Icosian
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: Unit icosians. The 120 unit icosians, which form the icosian group, are all even permutations of: In this case, the vector ("a", "b", "c", "d") refers to the quaternion "a" + "bi + "cj + "d"k, and φ represents the golden ratio ( + 1)/2. These 120 vectors form the H4 root system, with a Weyl group of order 14400. In addition to the 120 unit icosians forming  the vertices of a 600-cell, 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 lie in the "golden field", ("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") + ("c" + "d") + ("e" + "f") + ("g" + "h"). 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 formula_1 embeds as a subgroup of formula_2. Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.