Semi-regular Tessellations Research Articles

A digital distance on the kisrhombille tiling.

The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.

New quantum surface codes from semi-regular tessellations

New quantum surface codes from semi-regular tessellations

Fracture toughness of semi-regular lattices

Previous studies have shown that the kagome lattice has a remarkably high fracture toughness. This architecture is one of eight semi-regular tessellations, and this work aims to quantify the toughness of three other unexplored semi-regular lattices: the snub-trihexagonal, snub-square and elongated-triangular lattices. Their mode I fracture toughness was obtained with finite element simulations, using the boundary layer technique. These simulations showed that the fracture toughness KIc of a snub-trihexagonal lattice scales linearly with relative density ρ̄. In contrast, the fracture toughness of snub-square and elongated-triangular lattices scale as ρ̄1.5, an exponent different from other prismatic lattices reported in the literature. These numerical results were then compared with fracture toughness tests performed on Compact Tension specimens made from a ductile polymer and produced by additive manufacturing. The numerical and experimental results were in excellent agreement, indicating that our samples had a sufficiently large number of unit cells to accurately measure the fracture toughness. This result may be useful to guide the design of future experiments.

A Novel Design Method of Deployable Semi-Regular Tessellated Surfaces with Plano-Spherical Units

The design of movable systems gives an opportunity to create transformable designs which respond to the environmental, functional, cultural, and aesthetical needs of today's architecture. This paper proposes a method for designing a family of deployable structures which can be applied to semi-regular tessellated planar surfaces such as roofs, walls, and shading devices. The generated modular approach and adaptability provides a wide usage area and various combinations for these designs. The regular convex polygon modules are designed as a network of the triangular units. The triangular unit is designed using Bennett's overconstrained plano-spherical linkage topology. The polygonal modules are assembled to each other in one-uniform semi-regular tessellations. The assembly of adjacent regular convex polygons in each tessellation is examined to find a proper solution for no collision during deployment and to properly fit a surface without any gaps or overlaps in the deployed position. The assembly method for creating 1-DoF deployable surfaces and mobility calculations for a unit, the polygonal modules, and the assemblies are computed, and motion studies are demonstrated with CAD models and exemplified for a square module for motion tests in a prototype.

Unveiling students’ explorations of tessellations with Scratch through mathematical aesthetics

This study investigates sixth-grade students’ explorations of tessellations with Scratch through the mathematical aesthetics. For this purpose, the study analyzed the students’ explorations by considering the roles of mathematical aesthetics identified by Sinclair (2004): generative, evaluative, and motivational role. Six middle-school students were asked to create regular and semi-regular tessellations by using Scratch and interviewed individually about the investigation of tessellations. The findings indicated that the students’ generative aesthetic responses to mathematical tasks were most evident in their explorations with Scratch. The students recognized the aesthetic value of mathematical entities while working on tessellations. They never gave up studying on tessellations with Scratch and did not need any external motivation to complete the tasks. Implications for research and practice for teaching are drawn.

Mechanical Properties of Semi-Regular Lattices

The mechanical properties of seven semi-regular lattices were derived analytically for in-plane uniaxial compression and shear. These analytical expressions were then validated using Finite Element simulations. Our analysis showed that one topology is stretching-dominated; two are stretching-dominated in compression but bending-dominated in shear; and four are bending-dominated. To assess their potential, the properties of these seven semi-regular topologies were compared to regular lattices. We found the elastic buckling strength of the stretching-dominated semi-regular tessellation to be 43% higher than a regular triangular lattice. In addition, three of the four bending-dominated semi-regular topologies had a higher elastic modulus than a regular hexagonal lattice. In fact, one of these bending-dominated topologies was 85% stiffer and 11% stronger than a hexagonal lattice. This topology would be ideal for applications requiring a high stiffness and high energy absorption.

Chemical engineering of quasicrystal approximants in lanthanide-based coordination solids

Tessellation of self-assembling molecular building blocks is a promising strategy to design metal-organic materials exhibiting geometrical frustration and ensuing frustrated physical properties. Appearing in two-dimensional quasiperiodic phases, tilings consisting of five-vertex nodes are regarded as approximants for quasicrystals. Unfortunately, these structural motifs are exceedingly rare due to the complications of acquiring five-fold coordination confined to the plane. Lanthanide ions display the sufficient coordinative plasticity, and large ionic radii, to allow their incorporation into irregular molecule-based arrays. We herein present the use of ytterbium(II) as a five-vertex node in a two-dimensional coordination solid, YbI2(4,4′-bipyridine)2.5. The semi-regular Archimedean tessellation structure verges on quasicrystallinity and paves the way for lanthanide-based metal-organic materials with interesting photonic and magnetic properties.

On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. We revisit this problem by using some well-known formulations concerning regular hexagons. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. Archimedean {3,122} semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Efficiency remains satisfactory, as we show that, by producing the proposed irregular hexagon sensors from the same wafer as a regular hexagon, we can obtain almost the same SE.

Semiregular Tessellations with Pattern Blocks

Pattern blocks are multifunctional instructional tools with a variety of applications in various strands of mathematics (number sense, geometry, measurement, algebra, probability). The six pattern blocks are an equilateral triangle (green), a blue rhombus, an isosceles trapezoid (red), a regular hexagon (yellow), a square (orange), and a white rhombus. The sides of all pattern blocks are congruent, considered to be 1 unit in length for this article. Photograph 1 depicts a wall painting with squares and rhombuses found in Jersey City, New Jersey.

A quest for 2D lattice materials for actuation

In the last two decades, most of the studies in shape morphing technology have focused on the Kagome lattice materials, which have superior properties such as in-plane isotropy, high specific stiffness and strength, and low energy requirement for actuation of its members. The Kagome lattice is a member of the family of semi-regular tessellations of the plane. Two fundamental questions naturally arise: i-) What makes a lattice material suitable for actuation? ii-) Are there other tessellations more effective than the Kagome lattice for actuation? The present paper tackles both questions, and provides a clear answer to the first one by comparing an alternative lattice material, the hexagonal cupola, with the Kagome lattice in terms of mechanical/actuation properties. The second question remains open, but, hopefully easier to challenge owing to a newly-discovered criterion: for an n-dimensional (n=2,3) in-plane isotropic lattice material to be suitable for actuation, its pin-jointed equivalent must obey the generalised Maxwell’s rule, and must possess M=3(n−1) non strain-producing finite kinematic mechanisms.

On the behavior of growth of polygons in semi-regular hyperbolic tessellations

In this work we consider tessellations (or tilings) of the hyperbolic plane by copies of a semi-regular polygon with alternating angles and we study the behavior of the growth of the polygons, edges, and vertices when the distance increase from a fixed initial polygon.

Nonagons in the Hagia Sophia and the Selimiye Mosque

One of the classical topics in geometry is the ruler and compass construction of good approximations of the regular nonagon. We propose a method to choose a desired error of approximation, based on new linear third-order recurrence relations. It is related to patterns shown on the balustrade of minbar of Selimiye Mosque in Edirne (Turkey, 1569–1575), where apparently regular nonagons are placed over a net of regular hexagons. This kind of ornament also occurs in Hagia Sophia in the minbar, in freezes stucco carvings and in window grilles. In Medieval Islamic art and architecture the use of the nonagon is not frequent, but it is remarkable that this particular grid of interlocked nonagons and hexagons appears in the decorations of the works of Ottoman architect Mimar Sinan. The pattern is present in his masterpiece, the Selimiye Mosque, in the Hagia Sophia and in other works in the Istanbul city. Looking at practical ways for the construction of the pattern, we provide simple procedures to obtain angles close to 40° that could have been useful for a craftsman to realize the nonagonal geometric designs. In particular, almost regular nonagons are constructed using some elementary shapes that are related with semi-regular tessellations. We compare the patterns obtained through theoretical considerations to those displayed in the examples given above. Several hypotheses are proposed for the practical construction of the interlocked hexagonal patterns for nonagons.

Rainbow tetrahedra in Cayley graphs

Let Γn be the complete undirected Cayley graph of the odd cyclic group Zn. Connected graphs whose vertices are rainbow tetrahedra in Γn are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs G of largest degree 6, asymptotic diameter |V (G)| and almost all vertices with degree: (a) 6 in G; (b) 4 in exactly six connected subgraphs of the (3, 6, 3, 6)-semi-regular tessellation; and (c) 3 in exactly four connected subgraphs of the {6, 3}regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations.

Gauge theories, tessellations & Riemann surfaces

We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite graphs on the torus as well as the Platonic and Archimedean solids on the sphere. On higher genus they give rise to intricate patterns. Special attention will be paid to the master space and the moduli space of vacua of the gauge theory and to how their geometry is determined by the tessellations.

Quantifying membrane permeability of amphotericin B ion channels in single living cells

Recently, the structure–function relationships between amphotericin B (AmB) and ergosterol have been solved using synthetic techniques that require a mycosamine-mediated direct binding interaction between AmB and ergosterol to form AmB ion channels. However, studies to directly probe the AmB-induced membrane permeability changes have not been conducted. In the present work, we investigate the following fundamental question: does AmB induce concentration- and time-dependent permeability changes across ergosterol-containing membranes? Herein, we employ fluorescent dyes of known average diameter to quantify the diameters of AmB ion channels. In addition, we take a single-particle tracking approach to define the intracellular microrheology in the absence and presence of AmB ion channels. Present results show that increasing AmB concentration tends to increase the preferential accumulation of AmB ion channels in the presence of the excess membrane-embedded ergosterol. We found that AmB induces time-dependent membrane permeability; increases approaching 50% in both the velocity fluctuations and diffusion coefficients of vesicles occur on the same time scale as the efflux of potassium ions (≅30min). Furthermore, we propose a two-dimensional, semi-regular tessellation model to geometrically assess the pore size of the AmB ion channels in response to the AmB dose. This approach offers one possibility for the design of AmB ion channels with tunable aqueous pore size, which could provide an opportunity to replace damaged membrane water channels of the aquaporin family in future applications.

ON THE DENSEST PACKIN OF DISCS OF RADIUS 1 AND .2 -1

The main goal of this paper is to establish the long-sta di g conjecture that in the Euclidea plane no arrangeme t of discs of radius 1 and .2 -1 ca have larger packi g de sitythan that of the set of incircles of the semiregular (Archimedean)tessellation (8, 8, 4)

A semiregular image grid

This paper proposes and describes a new image grid, a grid that in some respects can be called pseudo-random. The grid is one of the semiregular tessellations, and is thus reasonably easy to handle. It is recursively decomposable. The notations and formulae necessary for using the grid are given, together with a simple resampling scheme. The new grid has no strong “principal directions”, thus it is reasonably rotation independent (in contrast to the square grid). The grid is mainly intended for computer graphics applications, where the appearance of a low-resolution image to a human is important. However, it could also be interesting to investigate the effects of standard morphological operations in this grid.

Continuous Transformations of Subdivided Periodic Surfaces

A unified morphological method for generating subdivisions of periodic surfaces is presented. The surfaces are characterized by symmetry, frequency, and the topology of subdivision. An open-ended hyper-cubic lattice is suggested as the structure space within which the subdivided surfaces can transform continuously from one to another by varying different independent parameters. For the purposes of illustration, eight parameters are specified to demonstrate an 8-dimensional model for the structure space. Additional types of subdivisions, and other geometric or physical parameters encountered in building, extend this space to a higher-dimensional one which acts as a systematic framework for generating a large variety of surface structures and their transformations. Geodesic spheres are special cases within this continuum which includes plane surfaces and the hyperbolic tessellations. An infinite class of semi-regular hyperbolic tessellations analogous to the Archimedean polyhedra and plane tessellations are presented, and higher frequency subdivisions of hyperbolic tessellations are suggested.

Tessellation graph characterization using rosettas

The graphs associated with regular and semi-regular plane tessellations are characterized in terms of certain algebraic structures associated with them. Each of the graphs has associated with it a rosetta, a configuration of lattice points and colored edges which is repeated throughout the plane.

Semi-regular tessellations

A semi-regular tessellation is one consisting of regular polygons of the same length of side, with the same ‘behaviour’ at each vertex. By this we mean that the polygons appear in the same order (though different senses are allowed) at each vertex. The three regular tessellations are for convenience included here as special cases of semi-regular ones. An example of a semi-regular tessellation is that with triangle–triangle–square–triangle–square in cyclic order, at each vertex. A shorthand notation (‘Schläfli symbol’) for this is 3.3.4.3.4.