Rubik's Abstract Polytopes

TL;DR

The paper develops a unified framework to generalize the Rubik's puzzle to arbitrary regular abstract polytopes by introducing the Rubik's construction ${\rm R}(\mathcal{P})$ and its group ${\rm GR}(\mathcal{P})$. It provides a wreath-product encoding that embeds ${\rm GR}(\mathcal{P})$ into a product of wreaths and proves that the image lies in the kernel of canonical characters, yielding coordinate-invariant constraints. The authors completely characterize the Rubik's group for the simplex ${\triangle^n}$, with explicit conditions on the component permutations and a detailed size formula, and extend the analysis to the hypercube ${\square^n}$, including a parity-coupled description and a corresponding size formula. They also develop extensions of facet moves to the whole polytope and relate non-rotational variants, hosotopes, and ditopes, culminating in a principled, dimension-agnostic model that parallels holonomy concepts in a discrete setting. The work unifies disparate Rubik-type groups and provides exact combinatorial and group-theoretic descriptions across dimensions, enabling algebraic and algorithmic insights into these discrete puzzles.

Abstract

We generalize the Rubik's cube, together with its group of configurations, to any abstract regular polytope. After discussing general aspects, we study the Rubik's simplex of arbitrary dimension and provide a complete description of the associated group. We sketch an analogous argument for the Rubik's hypercube as well.

Figures (7)

• Figure 1: The Rubik's Platonic solids.
• Figure 2: One the left, the Rubik's hyperbolic triangular tiling. On the right, the Rubik's $4$-dimensional hypercube projected to the $3$-dimensional space.
• Figure 3: Hasse diagram of the interval $F / G$ where $G, H_i$ and $F$ are faces of an abstract polytope of rank $i-1, i$ and $i+1$ respectively. The shape of the diagram motivates the name of the 'diamond' property.
• Figure 4: The Euclidean $3$-dimensional polytopes are known as the Platonic solids. There exist only five of them (up to isomorphism): tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
• Figure 5: A hosotope (left) and a ditope (right) of dimension $3$, also referred to as hosohedron and dihedron respectively.

Theorems & Definitions (40)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Example 2.1
• Example 2.2
• Definition 5
• Definition 6
• Example 3.1
• Definition 7