mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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. The algebra of quater ...

s with the same symmetry as the

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

. The term can be used to refer to two related, but distinct, concepts: * The icosian

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

: a

multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...

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 or morphism 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 the ...

to the

binary icosahedral group In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120. It is an extension of the icosahedral group ''I'' or (2,3,5) o ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

120. * The icosian

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

: all finite sums of the 120 unit icosians.

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'' + ''b''i + ''c''j + ''d''k, and φ represents the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

( + 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 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, hec ...

. Other

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

s of icosians correspond to the

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

, 16-cell and

24-cell In four-dimensional space, four-dimensional 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 "octa ...

Icosian ring

The icosians are a subset of quaternions of the form, (''a'' + ''b'') + (''c'' + ''d'')i + (''e'' + ''f'')j + (''g'' + ''h'')k, where the eight variables are

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s. 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''. 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 t ...

on ''L'', under which the lattice is isomorphic to the E8 lattice. This construction shows that the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

H_4

embeds as a subgroup of

E_8

. Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.

Notes

References

* John H. Conway,

Neil Sloane __NOTOC__ Neil James Alexander Sloane FLSW (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 cre ...

: ''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
{{Webarchive, url=https://web.archive.org/web/20110607165538/http://members.home.nl/fg.marcelis/Icosians%20and%20ADE.htm , date=2011-06-07 * Adam P. Gouche
Good fibrations
Quaternions John Horton Conway Finite groups Regular 4-polytopes E8 (mathematics)