Editing 600-cell (section)

===== 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.