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 ( + 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) + (c + d)i + (e + f)j + (g + h)k, where the eight variables are rational numbers.. 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)2 + (c + d)2 + (e + f)2 + (g + h)2. Its Euclidean norm is defined as u + v if the quaternion norm is u + v. 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
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 Icosians and ADE Adam P. Goucher Good fibrations