Related papers: Exploring the "Rubik's Magic" universe

Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta…

We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and non-orientable). We also give improved bounds for the topological complexity of unordered configuration spaces…

Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…

We develop infinitary analogues of the $N\times N\times N$ Rubik's cube. We'll be pushed to consider the possibility of transfinitely many twists and the foremost question we shall study is whether or not all infinite scrambles are…

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by…

We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This…

We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular…

A new methodological approach for the study of topology for shapes made of arrangements of lines, planes or solids is presented. Topologies for shapes are traditionally built on the classical theory of point-sets. In this paper, topologies…

A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…

A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some…

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a…

Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…

We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…

We create Resthetikhin-Turaev topological invariants of closed orientable three-manifolds from the quantum supergroup U_q(osp(1|2n)) at certain even roots of unity. To construct the invariants we develop tensor product theorems for finite…

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…

A long-standing question is what invariant sets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved…

We investigate the topology of the closure in a wonderful compactification of the set of unipotent-invariant bilinear forms.

Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…

Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than…

Building blocks and tiles are an excellent way of learning about geometry and mathematics in general. There are several versions of tiles that are either snapped together or connected with magnets that can be used to introduce topics like…