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:
The 120 unit icosians, which form the icosian group, are all even permutations of:
In this case, the vector refers to the quaternion a + bi + cj + dk, and Φ,φ represent the numbers /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, + i + j + k, where the eight variables are rational numbers. This quaternion is only an icosian if the vector is a point on a lattice L, which is isomorphic to an E8 lattice.
More precisely, the quaternion norm of the above element is 2 + 2 + 2 + 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.