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A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking…

We introduce a new operad-like structure that we call a reconnectad; the ``input'' of an element of a reconnectad is a finite simple graph, rather than a finite set, and ``compositions'' of elements are performed according to the notion of…

Periodic tilings can store information if individual tiles are decorated to lower their symmetry. Truchet tilings - the broad family of space-filling arrangements of such tiles - offer an efficient mechanism of visual data storage related…

Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of…

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological…

Controlling the topology of structures self-assembled from a set of heterogeneous building blocks is highly desirable for many applications, but is poorly understood theoretically. Here we show that the thermodynamic theory of self-assembly…

The Boerdijk-Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial translational or rotational symmetries. In this document, we develop a procedure by which this structure is modified to obtain both translational…

Graphs serve as powerful tools for modeling pairwise interactions in diverse fields such as biology, material science, and social networks. However, they inherently overlook interactions involving more than two entities. Simplicial…

Gyroscopic metamaterials --- mechanical structures composed of interacting spinning tops --- have recently been found to support one-way topological edge excitations. In these structures, the time reversal symmetry breaking that enables…

In the past year several constructions of non-invertible symmetries in Quantum Field Theory in $d\geq 3$ have appeared. In this paper we provide a unified perspective on these constructions. Central to this framework are so-called theta…

In this paper we characterize foams and tetrahedral structures in a unified way, by a simplified representation of both that conserves the system topology. The paper presents a workflow for an automated characterization of the topology of…

Anisotropy at the level of the inter-particle interaction provides the particles with specific instructions for the self-assembly of target structures. The ability to synthesize non-spherical colloids, together with the possibility of…

We compute and analyze isotropy subgroups of the complex orthogonal group with respect to the similarity transformation on itself and on skew-symmetric matrices. Their group structure is related to a group of certain nonsingular block…

This study introduces a novel approach to composite design by employing aperiodic monotiles, shapes that cover surfaces without translational symmetry. Using a combined computational and experimental approach, we study the fracture behavior…

We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out…

If particles interact according to isotropic pair potentials that favor multiple length scales, in principle a large variety of different complex structures can be achieved by self-assembly. We present, motivate, and discuss a conjecture…

We consider the incidence structure formed by the twelve pentagons given by the vertex neighborhoods of the icosahedron. Interpreting this structure purely in terms of coplanarity conditions, we show that -- up to projective equivalence --…

We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the…

This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in the three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a…

Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on…