geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a quaternionic polytope is a generalization of a

polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

in real space to an analogous structure in a

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 ...

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

, where each real dimension is accompanied by three imaginary ones. Similarly to

complex polytope In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collecti ...

s, points are not ordered and there is no sense of "between", and thus a quaternionic polytope may be understood as an arrangement of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on. Since the quaternions are non-

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

, a convention must be made for the multiplication of vectors by scalars, which is usually in favour of left-multiplication. As is the case for the complex polytopes, the only quaternionic polytopes to have been systematically studied are the regular ones. Like the real and complex regular polytopes, their symmetry groups may be described as reflection groups. For example, the regular quaternionic lines are in a one-to-one correspondence with the finite subgroups of ''U''₁(H): the

binary cyclic group In mathematics, the binary cyclic group of the ''n''-gon is the cyclic group of order 2''n'', C_, thought of as an extension of the cyclic group C_n by a cyclic group of order 2. Coxeter writes the ''binary cyclic group'' with angle-brackets, ⟨''n ...

binary dihedral group In group theory, a dicyclic group (notation Dic''n'' or Q4''n'', Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST) is a particular kind of non-abelian group of order 4''n'' (''n'' > 1). It is an extension of the ...

binary tetrahedral group In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of ...

binary octahedral group In mathematics, the binary octahedral group, name as 2O or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group ''O'' or (2,3 ...

, and

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) of or ...

References

Quaternions {{geometry-stub