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 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 (√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√5)+(c+d√5)i+(e+f√5)j+(g+h√5)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√5)2+(c+d√5)2+(e+f√5)2+(g+h√5)2. Its Euclidean norm is defined as u+v if the quaternion norm is u+v√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 H 4 {\displaystyle H_{4}} embeds as a subgroup of E 8 {\displaystyle E_{8}}. Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.

Notes