# Quaternionic polytope

A quaternionic polytope is a generalized polytope-like object whose containing space is n -dimensional module over quaternions, ${\displaystyle \mathbb {H} ^{n}}$. 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

### Quaternionic space

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, ${\displaystyle \mathbb {H} ^{n}}$, with left scalar multiplication.[note 1] Let ${\displaystyle \langle \cdot \mid \cdot \rangle }$ denote the standard unitary inner product on ${\displaystyle \mathbb {H} ^{n}}$:[1]

${\displaystyle \langle (x_{0},x_{1},\dots ,x_{n})\mid (y_{0},y_{1},\dots ,y_{n})\rangle =\sum _{i=0}^{n}x_{i}{\overline {y_{i}}}}$[note 2]

where ${\displaystyle {\overline {x_{i}}}}$ is the conjugate of ${\displaystyle x_{i}}$. Then we equip ${\displaystyle \mathbb {H} ^{n}}$ with the metric:[2]

${\displaystyle d(x,y)={\sqrt {\langle x-y\mid y-x\rangle }}}$

### Quaternionic polytopes

Let π be a set of affine subspaces of the ${\displaystyle \mathbb {H} ^{n}}$. 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 ${\displaystyle F\subsetneq G\subsetneq H}$.
• 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 ${\displaystyle \emptyset }$ and ${\displaystyle \mathbb {H} ^{n}}$.
• Every medial figure contains at least two elements.
• If ${\displaystyle \lambda <\nu -2}$ 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.