Editing 600-cell (section)

==== 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.