Heesch numbers of unmarked polyforms

A shape's Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of nontiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds

Grid-based Decorative Corners

Abstract I explore a space of geometric, decorative corner designs based on paths through a square grid. I discuss the problems of enumerating corners of a given size efficiently, and exploring them interactively in software. I then impose a higher-level connectivity constraint on corners and discuss the effect of this constraint on the mathematical and aesthetic properties of corner designs

The architecture of RNA polymerase fidelity

The basis for transcriptional fidelity by RNA polymerase is not understood, but the 'trigger loop', a conserved structural element that is rearranged in the presence of correct substrate nucleotides, is thought to be critical. A study just published in BMC Biology sheds new light on the ways in which the trigger loop may promote selection of correct nucleotide triphosphate substrates. See research article http://www.biomedcentral.com/1741-7007/8/5

A review of wetting versus adsorption, complexions, and related phenomena: the rosetta stone of wetting

This paper reviews the fundamental concepts and the terminology of wetting. In particular, it focuses on high temperature wetting phenomena of primary interest to materials scientists. We have chosen to split this review into two sections: one related to macroscopic (continuum) definitions and the other to a microscopic (or atomistic) approach, where the role of chemistry and structure of interfaces and free surfaces on wetting phenomena are addressed. A great deal of attention has been placed on thermodynamics. This allows clarification of many important features, including the state of equilibrium between phases, the kinetics of equilibration, triple lines, hysteresis, adsorption (segregation) and the concept of complexions, intergranular films, prewetting, bulk phase transitions versus “interface transitions”, liquid versus solid wetting, and wetting versus dewetting.Seventh Framework Programme (European Commission) (Grant FP7-NMP-2009-CSA-23348-MACAN

A chiral aperiodic monotile

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.Comment: 23 pages, 12 figure

An aperiodic monotile

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to "triskelion", added the name "turtle", added references, new H7/H8 rules (Fig 2.11), talk about reflection