Editing 600-cell (section)

=== Structure ===

==== Polyhedral sections ====
The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋.
Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏<sup>−1</sup>) beginning with a vertex}}{{Sfn|Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout|Parker|1998}}
These can be seen in the H3 [[Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}

:[[File:600-cell-polyhedral levels.png|640px]]

These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid).
Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane).
In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center.
But its own center is in the interior of the 600-cell, not on its surface.
V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell.
Thus V is the apex of a [[Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron.

{| class=wikitable
!colspan=2|Concentric Hulls
|-
|align=center|[[Image:Hulls of H4only-orthonormal.png|360px]]
|The 600-cell is projected to 3D using an orthonormal basis.
The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:<br>
<br>
1) two points at the origin<br>
2) two icosahedra<br>
3) two dodecahedra<br>
4) two larger icosahedra<br>
5) and a single icosidodecahedron<br>
<br>
for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices.
|-
|}

==== Golden chords ====
[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]]
The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]].
The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc.
The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
{{see also|24-cell#Hypercubic chords}}
The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏<sup>−1</sup>) beginning with a vertex; see column ''a''}} at eight different [[Chord (geometry)|chord]] lengths from each other.
These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}}
In ascending order of length, they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=The fractional square roots are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <math>\tfrac{1}{\phi} = \phi^{-1}</math>
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽<sup>2</sup> ≈ 0.382<br>
For example:
{{indent|7}}𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618|name=fractional square roots|group=}}

Notice that the four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons.
The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[golden ratio]] <big>φ</big> {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to <big>𝜋</big>:<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=John Carlos Baez|access-date=10 October 2022}}</ref><br>
: {{sfrac|𝜋|5}} = arccos ({{sfrac|φ|2}})
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing <big>φ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br>
: <big>φ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the <big>φ</big> = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is <big>φ</big>, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the <big>φ</big> chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <big>φ</big>{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

==== Boundary envelopes ====
[[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]].
The 3D surface made of 600 tetrahedra is visible.]]
The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell.
The other 96 vertices constitute a [[snub 24-cell]].
Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}}
The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells.
One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: tetrahedra of edge length {{sfrac|1|φ}} ≈ 0.618 instead of octahedra of edge length 1.
It encloses the {{radic|1}} edges of the 24-cells, which become invisible interior chords in the 600-cell, like the [[24-cell#Hypercubic chords|{{radic|2}} and {{radic|3}} chords]].

[[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell]] performing a [[24-cell#Simple rotations|simple rotation]].
The 3D surface made of 24 octahedra is visible.
It is also present in the 600-cell, but as an invisible interior boundary envelope.]]
Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of {{sfrac|1|φ}}, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[Tesseract#Radial equilateral symmetry|radially equilateral]].
Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[equilateral triangle]]s which all meet at the center.{{Efn|The long radius (center to vertex) of the [[24-cell]] is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[Golden triangle (mathematics)|golden triangle]]s which all meet at the center. All the [[regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[right triangle]]s which meet at the center.{{Efn|The [[orthoscheme]] is the generalization of the [[right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[golden ratio]] to its edge length; thus its radius is <big>φ</big> if its edge length is 1, and its edge length is {{sfrac|1|φ}} if its radius is 1.}}
Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[icosidodecahedron]], and the two-dimensional [[Decagon#The golden ratio in decagon|decagon]].
(The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.)
'''Radially golden''' polytopes are those which can be constructed, with their radii, from [[Golden triangle (mathematics)|golden triangles]].

The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes).
The shape of those interstices must be an [[Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}}
Therefore the successor may be constructed by placing [[Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor.
Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[cubic pyramid]]s.
But if we place 24 canonical [[octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell.
Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}}

==== Geodesics ====
The vertex chords of the 600-cell are arranged in [[geodesic]] [[great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§4 The planes of the 600-cell|pp=437-439}}

[[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[stereographic projection]] of the 600-cell's 72 central decagons onto their great circles.
Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]]
The {{radic|0.𝚫}} = 𝚽 edges form 72 flat regular central [[decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}}
Just as the [[icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10).
The 720 {{radic|0.𝚫}} edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, {{radic|3.𝚽}} apart.
As in the decagon and the icosidodecahedron, the edges occur in [[Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope.
The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[Clifford parallel]] geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[Clifford parallel]] great circles spanned by a twisted [[Annulus (mathematics)|annulus]].]][[Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[Hopf fibration|Hopf fiber bundles]] which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[Hopf fibration]]s, each filling the whole 600-cell. Each [[#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}

The {{radic|1}} chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{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}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The {{radic|1}} chords join vertices which are two {{radic|0.𝚫}} edges apart.
Each {{radic|1}} chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[triangular bipyramid]]), and passes through the center of the shared face.
As there are 1200 faces, there are 1200 {{radic|1}} chords, in 600 parallel pairs, {{radic|3}} apart.
The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell|Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's "points" and "lines" are axes and hexagons, respectively.
The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs.
Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}}
The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}

The {{radic|1.𝚫}} chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}}
The {{radic|1.𝚫}} chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon.
The {{radic|1.𝚫}} chords join vertices which are two {{radic|0.𝚫}} edges apart on a geodesic great circle.
The 720 {{radic|1.𝚫}} chords occur in 360 parallel pairs, {{radic|2.𝚽}} = φ apart.

The {{radic|2}} chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex).
The {{radic|2}} chords join vertices which are three {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart).
There are 600 {{radic|2}} chords, in 300 parallel pairs, {{radic|2}} apart.
The 450 great squares (225 [[completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}

The {{radic|2.𝚽}} = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex.
The third edge (base) of each isosceles triangle is of length {{radic|3.𝚽}}.
The {{radic|2.𝚽}} chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 720 distinct {{radic|2.𝚽}} chords, in 360 parallel pairs, {{radic|1.𝚫}} apart.

The {{radic|3}} chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells).
Each set of 32 triangles consists of the 96 {{radic|3}} chords and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The {{radic|3}} chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The {{radic|3}} chords join vertices which are four {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart on a geodesic great circle).
Each {{radic|3}} chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 {{radic|1}} cubic cells.
The 1200 {{radic|3}} chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each {{radic|3}} chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}}
There are 1200 {{radic|3}} chords, in 600 parallel pairs, {{radic|1}} apart.

The {{radic|3.𝚽}} chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex.
The third edge (base) of each isosceles triangle is an edge of the pentagon of length {{radic|1.𝚫}}, so these are [[Golden triangle (mathematics)|golden triangles]]. The {{radic|3.𝚽}} chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 720 distinct {{radic|3.𝚽}} chords, in 360 parallel pairs, {{radic|0.𝚫}} apart.

The {{radic|4}} chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell.
The {{radic|4}} chords join opposite vertices which are five {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells.

The sum of the squared lengths{{Efn|The sum of 0.𝚫・720 + 1・1200 + 1.𝚫・720 + 2・1800 + 2.𝚽・720 + 3・1200 + 3.𝚽・720 + 4・60 is 14,400.}} of all these distinct chords of the 600-cell is 14,400 = 120<sup>2</sup>.{{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}}}}
These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[completely orthogonal]] to a great 30-gon{{Efn|A ''[[triacontagon]]'' or 30-gon  is a thirty-sided polygon.
The triacontagon is the largest regular polygon whose interior angle is the sum of the [[interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[equilateral triangle]] (60°) and the [[regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell.
The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly.
The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords.
It does not touch any edges or vertices, but it does hit faces.
It is the central axis of a spiral skew 30-gram, the [[Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}}
Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all.
There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}}

All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells).
These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[icosidodecahedron]].
There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.{{Efn|Two angles are required to fix the relative positions of 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.)"}}
Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space.
Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens|Seidel|1973}}
Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together.
Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration.
Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles.
Vectors in 4-space may be combined by [[Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[William Rowan Hamilton|Hamilton]].{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; 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 <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br>
<small><math display=block>\begin{pmatrix}
w_2\\
x_2\\
y_2\\
z_2
\end{pmatrix}
*
\begin{pmatrix}
w_1\\
x_1\\
y_1\\
z_1
\end{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}
</math></small>}}
The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}}
Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}}
Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell.
Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}}
Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}}
Planes which are isoclinic have [[Clifford parallel]] great circles.{{Efn|name=Clifford parallels}}
A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}}
Each great square plane is [[completely orthogonal]] to another great square plane.
Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one {{radic|4}} long diameter): a great [[digon]] plane.{{Efn|In the 24-cell each great square plane is [[completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[digon]] plane.|name=digon planes}}
Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane.
Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere.
Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere.
Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell.
In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}}

==== Fibrations of great circle polygons ====
Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}}
Each [[fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S<sub>3</sub>|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}}
The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets.
The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell|Semple|1971|loc=§4. Isoclinic planes in Euclidean space E<sub>4</sub>|pp=6-7}}

===== Decagons =====
[[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[Schläfli double six]] configuration 30<sub>2</sub>12<sub>5</sub> characteristic of the H<sub>4</sub> polytopes.{{^|Efn|name=Schläfli double six}} The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]]
The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.

Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}}
Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations.
The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}}

The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[Hopf fibration]]s make the 600-cell a [[Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30<sub>2</sub>12<sub>5</sub>.{{^|Efn|name=Schläfli double six}}
It is called the [[Schläfli double six]] configuration after [[Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3<sup>2, 2, 1</sup>] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}

===== Hexagons =====
The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons.
Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon.
Each octahedral cell occupies only one cell ring in each of the 4 fibrations.
The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration.

The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}}
It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. 
The Hopf map of this fibration is the [[dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}
Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations.
The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells.

===== Squares =====
The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares.
Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each.
Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations.
The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration.

The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells.
It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the {{radic|2}} tetrahedral cells of the 75 inscribed 16-cells, ''not'' the {{radic|0.𝚫}} tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} 
The Hopf map of this fibration is the [[icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}
Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations.

===== Clifford parallel cell rings =====
The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, studied cell rings in the general case of their geometry and [[group theory]], identifying each cell ring as a [[polytope]] in its own right which fills a three-dimensional manifold (such as the [[3-sphere]]) with its corresponding [[Honeycomb (geometry)|honeycomb]].{{Efn|name=orthoscheme ring}}
He found that cell rings follow [[Petrie polygon]]s and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms.
Specifically, he found that the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}}
Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[Clifford torus]], showed how the honeycombs correspond to [[Hopf fibration]]s, and made decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S<sub>3</sub>|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces.
Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}}
The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells.
They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}}
The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell).
The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell.
The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells.
In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}}
This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}}
The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings.
Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration).
In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}}
The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration.
All the fibrations have this two level structure with ''subfibrations''.

The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon.
Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.

The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square.
Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}}

The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope.
Each fibration is the entire 4-polytope.}}
The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}}
On this map each great circle fiber appears as a single point.
The base of a great decagon fibration of the 600-cell is the [[icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}
To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration.
But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell.
Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell.
Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Sfn|Sadoc|Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,{{Efn|name=Clifford parallels}} along which the molecules can be aligned without any conflict between compactness and [[torsion of a curve|torsion]].... The fibres of this [[Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[Clifford parallel]]s. Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint."}}|name=Hopf fibration base}} of these fibrations on the 2-sphere).
Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 red faces of the [[Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration'').
The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}}