Icosian

For Hamilton's use of icosian, see icosian game and icosian calculus.

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 (√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 $H_{4}$ embeds as a subgroup of $E_{8}$. 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 Archived 2011-06-07 at the Wayback Machine
• Adam P. Goucher Good fibrations