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, consisting of the 120 unit icosians, comprises the distinct even permutations of 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.
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.
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].