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:

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 + bi + cj + dk, 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)i + (c + d√5)j + (e + f√5)k + (g + h√5), 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 the E8 lattice.

References

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