Editing 600-cell (section)

=== Constructions ===
The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}}
Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial.
The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize.

==== Gosset's construction ====
[[Thorold Gosset]] discovered the [[Semiregular polytope|semiregular 4-polytopes]], including the [[snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius.
Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form.
In the first, more complex step (described [[Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}
In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}}

The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}}
The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells.
The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.

Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps.
The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation.
Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[midsphere]] to construct its outer [[Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual.
That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell.

==== Cell clusters ====
Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]],{{Efn|name=tetrahedral cell adjacency}} or how they lie on the underlying surface envelope of the 24-cell's octahedral cells.
For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.

Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences....
[Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains.
Coxeter made full use of it, and expected the reader to use it....
Visualization is one of the most powerful interiorization techniques.
It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module.
Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional space that we could actually "walk around in" and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}}
In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.

===== Icosahedra =====
[[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[Regular icosahedron#Symmetries|snub octahedron]].
This term refers specifically to a [[Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}}
Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces.
The apex of the [[icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible).
The four cells lie in different hyperplanes.|alt=|200x200px]]
The [[vertex figure]] of the 600-cell is the [[icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center.
Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there.
However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices.
Thus the vertex icosahedron is actually a canonical [[icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|The icosahedron is not radially equilateral in Euclidean 3-space, but an icosahedral pyramid is radially equilateral in the curved 3-space of the 600-cell's surface (the [[3-sphere]]). In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the [[icosahedral pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that curved 3-space''. In Euclidean 4-space 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=vertex icosahedral pyramid|group=}}
Twenty tetrahedral cells meet at each vertex, forming an [[icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron.
The 600-cell has a [[dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}}

An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra.
Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five).
Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells.
Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}}

The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell.
The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[snub 24-cell]], which has exactly the same [[Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing.

The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces.
Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the "[[Regular icosahedron#Symmetries|snub octahedron]]" icosahedron all look the same, with two yellow and three blue faces.
Each pentagon has five distinct rotational orientations.
Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}}
Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell.
Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}

The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids.
Their apexes are the vertices of a [[24-cell#Hexagons|great circle hexagon]].
This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[triangular bipyramid]]) is a hexagon edge (a 24-cell edge).
There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a {{radic|1}} octahedral cell, but in the larger {{radic|2}} octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron.
The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]].
The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}}
The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}}
The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}}
The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 8 vertices of the octahedron;{{Sfn|Itoh|Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[Jitterbug transformation|Jitterbug]] by [[Buckminster Fuller]]."}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}}
The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca|Al-Mukhaini|Koca|Al Qanobi|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[3-sphere|S<sup>3</sup>]] is similar to the [[kinematics of the cuboctahedron]] and the [[Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[Three-dimensional space|R<sup>3</sup>]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[Buckminster Fuller]].<ref>{{cite journal
 | last = Verheyen | first = H. F.
 | doi = 10.1016/0898-1221(89)90160-0
 | issue = 1–3
 | journal = [[Computers and Mathematics with Applications]]
 | mr = 0994201
 | pages = 203–250
 | title = The complete set of Jitterbug transformers and the analysis of their motion
 | volume = 17
 | year = 1989| doi-access = free
 }}</ref>}}

The tetrahedral cells are face-bonded into [[Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[Boerdijk-Coxeter helix]].
This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]].
In the 600-cell we find them bent in the fourth dimension into geodesic rings.
Each ring has 30 cells and touches 30 vertices.
The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}}
5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge).
A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[Hopf fibration]].
There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".|name=Boerdijk–Coxeter helix}}
The three helices are geodesic "straight lines" of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other.
Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex.
Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[Skew lines|skew edges]] of the tetrahedron.
Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron.
However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[Hopf fibration|Hopf fiber bundle]] of 12.
Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]].

The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same.
One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}}
Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}}

===== Octahedra =====
There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell.

Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells.
The central cell is the first section of the 600-cell beginning with a cell.
By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell.

First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length {{radic|1}}.
Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs.
They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length {{radic|1}}.
They form a tetrahedron of edge length {{radic|1}}, which is the second section of the 600-cell beginning with a cell.{{Efn|The {{radic|1}} tetrahedron has a volume of 9 {{radic|0.𝚫}} tetrahedral cells.
In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it.
The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the {{radic|1}} tetrahedron.
The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}}
There are 600 of these {{radic|1}} tetrahedral sections in the 600-cell.

With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster.
The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length {{radic|1}}, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}}
As partially filled so far (by 17 tetrahedral cells), this {{radic|1}} octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each {{radic|1}} edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells).
In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces.
Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}}
Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A {{radic|1}} octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell.
The same {{radic|1}} octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point {{radic|1}} octahedral section, a 4-point {{radic|1}} tetrahedral section, and a 4-point {{radic|0.𝚫}} tetrahedral section.
In the curved three-dimensional space of the 600-cell's surface, the {{radic|1}} octahedron surrounds the {{radic|1}} tetrahedron which surrounds the {{radic|0.𝚫}} tetrahedron, as three concentric hulls.
This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}}

Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical {{radic|1}} [[octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter {{radic|0.𝚫}} edges, replacing the apex with four vertices.
The truncation has also created another four vertices (arranged as a {{radic|1}} tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with {{radic|0.𝚫}} edges.
The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all.
The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two {{radic|1}} edges (and just one of those routes ran through the single apex).
The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three {{radic|0.𝚫}} edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length {{sfrac|1|φ}} ≈ 0.618.

===== Union of two tori =====
There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."}} and the [[#Decagons|decagonal fibrations]] of the 600-cell.
An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines".

[[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}}
Its opposite edges are identified, forming a [[duocylinder]].]]
The [[120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]].
Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex.
The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}

Start by assembling five tetrahedra around a common edge.
This structure looks somewhat like an angular "flying saucer".
Stack ten of these, vertex to vertex, "pancake" style.
Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron.
You can view this as five vertex-stacked [[icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet.
This 10-cell ring is shaped like a [[pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center.
This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}}
Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}}
The surface is the same as that of ten stacked [[pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}}
Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices.
Stack another tetrahedron on each exposed face.
This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge.
The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}}
The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons).
These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes).

Build a second identical torus of 250 cells that interlinks with the first.
This accounts for 500 cells.
These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges.
This latter set of 100 tetrahedra are on the exact boundary of the [[duocylinder]] and form a [[Clifford torus]].{{Efn|A [[Clifford torus]] is the [[Hopf fibration|Hopf fiber bundle]] of a distinct [[SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[3-sphere]], involving all of its points. The [[SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[Cartesian product]] of two [[completely orthogonal]] [[great circle]]s. It is a filled [[doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}}
They can be "unrolled" into a square 10×10 array.
Incidentally this structure forms one tetrahedral layer in the [[tetrahedral-octahedral honeycomb]].
There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy "egg crate" square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} How can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}}
In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[triangular bipyramid]] composed of two tetrahedra.

This decomposition of the 600-cell has [[Coxeter notation|symmetry]] [10,2<sup>+</sup>,10], order 400, the same symmetry as the [[grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H<sub>2</sub> × H<sub>2</sub>}}
The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}}

The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells.

===== Boerdijk–Coxeter helix rings =====
The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[Hopf fibration]] which fills the entire 600-cell. Each ring of 30 face-bonded tetrahedra is a cylindrical [[Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension.

{| class="wikitable" width="600"
|[[File:600-cell tet ring.png|200px]]<br>A single 30-tetrahedron [[Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}}
|[[File:600-cell Coxeter helix-ring.png|200px]]<br>A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}}
|[[File:Regular_star_polygon_30-11.svg|200px]]<br>The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[Skew polygon|skew]] {30/11} [[star polygon]] with a [[winding number]] of 11 called a [[Triacontagon#Triacontagram|triacontagram<sub>11</sub>]], a continuous tight corkscrew [[helix]] bent into a loop of 30 edges (the magenta edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[Density (polytope)|characterized]] as the [[Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}}
|-
|colspan=3|[[File:Coxeter_helix_edges.png|625px]]<br>The 30-vertex, 30-tetrahedron [[Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''.
|}

The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three cyan Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}}
The 30 magenta edges joining these vertex pairs form a helical [[Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[Petrie polygon]] is a skew [[Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h<sub>1</sub> h<sub>2</sub>''.}}|name=Petrie polygon in 30-cell ring}}
The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell]], the 600-cell's [[dual polytope]].{{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 [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 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 discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}}
The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}}

The 15 orange edges and 15 yellow edges form separate 15-gram helices.
Each orange or yellow edge crosses between two cyan great decagons.
Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}}
Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}}
The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between.
A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop.
The two connected loops forms a single [[Möbius loop]], a skew {15/2} [[pentadecagram]].
The pentadecagram is not shown in these illustrations (but [[#Decagons and 5𝝅 polygrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations.
Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}}
Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}}
They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}}

Five of these 30-cell [[Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H<sub>2</sub> × H<sub>2</sub>}}
Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[chiral]] objects; they either spiral clockwise (right) or counterclockwise (left).
The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings.
Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]].
Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration.
In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings.
Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}}

==== Radial golden triangles ====
The 600-cell can be constructed radially from 720 golden triangles of edge lengths {{radic|0.𝚫}} {{radic|1}} {{radic|1}} which meet at the center of the 4-polytope, each contributing two {{radic|1}} radii and a {{radic|0.𝚫}} edge.
They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral {{radic|0.𝚫}} bases (the faces of the 600-cell).
These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular {{radic|0.𝚫}} tetrahedron bases (the cells of the 600-cell).

==== Characteristic orthoscheme ====
{| class="wikitable floatright"
!colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}}
|-
!align=right|𝒍
|align=center|<small><math>\tfrac{1}{\phi} \approx 0.618</math></small>
|align=center|<small>36°</small>
|align=center|<small><math>\tfrac{\pi}{5}</math></small>
|align=center|<small>164°29′</small>
|align=center|<small><math>\pi-2\text{𝟁}</math></small>
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small>
|align=center|<small>22°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437</math></small>
|align=center|<small>18°</small>
|align=center|<small><math>\tfrac{\pi}{10}</math></small>
|align=center|<small>36°</small>
|align=center|<small><math>\tfrac{\pi}{5}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small>
|align=center|<small>17°44′40″</small>
|align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{4\phi^2}} \approx 0.535</math></small>
|align=center|<small>22°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_1R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{4\phi^2}} \approx 0.309</math></small>
|align=center|<small>18°</small>
|align=center|<small><math>\tfrac{\pi}{10}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_2R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{12\phi^2}} \approx 0.178</math></small>
|align=center|<small>17°44′40″</small>
|align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R^4/l</math></small>
|align=center|<small><math>1</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}} \approx 0.951</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_3R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
|
|
|
|
|
|-
!align=right|<small><math>\text{𝜼}</math></small>
|align=center|
|align=center|<small>37°44′40″</small>
|align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small>
|align=center|
|align=center|
|}
Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[orthoscheme]] is a [[chiral]] irregular [[simplex]] with [[right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[facet (geometry)|facet]]s (its ''mirror walls'').
Every regular polytope can be dissected radially into instances of its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center.
The characteristic orthoscheme has the shape described by the same [[Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}
The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[Coxeter-Dynkin diagram]] {{CDD|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets.
It is an irregular [[Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]].
The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius.
The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center.
Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme.
The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 600-cell has unit radius and edge length <small><math>\text{𝒍} = \tfrac{1}{\phi} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell).
The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.

==== Reflections ====
The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[polychoron]] consists of 3-dimensional cells.}}
Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections.
Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."}}
For example, a full isoclinic rotation of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram<sub>2</sub> geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}}
Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q<sup>''q''</sup> R<sup>''r''</sup> T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q<sup>2</sup> is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q<sup>''q''</sup> R<sup>''r''</sup><br>
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q<sup>''q''</sup> R<sup>''r''</sup> T<br>
where 2''q'' + ''r'' + 1 ≤ ''n''.<br>
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q<sup>2</sup>, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}}

==== Weyl orbits ====
Another construction method uses [[#Symmetries|quaternions]] and the [[Icosahedral symmetry]] of [[Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4):
: O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
: O(1000) : V1
: O(0010) : V2
: O(0001) : V3

[[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]]

With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell]] of order 14400.

Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct:

* the [[snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math>
* the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> 
* the [[120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math>