Icosian

Template:For Template:Short description Template:No footnotes 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 (Template:Radic + 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 + bTemplate:Radic) + (c + dTemplate:Radic)i + (e + fTemplate:Radic)j + (g + hTemplate:Radic)k, where the eight variables are rational numbersTemplate:Refn. 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 + bTemplate:Radic)² + (c + dTemplate:Radic)² + (e + fTemplate:Radic)² + (g + hTemplate:Radic)². Its Euclidean norm is defined as u + v if the quaternion norm is u + vTemplate:Radic. 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 Template:Webarchive
• Adam P. Goucher Good fibrations