Editing 600-cell (section)

==== 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.
|-
|}