Related papers: Topologically Interlocking Blocks inside the Tetro…
Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus…
Combinatorial mechanical metamaterials are made of anisotropic, flexible blocks, such that multiple metamaterials may be constructed using a single block type, and the system's response depends on the frustration (or its absence) due to the…
We consider defining the embedding of a triangle mesh into $R^3$, up to translation, rotation, and scale, by its vector of dihedral angles. Theoretically, we show that locally, almost everywhere, the map from realizable vectors of dihedrals…
The chain of algebraic geometry constructions permits to transfer from the minimal surface with zero instability index, and from the lattice over the ring of cyclotomic integers to the tetra-block helix. The tetra-block is the 7-vertex…
We explore the use of templated self-assembly to facilitate the formation of complex target structures made from patchy particles. First, we consider the templating of high-symmetry shell structures around a spherical core particle. We find…
We investigate the motions of a bar structure consisting of two congruent tetrahedra, whose edges in their basic position form the face diagonals of a rectangular parallelepiped. The constraint of the motion is that the originally…
Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids are used within the segmentation framework to encode a hierarchy of partitions. The different graph models used within the irregular…
In 1957, Steinhaus proved that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop does not exist. Over the years, various modifications of this statement have been considered and analysed. Weakening the statement…
We investigate various topological spaces and varieties which can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the…
The space of elliptic modular forms of fixed weight and level can be identfied with a space of intertwining operators, from a holomorphic discrete series representation of SL2(R) to a space of automorphic forms. Moreover, multiplying…
Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction…
We present a new numerical scheme to study systems of non-convex, irregular, and punctured particles in an efficient manner. We employ this method to analyze regular packings of odd-shaped bodies, not only from a nanoparticle but also both…
A mechanically-based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density…
The self-assembly of hard polyhedral particles confined to a flat interface is studied using Monte Carlo simulations. The particles are pinned to the interface by restricting their movement in the direction perpendicular to it while…
The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are…
Different novel organic-chemical possibilities for tetrahedral building units are considered, with attention to their utility in constructing different super-structures. As a representative construction we consider the use of sets of 20…
Illumination complexes are examples of 'flat polyhedral complexes' which arise if several copies of a convex polyhedron (convex body) Q are glued together along some of their common faces (closed convex subsets of their boundaries). A…
We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal…
Over the complex numbers, the complement of a collection of hyperplanes is a widely-studied object; the cohomology ring, in particular, is known to have a structure depending only on the combinatorial properties of the intersection of…
Topologically Interlocked Structures (TIS) are structural assemblies that achieve stability and carrying capacity through the geometric arrangement of interlocking blocks, relying solely on normal and friction forces for load transfer.…