Editing 600-cell (section)

==== 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}}