Editing 120-cell (section)

==== √8 radius coordinates ====
The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length  4−2φ = 3−{{radic|5}} ≈ 0.764.

In this frame of reference, its 600 vertex coordinates are the {[[permutations]]} and {{bracket|[[even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} 
{| class=wikitable
|-
!24
| ({0, 0, ±2, ±2})
| [[24-cell#Squares|24-cell]]
| rowspan=7 | 600-point 120-cell
|-
!64
| ({±φ, ±φ, ±φ, ±φ<sup>−2</sup>})
|
|-
!64
| ({±1, ±1, ±1, ±{{radic|5}}<nowiki />})
|
|-
!64
| ({±φ<sup>−1</sup>, ±φ<sup>−1</sup>, ±φ<sup>−1</sup>, ±φ<sup>2</sup>})
|
|-
!96
| ([0, ±φ<sup>−1</sup>, ±φ, ±{{radic|5}}])
| [[Snub 24-cell#Coordinates|Snub 24-cell]]
|-
!96
| ([0, ±φ<sup>−2</sup>, ±1, ±φ<sup>2</sup>])
| [[Snub 24-cell#Coordinates|Snub 24-cell]]
|-
!192
| ([±φ<sup>−1</sup>, ±1, ±φ, ±2])
|
|}

where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618.