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Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldán show that there exist 2690 connected closed cubical surfaces up to isomorphism in the 5-cube. They give a classification in terms of their genus 𝑔 for closed orientable cubical surfaces, and their demigenus 𝑘 for a closed non-orientable cubical surface. In this paper we present the definition of a cubical surface and we visualize the projection to $R^3$ of a torus, a genus two torus, the projective plane, and the Klein bottle. We use reinforcement learning techniques to obtain configurations optimized for 3D-printing. 

How can we visualize all the surfaces that can be made from the faces of the tesseract? In recent work, Aveni, Govc, and Roldán showed that the torus and the sphere are the only closed surfaces that can be realized by a subset of two-dimensional faces of the tesseract. They also gave an exhaustive list of all the isomorphic types of embedings of these two surfaces. Here, we generate 3D models of all these surfaces. We also exhibit, with the help of some hyper-ants, the minimum realization of the Möbius strip on the tesseract. 

Abstract The Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in $$\mathbb {R}^d$$ $R^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at timetgrow at a rate between$$t^{(d-1)/d}$$ $t^{(d-1)/d}$and$$P_d(t)$$ $P_d\left(t\right)$, where$$P_d(t)$$ $P_d\left(t\right)$is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes. 

What is the maximum number of holes enclosed by a $$d$$-dimensional polyomino built of $$n$$ tiles? Represent this number by $$f_d(n)$$. Recent results show that $$f_2(n)/n$$ converges to $1/2$. We prove that for all $$d \geq 2$$ we have $$f_d(n)/n \to (d-1)/d$$ as $$n$$ goes to infinity. We also construct polyominoes in $$d$$-dimensional tori with the maximal possible number of holes per tile. In our proofs, we use metaphors from error-correcting codes and dynamical systems. 

Abstract How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that $$\lfloor {\frac{n}{2}} \rfloor $$ ⌊ n 2 ⌋ rooks or $$\lfloor {\frac{n}{3}} \rfloor $$ ⌊ n 3 ⌋ queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d -dimensional rooks and queens on d -dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.