In mathematics, the icosians are a specific set of Hamiltonian quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s with the same symmetry as the 600-cell. The term can be used to refer to two related, but distinct, concepts:

• The icosian group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
  : a multiplicative group
  Multiplicative group
  In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
  
   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
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
  : 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, ±ψ, ±φ)

In this case, the vector (a, b, c, d) refers to the quaternion a + bi + cj + dk. These 120 vectors form the H4 root system, with a Weyl group

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

 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

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

, 16-cell

16-cell

In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....

 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 number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s. Interestingly, this quaternion is only an icosian if the vector (a, b, c, d, e, f, g, h) is a point on the E8 lattice

E8 lattice

In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...

.