# Quaternionic polytope

A **quaternionic polytope** is a generalized polytope-like object whose containing space is n -dimensional module over quaternions, . Quaternionic polytopes are very similar to complex polytopes in definition and structure. Neither group are polytopes in the abstract sense, as they do not obey the diamond property.

## Definition[edit | edit source]

### Quaternionic space[edit | edit source]

First we define the space for quaternionic polytopes based on the quaternions. The space determines what transformations are symmetries which is required for concepts like regularity. Let us have the n -dimensional free module over quaternions, , with left scalar multiplication.^{[note 1]}
Let denote the standard unitary inner product on :^{[1]}

^{[note 2]}

where is the conjugate of . Then we equip with the metric:^{[2]}

### Quaternionic polytopes[edit | edit source]

Let π be a set of affine subspaces of the . We introduce some terminology:

- ΞΌ -dimensional elements of π are called ΞΌ -
**flats**. - Two flats are
**incident**if one is a proper subspace of the other. - For any two flats F and H , the
**medial figure**of F and H is the set of all flats G such that . - A set of flats π§ is
**connected**if for any two flats, F and H , in π§ there is a sequence of flats, S , beginning with F and ending with H , such that every pair of consecutive flats in S are incident.

Then π is a **quaternionic polytope** iff it satisfies the following properties:^{[3]}

- π contains the flats and .
- Every medial figure contains at least two elements.
- If then the medial figure of a Ξ» -flat and a Ξ½ -flat is connected.

In other words the abstract structure of P is an incidence complex.

## External links[edit | edit source]

- Wikipedia contributors. "Quaternionic polytope".

## Notes[edit | edit source]

- β Since quaternion mutliplication is non-commutative left and right scalar multiplication are distinct. However as long as one is chosen consistently it makes no difference to the result. This article uses left scalar multiplication for consistency.
- β If using a right scalar multiplication the product should be instead.

## References[edit | edit source]

- β Hoggar (1980:219)
- β Cuypers (1995:312)
- β Cuypers (1995:313)

## Bibliography[edit | edit source]

- Hoggar, S.G. (1980). Davis, Chandler; GrΓΌnbaum, Branko; Sherk, F.A. (eds.). "Two Quaternionic 4-polytopes".
*The Geometric Vein: The Coxeter Festschrift*: 219β229. ISBN 9781461256489. Retrieved 2023-10-31. - Cuypers, Hans (1995). "Regular quaternionic polytopes".
*Linear Algebra and Its Applications*. 226-228: 311β329. doi:10.1016/0024-3795(95)00149-L.

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