Editing 600-cell (section)

==== Unit radius Cartesian coordinates ====
The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length {{sfrac|1|φ}} ≈ 0.618 (where φ = {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows:

8 vertices obtained from

:(0, 0, 0, ±1)

by permuting coordinates, and 16 vertices of the form:

:(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})

The remaining 96 vertices are obtained by taking [[even permutation]]s of

:(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ<sup>−1</sup>|2}}, 0)

Note that the first 8 are the vertices of a [[16-cell]], the second 16 are the vertices of a [[tesseract]], and those 24 vertices together are the vertices of a [[24-cell]].
The remaining 96 vertices are the vertices of a [[snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}

When interpreted as [[#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[icosian]]s.

In the 24-cell, there are [[24-cell#Squares|squares]], [[24-cell#Hexagons|hexagons]] and [[24-cell#Triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}}
In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint.
In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}}
In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}
Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]].

The 60 axes and 75 16-cells of the 600-cell constitute a [[Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60<sub>5</sub>75<sub>4</sub> to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}}
Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs.