Gabe Cunningham
YOU?
Author Swipe
View article: Polytopality criteria for the mix of polytopes and maniplexes
Polytopality criteria for the mix of polytopes and maniplexes Open
The mix of two maniplexes is the minimal maniplex that covers both. This construction has many important applications, such as finding the smallest regular cover of a maniplex. If one of the maniplexes is an abstract polytope, a natural qu…
View article: Graph Powers of Groups
Graph Powers of Groups Open
The Lights Out Puzzle, played on a graph $Γ$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements of a …
View article: Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464
Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464 Open
The Parks Puzzle is a paper-and-pencil puzzle game that is classically played on a square grid with different colored regions (the parks). The player needs to place a certain number of "trees" in each row, column, and park such that none a…
View article: Reflexible covers of prisms
Reflexible covers of prisms Open
The Tomotope provided the first well understood example of an abstract 4-polytope whose connection (monodromy) group was not a string C-group, and which also did not have a unique minimal regular cover. Conversely, we know that if the conn…
View article: Finite 3-orbit polyhedra in ordinary space, II
Finite 3-orbit polyhedra in ordinary space, II Open
We enumerate the 188 3-orbit skeletal polyhedra in $${\mathbb {E}}^3$$ with irreducible symmetry group. The analysis is carried out by determining the polyhedra having each irreducible finite group of isometries as their symmetry grou…
View article: Cayley extensions of maniplexes and polytopes
Cayley extensions of maniplexes and polytopes Open
A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define $\mathcal{M}$ to be a \emph{Cayley extension} of $\ma…
View article: Stratified operations on maniplexes
Stratified operations on maniplexes Open
There is an increasingly extensive literature on the problem of describing the connection (monodromy) groups and automorphism groups of families of polytopes and maniplexes that are not regular or reflexible. Many such polytopes and manipl…
View article: Tight Chiral Polytopes
Tight Chiral Polytopes Open
A chiral polytope with Schläfli symbol $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags, and it is called \emph{tight} if the number of flags meets this lower bound. The Schläfli symbols of tight chiral polyhedra were cl…
View article: Flat extensions of abstract polytopes
Flat extensions of abstract polytopes Open
We consider the problem of constructing an abstract $(n+1)$-polytope $Q$ with $k$ facets isomorphic to a given $n$-polytope $P$, where $k \geq 3$. In particular, we consider the case where we want $Q$ to be $(n-2,n)$-flat, meaning that eve…
View article: Internal and external duality in abstract polytopes
Internal and external duality in abstract polytopes Open
We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, …
View article: Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes
Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes Open
An abstract polytope is flat if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regula…
View article: Internal and external duality in abstract polytopes
Internal and external duality in abstract polytopes Open
We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, …
View article: Internal and external duality in abstract polytopes
Internal and external duality in abstract polytopes Open
We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, …
View article: Open problems on k-orbit polytopes
Open problems on k-orbit polytopes Open
We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.