6~7
|
|colspan=2|
|rowspan=2|[[File:81.1° × 98.9° great rectangle.png|100px]]
|rowspan=2|Great rectangles<br> in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|17<sub>0</sub><br><br>#7~8~8
|- style="background: gainsboro;"|
|81.1~°
|{{radic|1.69~}}
|1.300~
|98.9~°
|{{radic|2.31~}}
|1.520~
|- style="background: gainsboro; height:50px"|
|rowspan=2|14<sub>0</sub><br><br>#6~7~7
|
|colspan=2|
|rowspan=2|[[File:84.5° × 95.5° great rectangle.png|100px]]
|rowspan=2|Great rectangles<br> in <big>☐</big> planes
|
|colspan=2|
|rowspan=2|16<sub>0</sub><br><br>#7~8
|- style="background: gainsboro;"|
|84.5~°
|{{radic|0.81~}}
|1.345~
|95.5~°
|{{radic|2.19~}}
|1.480~
|- style="background: seashell;"|
|rowspan=2|15<sub>0</sub><br><br>#7
|<math>\pi / 2</math>
|colspan=2|
|rowspan=2|[[File:Great square rectangle.png|100px]]
|rowspan=2|4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}}<br>
in 4050 <big>☐</big> planes
|<math>\pi / 2</math>
|colspan=2|
|rowspan=2|15<sub>0</sub><br><br>#7
|- style="background: seashell;"|
|90°
|{{radic|2}}
|1.414~
|90°
|{{radic|2}}
|1.414~
|}