mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the icosians are a specific set of Hamiltonian

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

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: a multiplicative group of 120 quaternions, positioned at the vertices of a 600-cell of unit radius. This group is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

to the binary icosahedral group of

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

120. * The icosian

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

: 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, ±1''/φ'', ±''φ'') In this case, the vector (''a'', ''b'', ''c'', ''d'') refers to the quaternion ''a'' + ''b''i + ''c''j + ''d''k, and φ represents the golden ratio ( + 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 In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...

. Other subgroups of icosians correspond to the tesseract,

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...

and

24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...

Icosian ring

The icosians lie in the ''golden field'', (''a'' + ''b'') + (''c'' + ''d'')i + (''e'' + ''f'')j + (''g'' + ''h'')k, 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 a lattice ''L'', which is isomorphic to an E8 lattice. More precisely, the quaternion norm of the above element is (''a'' + ''b'')² + (''c'' + ''d'')² + (''e'' + ''f'')² + (''g'' + ''h'')². Its Euclidean norm is defined as ''u'' + ''v'' if the quaternion norm is ''u'' + ''v''{{radic, 5. This Euclidean norm defines a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

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 John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

Neil Sloane __NOTOC__ Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator ...

: ''Sphere Packings, Lattices and Groups (2nd edition)'' * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss: ''The Symmetries of Things (2008)'' * Frans Marceli
Icosians and ADE
* Adam P. Gouche
Good fibrations
Quaternions John Horton Conway