Three-dimensional Tessellations Research Articles

Augmented Aztec bipyramid and dicube tilings

We consider the enumeration of dicube tilings, where each tiling represents a three-dimensional tessellation of a polycube using dicubes. While the enumeration of domino tilings of polycubes like the Aztec diamond and the augmented Aztec diamond is well studied, we focus on the three-dimensional analogue, the augmented Aztec bipyramid. This polycube consists of unit cubes and resembles a Platonic octahedron. In this paper, we find a bijection between dicube tilings of the augmented Aztec bipyramid and three-dimensional Delannoy paths, and use this correspondence to determine the number of dicube tilings of the augmented Aztec bipyramid.

Three-Dimensional Mobile Assemblies Based on Threefold-Symmetric Bricard Linkages

Abstract Multiple mobile assemblies have been created based on two-dimensional tessellations of linkages for deployable structures. However, few three-dimensional tessellations of linkages have been created, especially mobile assemblies with bifurcation. Here, we proposed four types of mobile assemblies of kaleidocycles, a special type of threefold-symmetric Bricard linkages, based on cubic cellulation and symmetry. Kinematic analysis of them is carried out based on the matrix method and numerical method. Two assemblies have bifurcations with two motion paths following cuboid symmetry and tetrahedral symmetry, respectively. Meanwhile, the other two have one motion path with a single degree-of-freedom. The designing process facilitates the creation of new mobile assemblies under symmetry, and the four assemblies have the potential application for designing metamaterials.

From three-dimensional tessellations to lightweight filling materials for additively manufactured structures: Concept, simulation, and testing

To improve the structural efficiency and reduce costs, most additively manufactured parts are printed as thin shells filled with a lightweight cellular material. The filling material reacts to secondary stresses and provides distributed support for the load-carrying outer shell. In a recent publication, the authors have proposed a two-step method to design lightweight filling metamaterials that are intrinsically strong and stiff. Conceptually, the space is first divided into repetitive volumes according to known three-dimensional tessellation schemes. The tessellation is then replaced by a kinematically rigid wireframe with beams along the edges and across the faces of the native volumes. Elaborating on that idea, the present paper pursues three objectives: (a) show the variety of material designs that derive from the tessellation-wireframe approach; (b) characterize the materials through finite element analyses on full-scale models and compare them with former predictions based on scaling and homogenization techniques; (c) validate the numerical results against experimental tests on a selection of prototype structures. Good correlation is revealed between theoretical predictions, computational models, and experimental data.

Mechanical design and modelling of lightweight additively manufactured lattice structures evolved from regular three-dimensional tessellations

To lower costs and improve efficiency, most additively manufactured parts are printed as thin shells filled with a lightweight cellular material, which reacts to secondary stresses and provides distributed support to the load-carrying outer casing. This paper proposes a two-step method to design filling metamaterials that are intrinsically strong, stiff and lightweight. First, the space is divided into repetitive volumes according to available three-dimensional tessellation schemes. The tessellation is then transformed into a trabecular wireframe by converting each unit volume into a kinematically-rigid open cell with edge beams and bracings. Dimensional analysis allows the lattice structures to be characterized mechanically and elastically with a finite number of simple computational analyses. The paper shows that the properties of metamaterials with triangular, square and hexagonal prismatic cells compare favourably with state-of-the art porous materials like foams and honeycombs.

Application of Tessellation in Architectural Geometry Design

Tessellation plays a significant role in architectural geometry design, which is widely used both through history of architecture and in modern architectural design with the help of computer technology. Tessellation has been found since the birth of civilization. In terms of dimensions, there are two- dimensional tessellations and three-dimensional tessellations; in terms of symmetry, there are periodic tessellations and aperiodic tessellations. Besides, some special types of tessellations such as Voronoi Tessellation and Delaunay Triangles are also included. Both Geometry and Crystallography, the latter of which is the basic theory of three-dimensional tessellations, need to be studied. In history, tessellation was applied into skins or decorations in architecture. The development of Computer technology enables tessellation to be more powerful, as seen in surface control, surface display and structure design, etc. Therefore, research on the application of tessellation in architectural geometry design is of great necessity in architecture studies.

Development of teaching materials to learn crystallographic symmetry

Authors have developed proper polyhedron models to enable people to learn the concept of three-dimensional symmetry. Touching and operating the symmetry elements of the proper polyhedron enables people to understand symmetry. In this study, authors made three-dimensional tessellation models. Certain polyhedra can be stacked in a regular periodic pattern to fill three-dimensional space completely. Figures show our models. The cube (Fig. (a)) is the only regular polyhedron to fill three-dimensional space completely. The cube is a Voronoi region of the simple cubic lattice (sc). The truncated octahedron (Fig. (b)) is the only Archimedean solid to fill three-dimensional space completely. The truncated octahedron is a Voronoi region of the body-centered cubic lattice (bcc). The rhombic dodecahedron (Fig. (c)) is the only Catalan solid (or Archimedean dual) to fill three-dimensional space completely. The rhombic dodecahedron is a Voronoi region of the face-centered cubic lattice (fcc). Figs. (a), (b), and (c) show three kinds of their aggregate respectively. In each of left-hand aggregate, there is a two-fold rotational axis along a vertical direction. In each of central aggregate, there is a three-fold rotational axis along a vertical direction. In each of right-hand aggregate, there is a four-fold rotational axis along a vertical direction. Fig. (d) is a nontrivial polyhedron to fill three-dimensional space completely. The external shape of the polyhedron was designed as a tree shape. We call such a model three-dimensional Escher shape (3DES) [1]. This can be stacked in a regular periodic pattern too.

Influence of voids on damage mechanisms in carbon/epoxy composites determined via high resolution computed tomography

A multi-scale computed tomography (CT) technique has been used to determine the material structure and damage mechanisms in hydrostatically loaded composite circumferential structures. Acoustic emission sensing was used to locate macroscopically regions of high damage under load to inform the computed tomography. The resultant images allow direct three-dimensional analysis of voids, fibre breaks and cracking, for which a high level of confidence can be placed in the results when compared to other indirect and/or surface-based methods. Ex situ analysis of loaded samples revealed matrix cracking in the longitudinally wound plies, whilst fibre breaks were observed in the circumferentially wound plies. The matrix cracking within the longitudinally wound plies is shown to interact directly with intralaminar voids. The correlation of voids with fibre breaks in the circumferentially wound plies is less distinct. A three-dimensional tessellation technique was used to analyse the spatial distribution of the voids and to compare with single fibre break locations. Whilst there was no first order correlation between fibre break densities and void volume fractions or void dimensions, a distinct correlation was found between voids and nearest neighbouring fibre breaks, where 2.6-5 times more fibre breaks occurred immediately adjacent to a void than would be expected for randomly distributed breaks. (C) 2013 Elsevier Ltd. All rights reserved.

X-ray CT observation of a “mosaic-tiling” structure and a “worm-like” structure in the ternary polymer blends

Two novel phase-separation morphology are reported in the ternary polymer blends of poly(propylene)/nylon-12/poly(lactic acid) by utilizing a high-contrast X-ray computerized tomography. One is a “mosaic-tiling” structure, where each of three components forms equally-sized droplets in a three-dimensional tessellation manner. The other morphology is a “worm-like” structure. Two components of the three form successive droplets like stripe socks. These results strongly suggest the wide variety of the phase-separation morphologies in ternary polymer blends.

One cognitive psychologist’s quest for the structural grounds of music cognition.

ABSTRACT-Here, I describe how I came to believe that the musical scales and tonal systems of diverse human cultures reflect both (a) underlying, universal principles of cognition, including a modality-general time-distance law, and (b) underlying cognitive structures corresponding to geometrically regular helical and toroidal configurations in higher-dimensional spaces. I review supportive evidence obtained, first, by means of methods (which I and my associates developed at the Bell Labs) of multidimensional scaling and of synthesizing tones that form the height-suppressed chroma circle and, second, by means of the probe-tone method (which Krumhansl and I developed at Stanford) for quantifying tonal hierarchies. I argue that two quite independent kinds of considerations-acoustical and abstract structural-converge in establishing the perceptual-cognitive optimality of the extended (12-tone) diatonic system. I then venture some speculations about the sources of the emotional power of music. At the end, I append an annotated list of my former students, coworkers, and associates who contributed to the work described here. PRELIMINARY REMARK Since retiring from Stanford and closing my psychological laboratory some dozen years ago, I have focused my professional endeavors on thinking and writing about more theoretical and philosophical issues of cognitive science (see Shepard, 2001, 2004, 2008). Accordingly, instead of presenting any new empirical findings of my own, I shall attempt to do two things: The first is to offer some reminiscences about how I have off and on sought to extend my general approach to cognitive science on to the investigation, specifically, of music perception and cognition. The second is to provide a relatively coherent overview of what I retrospectively regard as the most significant conclusions to be drawn from the work that my students and associates (at the Bell Labs and at Stanford) have carried out relating to auditory perception and music cognition. THE EMERGENCE OF MY INTERESTS IN SCIENCE, GEOMETRY, AND MUSIC My early scholastic record was not promising - from the first grade (which I was required to repeat) through my freshman year of college (from which I was required to take a two-quarter leave before being granted a probationary re-entry). I was always fascinated by things mechanical and geometrical and loved to tinker, to draw, and to explore the wonderful sounds produced by depressing various combinations of keys on the piano. But I was typically unable to tear myself away from these interests to undertake the less interesting work assigned by my teachers. I found the obligatory elementary school music class to be wholly devoid of intellectual content, and I was too shy and disdainful to join in the singing of what I regarded as silly songs. All of this changed following my probationary period at Stanford, when I was finally permitted to register for more advanced courses whose subject matters - particularly in science, math, and philosophy - did finally inspire me to work toward conceptual mastery and to explore various implications, extensions, and applications on my own. (These included the construction and study of two- and three-dimensional tessellations of constant curvature and of three- and four-dimensional regular polytopes.) I also began attending the Tuesday afternoon organ concerts in Stanford's Memorial Church (by Stanford's organist Herbert Nanny and by other, visiting organists). I was immediately enthralled by the architectonic grandeur of Bach's preludes, toccatas, and fugues and by the harmonically rich and splendorously driving 19th- and 20th-century organ music of Franck, Vierne, Widor, Dupre, and others of the French school. Later, while a graduate student in psychology at Yale, I became spellbound by the late string quartets of Beethoven and, soon thereafter, the quartets of Bartok. At an abstract level, it was the structural elegance and symmetries that appealed to me both in such geometrical objects as regular tessellations and polytopes and in contrapuntal music. …

Illustrating Abu al-Wafā' Būzjānī: Flat Images, Spherical Constructions

This article analyzes two-dimensional images documented in the work of fourth/tenth century mathematician Abu al-Wafā' Būzjānī, which appear in his treatise, On Those Parts of Geometry Needed by Craftsmen. These images record three-dimensional tessellations of the sphere, geometric constructions, which may have served as a basis for architectural monuments and the interior design of domes in Islamic Iran. Such designs were very likely the result of collaborations among mathematicians and artists. As explained here, the construction of the icosahedron on a sphere, as presented in Būzjānī's treatise, is not mathematically correct. But the construction of the spherical dodecahedron is precise. Būzjānī's studies of the sphere as a three-dimensional form deserve further consideration as an important component in the development of mathematical thinking.

Spatiotemporal Analysis of Surface Changes by Tessellations

This paper develops a method for analyzing the change of surface, a continuous function defined over the two‐dimensional space. The surface change is represented by a scalar field in a spatiotemporal region. The region is divided into subregions called hills and waves which are based on the local form of the surface. They are both three‐dimensional tessellations of the spatiotemporal region, and provide an overview of the surface change. To summarize the tessellations, four operations are proposed: section, projection, merging, and graph representation. They extract useful information from tessellations to describe the structure of surface change. To test the validity of the method, the change of a retail cluster in Shinjuku and Shibuya area in Tokyo is analyzed. The empirical study yields some interesting findings that help us understand changes in the spatial structure of retailing.

Geometric Measures for Random Curved Mosaics ofRd

This paper deals with stationary random mosaics of Rd with general cell shapes. As geometric measures concentrated on the i-skeleton (i = 0, 1,…,d) the i-dimensional surface area (volume) measure and (i — 1) different curvature measures are chosen. The corresponding densities are calculated as well as for the mosaics and their superpositions in terms of mean cell parameters and mean cell numbers. This leads to various relations between the characteristic which are applied, in particular, to two- and three-dimensional tessellations. A comparison with known formulas for mosaics with convex cells in R2 and R3 is given.