{{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 quaternions with the same symmetry as the 600-cell. The term can be used to refer to two related, but distinct, concepts:
* The '''icosian group''': a multiplicative group of 120 quaternions, positioned at the vertices of a 600-cell of unit radius. This group is isomorphic to the binary icosahedral group of order 120. * The '''icosian ring''': all finite sums of the 120 unit icosians.
==Unit icosians== The icosian group, consisting of the 120 unit icosians, comprises the distinct even permutations of
* ½(±2, 0, 0, 0) (resulting in 8 icosians), * ½(±1, ±1, ±1, ±1) (resulting in 16 icosians), * ½(0, ±1, ±1''/φ'', ±''φ'') (resulting in 96 icosians).
In this case, the vector (''a'', ''b'', ''c'', ''d'') refers to the quaternion ''a'' + ''b'''''i''' + ''c'''''j''' + ''d'''''k''', and φ represents the golden ratio ({{radic|5}} + 1)/2. These 120 vectors form the vertices of a 600-cell, whose symmetry group is the Coxeter group H4 of order 14400. In addition, the 600 icosians of norm 2 form the vertices of a 120-cell. Other subgroups of icosians correspond to the tesseract, 16-cell and 24-cell.
==Icosian ring== The icosians are a subset of quaternions of the form, (''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 numbers.{{refn|group=note| The complex numbers of the form ''a'' + ''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'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''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'' + ''b''{{radic|5}})<sup>2</sup> + (''c'' + ''d''{{radic|5}})<sup>2</sup> + (''e'' + ''f''{{radic|5}})<sup>2</sup> + (''g'' + ''h''{{radic|5}})<sup>2</sup>. Its Euclidean norm is defined as ''u'' + ''v'' if the quaternion norm is ''u'' + ''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)