Icosian

In today's world, Icosian has become a topic of great relevance and interest to a wide variety of people. Whether due to its impact on society, its importance in the professional field or its historical relevance, Icosian has captured the attention of individuals of all ages and backgrounds. This article seeks to fully explore the meaning and implications of Icosian, offering both an overview and a detailed analysis of its different aspects. Over the next few lines, we will delve into the fascinating world of Icosian, with the aim of providing a complete and enriching perspective on this topic of undoubted importance in the current panorama.

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:

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[note 1]. 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.

Notes

  1. ^ The complex numbers of the form a + b√5 , where a and b are both rational, are sometimes referred to as the golden field owing to their connection with the Golden ratio.

References