# 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
**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 120 unit icosians, which form the icosian group, are all even permutations of:

- 8 icosians of the form ½(±2, 0, 0, 0)
- 16 icosians of the form ½(±1, ±1, ±1, ±1)
- 96 icosians of the form ½(0, ±1, ±
*Φ*, ±*φ*)

In this case, the vector (*a*, *b*, *c*, *d*) refers to the quaternion *a* + *b***i** + *c***j** + *d***k**, and Φ,φ represent the numbers (√5 ± 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*√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 embeds as a subgroup of . Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.

## 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