Icosian#Unit icosians
{{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 + bi + cj + dk, 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}})2 + (c + d{{radic|5}})2 + (e + f{{radic|5}})2 + (g + h{{radic|5}})2. 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 embeds as a subgroup of . Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.
Notes
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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]