View source for Icosian

{{For|Hamilton's use of icosian|icosian game|icosian calculus}}
{{short description|Specific set of Hamiltonian quaternions with the same symmetry as the 600-cell}}
{{no footnotes|date=September 2021}}
In [[mathematics]], the '''icosians''' are a specific set of Hamiltonian [[quaternion]]s with the same symmetry as the [[600-cell]]. The term can be used to refer to two related, but distinct, concepts:

* The '''icosian [[Group (mathematics)|group]]''': a [[multiplicative group]] of 120 quaternions, positioned at the vertices of a 600-cell of unit radius. This group is [[group isomorphism|isomorphic]] to the [[binary icosahedral group]] of [[order of a group|order]] 120.
* The '''icosian [[Ring (mathematics)|ring]]''': all finite sums of the 120 unit icosians.

==Unit icosians==
The icosian group, consisting of the 120 unit icosians, comprises the distinct [[even permutation]]s of

* ½(±2,&nbsp;0,&nbsp;0,&nbsp;0) (resulting in 8 icosians),
* ½(±1,&nbsp;±1,&nbsp;±1,&nbsp;±1) (resulting in 16 icosians),
* ½(0,&nbsp;±1,&nbsp;±1''/&phi;'',&nbsp;±''&phi;'') (resulting in 96 icosians).

In this case, the vector (''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;''d'') refers to the quaternion ''a''&nbsp;+&nbsp;''b'''''i'''&nbsp;+&nbsp;''c'''''j'''&nbsp;+ ''d'''''k''', and &phi; represents the [[golden ratio]] ({{radic|5}}&nbsp;+&thinsp;1)/2. These 120 vectors form the vertices of a 600-cell, whose symmetry group is the Coxeter group [[H4 (mathematics)|H4]] of order 14400. In addition, the 600 icosians of norm 2 form the vertices of a [[120-cell]]. Other [[subgroup]]s of icosians correspond to the [[tesseract]], [[16-cell]] and [[24-cell]].

==Icosian ring==
The icosians are a subset of quaternions of the form, (''a''&nbsp;+&nbsp;''b''{{radic|5}})&nbsp;+&nbsp;(''c''&nbsp;+&nbsp;''d''{{radic|5}})'''i'''&nbsp;+&nbsp;(''e''&nbsp;+&nbsp;''f''{{radic|5}})'''j'''&nbsp;+&nbsp;(''g''&nbsp;+&nbsp;''h''{{radic|5}})'''k''', where the eight variables are [[rational number]]s{{refn|group=note| The complex numbers of the form ''a''&nbsp;+&nbsp;''b''{{radic|5}} , where ''a'' and ''b'' are both rational, are sometimes referred to as ''the golden field'' owing to their connection with the Golden ratio.}}. This quaternion is only an icosian if the vector (''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;''d'',&nbsp;''e'',&nbsp;''f'',&nbsp;''g'',&nbsp;''h'') is a point on a lattice ''L'', which is isomorphic to an [[E8 lattice]].

More precisely, the quaternion norm of the above element is (''a''&nbsp;+&nbsp;''b''{{radic|5}})<sup>2</sup>&nbsp;+&nbsp;(''c''&nbsp;+&nbsp;''d''{{radic|5}})<sup>2</sup>&nbsp;+&nbsp;(''e''&nbsp;+&nbsp;''f''{{radic|5}})<sup>2</sup>&nbsp;+&nbsp;(''g''&nbsp;+&nbsp;''h''{{radic|5}})<sup>2</sup>.  Its Euclidean norm is defined as ''u''&nbsp;+&nbsp;''v'' if the quaternion norm is ''u''&nbsp;+&nbsp;''v''{{radic|5}}.  This Euclidean norm defines a [[quadratic form]] on ''L'', under which the lattice is isomorphic to the [[E8 lattice]].

This construction shows that the [[Coxeter group]] <math>H_4</math> embeds as a subgroup of <math>E_8</math>.  Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.

==Notes==
{{reflist|group=note}}

==References==
* [[John H. Conway]], [[Neil Sloane]]: ''Sphere Packings, Lattices and Groups (2nd edition)''
* John H. Conway, [[Heidi Burgiel]], [[Chaim Goodman-Strauss]]: ''The Symmetries of Things (2008)''
* Frans Marcelis [http://members.home.nl/fg.marcelis/Icosians%20and%20ADE.htm Icosians and ADE] {{Webarchive|url=https://web.archive.org/web/20110607165538/http://members.home.nl/fg.marcelis/Icosians%20and%20ADE.htm |date=2011-06-07 }}
* Adam P. Goucher [https://cp4space.wordpress.com/2012/09/27/good-fibrations/ Good fibrations]

[[Category:Quaternions]]
[[Category:John Horton Conway]]
[[Category:Finite groups]]
[[Category:Regular 4-polytopes]]
[[Category:E8 (mathematics)]]