Editing 600-cell (section)

== Symmetries ==
The [[icosian]]s are a specific set of Hamiltonian [[quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}}
The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[rational number]]s.{{Sfn|Steinbach|1997|p=24}}
The finite sums of the 120 [[Icosian#Unit icosians|unit icosians]] are called the [[Icosian#Icosian ring|icosian ring]].

When interpreted as [[quaternion]]s,{{Efn|In [[Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[quaternion]] is simply a (w, x, y, z) Cartesian coordinate.|name=quaternions}} the 120 vertices of the 600-cell form a [[group (mathematics)|group]] under quaternionic multiplication.
This group is often called the [[binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}}
It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[invariant subgroup]], namely as the subgroup ''2I<sub>L</sub>'' of quaternion left-multiplications and as the subgroup ''2I<sub>R</sub>'' of quaternion right-multiplications.
Each rotational symmetry of the 600-cell is generated by specific elements of ''2I<sub>L</sub>'' and ''2I<sub>R</sub>''; the pair of opposite elements generate the same element of ''RSG''.
The [[Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''.
We have the isomorphism ''RSG ≅ (2I<sub>L</sub> × 2I<sub>R</sub>) / {Id, -Id}''.
The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200.
The [[quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[rotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|2015|p=1|loc="''[[Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions."}}

The binary icosahedral group is [[isomorphic]] to [[special linear group|SL(2,5)]].

The full [[symmetry group]] of the 600-cell is the [[H4 (mathematics)|Coxeter group H<sub>4</sub>]].{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§2 The Labeling of H<sub>4</sub>}}
This is a [[group (mathematics)|group]] of order 14400.
It consists of 7200 [[Rotation (mathematics)|rotations]] and 7200 rotation-reflections.
The rotations form an [[invariant subgroup]] of the full symmetry group.
The rotational symmetry group was first described by S.L. van Oss.{{Sfn|Oss|1899||pp=1-18}}
The H<sub>4</sub> group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'.
In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly.
Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes....
This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}}