#
Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators^{ †}

^{†}

## 1. Introduction

## 2. Maps

## 3. Polytopes

- (1)
- $\mathcal{F}$ contains a unique maximal and a unique minimal element.
- (2)
- All maximal chains (these are called $flags$) have the same length. This allows us to assign a “rank” or “dimension” to each face. The unique minimal face (usually called “∅”) is given rank $-1$.
- (3)
- If faces $a<b<c$ are consecutive in some chain, then there exists exactly one ${b}^{\prime}\ne b$ such that $a<{b}^{\prime}<c$. (This axiom is usually called the diamond condition).
- (4)
- For any $a\le c$, the section $[a,c]$ is the sub-poset consisting of all faces b such that $a\le b\le c$. We require it to be true in any section that if ${f}_{1}$ and ${f}_{2}$ are any two flags of the section, then there is a sequence of flags of the section, beginning at ${f}_{1}$ and ending at ${f}_{2}$, such that any two consecutive flags differ in exactly one rank. This condition is called strong flag connectivity.

## 4. Maps and Polytopes

## 5. Complexes and Maniplexes

- (1)
- Each ${r}_{i}$ is a partition of $\mathrm{\Omega}$ into sets of size 2.
- (2)
- If $i\ne j$, then ${r}_{i}$ and ${r}_{j}$ are disjoint.
- (3)
- $\mathcal{M}$ is connected. We will explain this after some discussion of notation.

- (1)
- as a partition,
- (2)
- as a function,
- (3)
- as a permutation,
- (4)
- as a set of edges colored i in a graph $\mathrm{\Gamma}$,
- (5)
- in particular, a perfect matching on the set $\mathrm{\Omega}$.

- (1)
- Each facet is an n-maniplex.
- (2)
- For any subfacet a, ${r}_{n+1}$ (considered as a function), restricted to a, is an isomorphism from a to some subfacet ${a}^{\prime}$ (${a}^{\prime}$ need not be different from a, though it often is).

**Example:**Every 0-complex is a 0-maniplex, isomorphic to this one: $\mathrm{\Omega}=\{1,2\},{r}_{0}=\left\{\{1,2\}\right\}.$

**Example:**Every 1-complex is a 1-maniplex, and can be regarded geometrically as a polygon or 2-polytope. $\mathrm{\Omega}=\{1,2,3,4,5,6,7,8\}$, ${r}_{0}=\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}$, ${r}_{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{\{2,3\},\{4,5\},\{6,7\},\{8,1\}\}$. This maniplex has four vertices ($\{2,3\},\{4,5\}$, etc.), and four edges ($\{1,2\},\{3,4\},\cdots $); thus it is a 4-gon.

**Example:**A 2-maniplex. The 2-maniplex shown in Figure 4 has four faces (the 0–1 cycles), four vertices (the 1–2 cycles) and 6 edges (the 0–2 cycles). Each vertex shares an edge with each other vertex and so has degree 3. Similarly each face has three sides and so this maniplex is the tetrahedron.

**Fact:**Every map can be considered as a 2-maniplex.

## 6. Non-Polytopal Maps

## 7. Symmetry in Maniplexes

## 8. Operators on maniplexes

## 9. Conclusions

- (1)
- Can we classify all rotary maniplexes having one facet? Two facets? In maps, the classifications of 1-face and 2-face maps are easy and all such maps are reflexible. In maniplexes, several examples of chiral 1-facet and 2-facet 3-maniplexes are known.
- (2)
- What conditions on a polytope will guarantee that all of its derivates are polytopal as well?
- (3)
- Given a rotary map $\mathcal{M}$, what are all the rotary 3-maniplexes having facets isomorphic to $\mathcal{M}$?
- (4)
- Given a graph $\mathrm{\Gamma}$, what are all the rotary maniplexes whose 1-skeleton is isomorphic to $\mathrm{\Gamma}$?

## Acknowledgement

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**MDPI and ACS Style**

Wilson, S. Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators. *Symmetry* **2012**, *4*, 265-275.
https://doi.org/10.3390/sym4020265

**AMA Style**

Wilson S. Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators. *Symmetry*. 2012; 4(2):265-275.
https://doi.org/10.3390/sym4020265

**Chicago/Turabian Style**

Wilson, Steve. 2012. "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators" *Symmetry* 4, no. 2: 265-275.
https://doi.org/10.3390/sym4020265