A Local Characterization of Combinatorial Multihedrality in Tilings

A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.Comment: 10 pages (to appear in Contributions to Discrete Mathematics

Combinatorial Space Tiling

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science

Fluidization and Segregation in Confluent Models for Biological Tissues

Collective tissue dynamics, more specifically a tissue\u27s ability to fluidize and segregate, is imperative for proper embryonic development and normal physiological functioning. In this thesis, I use vertex models to understand how cell-scale properties govern large-scale collective behavior. I begin with the process of fluidization in ordered monolayers. By perturbing beyond the linear regime, I show that in confluent tissues the linear response does not correctly predict the non-linear behavior, which, in this case, is the ability to exchange neighbors and fluidize. We also construct a simple analytic ansatz that can predict the non-linear behaviour responsible for cellular motion in tissues. Shifting from fluidization to segregation, I next focus on two-dimensional (2D) binary mixtures. I show that a difference in cellular shape or size is insuffcient to induce an emergent interfacial tension, and this leads to large-scale mixing. However, shape disparity can induce a small-scale demixing over a few cell diameters. We report a very similar de-mixing observed in an experimental co-culture of differently shaped Keratinocytes. This can be understood by examining the non-reciprocal energy barriers for neighbor exchanges at the interface, leading to micro-segregation. We next move on to three-dimensional (3D) binary mixtures that have an explicit interfacial tension between two distinct cell types. We find that they can undergo complete segregation, imparting unique geometric properties to cells at the interface. To understand the feedback between interfacial tension and cellular geometry, we develop simple toy models to probe the system\u27s response to perturbations in cellular topology along the interface. Neighbor exchange processes in confluent tissues also involve perturbing the underlying topology with neighboring cells, and therefore are heavily regulated by the cell shape and inhomogeneity in surface tension. In all of the above cases, these local barriers govern the onset of unique collective behavior like-fluidization, microdemixing and novel geometric signatures

Stinging the Predators: A collection of papers that should never have been published

This ebook collects academic papers and conference abstracts that were meant to be so terrible that nobody in their right mind would publish them. All were submitted to journals and conferences to expose weak or non-existent peer review and other exploitative practices. Each paper has a brief introduction. Short essays round out the collection

The Local Theorem for Monotypic Tilings

A locally finite face-to-face tiling T of euclidean d-space E d is monotypic if each tile of T is a convex polytope combinatorially equivalent to a given polytope, the combinatorial prototile of T. The paper describes a local characterization of combinatorial tile-transitivity of monotypic tilings in E d; the result is the Local Theorem for Monotypic Tilings. The characterization is expressed in terms of combinatorial symmetry properties of large enough neighborhood complexes of tiles. The theorem sits between the Local Theorem for Tilings, which describes a local characterization of isohedrality (tile-transitivity) of monohedral tilings (with a single isometric prototile) in E d, and the Extension Theorem, which gives a criterion for a finite monohedral complex of polytopes to be extendable to a global isohedral tiling of space