Related papers: Topologically Interlocking Blocks inside the Tetro…

We give a complete description of all convex polyhedra whose surface can be constructed from several congruent regular pentagons by folding and gluing them edge to edge. Our method of determining the graph structure of the polyhedra from a…

The number of distinguishable inherent structures of a liquid is the key component to understanding the thermodynamics of glass formers. In the case of hard potential systems such as hard discs, spheres and ellipsoids, an inherent structure…

If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we…

Multilevel self-assembly involving small structured groups of nano-particles provides new routes to development of functional materials with a sophisticated architecture. Apart from the inter-particle forces, the geometrical shapes and…

We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry protected topological states can be…

Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…

We introduce and discuss the concept of modular entanglement. This is the entanglement that is established between the end points of modular systems composed by sets of interacting moduli of arbitrarily fixed size. We show that end-to-end…

Manipulating the way in which colloidal particles self-organise is a central challenge in the design of functional soft materials. Meeting this challenge requires the use of building blocks that interact with one another in a highly…

Polyominoes are a subset of polygons which can be constructed from integer-length squares fused at their edges. A system of polygons P is interlocked if no subset of the polygons in P can be removed arbitrarily far away from the rest. It is…

A new tetrahedral structure model was developed and the geometrical structure of jammed disordered packings of monodisperse spheres with different friction coefficients was systematically characterized. An intrinsic structure feature is…

Nanoparticles with "sticky patches" have long been proposed as building blocks for the self-assembly of complex structures. The synthetic realizability of such patchy particles, however, greatly lags behind predictions of patterns they…

The formation of quasi-spherical cages from protein building blocks is a remarkable self-assembly process in many natural systems, where a small number of elementary building blocks are assembled to build a highly symmetric icosahedral…

Space-filling building blocks of diverse shape permeate nature at all levels of organization, from atoms to honeycombs, and have proven useful in artificial systems, from molecular containers to clay bricks. But, despite the wide variety of…

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a…

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…

In this paper we present algorithms for collective construction systems in which a large number of autonomous mobile robots trans- port modular building elements to construct a desired structure. We focus on building block structures…

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a…

Tutte's embedding theorem states that every 3-connected graph without a $K_5$ or $K_{3,3}$ minor (i.e. a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of…

The topic of totally separable sphere packings is surveyed with a focus on regular constructions, uniform tilings, and contact number problems. An enumeration of all regular totally separable sphere packings in $\mathbb{R}^2$,…

We consider the configuration space of ordered points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that…