Editing 120-cell (section)

===Cartesian coordinates===
Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen.

==== √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.

==== Unit radius coordinates ====
The unit-radius 120-cell has edge length {{Sfrac|1|φ<sup>2</sup>{{Radic|2}}}} ≈ 0.270.

In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone|Pileio|Levitt|2010|p=1442|loc=Table 3}} are the {[[permutations]]} and {{bracket|[[even permutation]]s}} in the left column below:
{| class="wikitable" style=width:720px
|-
!rowspan=3|120
!8
|style="white-space: nowrap;"|({±1, 0, 0, 0})
|[[16-cell#Coordinates|16-cell]]
| rowspan="2" |[[24-cell#Hexagons|24-cell]]
| rowspan="3" |[[600-cell#Coordinates|600-cell]]
| rowspan="10" style="white-space: nowrap;"|120-cell
|-
!16
|style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2
|[[Tesseract#Radial equilateral symmetry|Tesseract]]
|-
!96
|style="white-space: nowrap;"|([0, ±φ<sup>−1</sup>, ±1, ±φ]) / 2
|colspan=2|[[Snub 24-cell#Coordinates|Snub 24-cell]]
|-
!rowspan=7|480
!colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]]
!5-point [[5-cell#Coordinates|5-cell]]
![[24-cell#Squares|24-cell]]
![[600-cell#Coordinates|600-cell]]
|-
!32
|style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ<sup>−2</sup>]) / {{radic|8}}
|rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br>
(−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br>
(−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br>
(−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br>
(−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4
|rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0})
|rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br>
({±1, ±1, ±1, ±1}) / 2<br>
([0, ±φ<sup>−1</sup>, ±1, ±φ]) / 2
|-
!32
|style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}}
|-
!32
|style="white-space: nowrap;"|([±φ<sup>−1</sup>, ±φ<sup>−1</sup>, ±φ<sup>−1</sup>, ±φ<sup>2</sup>]) / {{radic|8}}
|-
!96
|style="white-space: nowrap;"|([0, ±φ<sup>−1</sup>, ±φ, ±{{radic|5}}]) / {{radic|8}}
|-
!96
|style="white-space: nowrap;"|([0, ±φ<sup>−2</sup>, ±1, ±φ<sup>2</sup>]) / {{radic|8}}
|-
!192
|style="white-space: nowrap;"|([±φ<sup>−1</sup>, ±1, ±φ, ±2]) / {{radic|8}}
|-
|colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors  <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br>
<small><math display=block>\begin{pmatrix}
w_2\\
x_2\\
y_2\\
z_2
\end{pmatrix}
*
\begin{pmatrix}
w_1\\
x_1\\
y_1\\
z_1
\end{pmatrix}
=
\begin{pmatrix}
{w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\
{w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\
{w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\
{w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1}
\end{pmatrix}
</math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone|Pileio|Levitt|2010|loc=Table 3|p=1442}}}}
|}

The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}}