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The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy
Abstract
A translation is given of Truchet's 1704 paper showing that an infinity of patterns can be generated by the assembly of a single half—colored tile in various orientations. It embodies an early representation of the principles of combinatorial theory and of crystallographic symmetry including color symmetry. Simple rules of the topology of separation and junction are used to extend Truchet's concept of directional choice and, by relaxing symmetry rules, to generate diagrams illustrating field/ground relations, the hierarchy of structural freedom and the origin and nature of structural order and disorder in general.
... In 1704, Sebastien Truchet published a short paper "Mémoire sur les Combinaisons" in which he examined the feasible patterns from a set of square tiles, each split diagonally into two colored triangles (Truchet 1704; Reimann 2009) (Fig. 1). He studied the "graphical treatment of combinatorics", which was largely revolutionary in mathematics at the time (Smith and Boucher 1987). In 1987, C.S. Smith published "The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy", in which he analyzed Truchet's tiling pattern and introduced the now recognized variation of Truchet's original tile (Smith and Boucher 1987). ...
... He studied the "graphical treatment of combinatorics", which was largely revolutionary in mathematics at the time (Smith and Boucher 1987). In 1987, C.S. Smith published "The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy", in which he analyzed Truchet's tiling pattern and introduced the now recognized variation of Truchet's original tile (Smith and Boucher 1987). In this variation, Smith used arcs in diagonal corners, replacing the two triangles of Truchet's original tile. ...
... Sebastian Truchet's original tile: a Truchet's bi-colored tiles, b Possible combinations of two tiles, c Example tiling pattern. Tiles redrawn from (Smith and Boucher 1987) From Graphical Treatment of Combinatorics to Tiling Grammars (Smith and Boucher 1987) ...
This research employs shape grammars to generate variations of the Truchet tile to characterize a two-dimensional and three-dimensional generative tiling system.
... French clergyman Se´bastienSe´bastien Truchet (1657–1729) is best known for two mathematical innovations; the invention of the point system for measuring the size of typographic characters, and his description of a certain type of artistic tiling. In a short paper titled ''Memoir sur les Combinaisons'' published in 1704, Truchet described a ceramic tile he had recently seen consisting of a square divided by a diagonal line between opposite corners into two coloured parts [1], and went on to provide a combinatorial analysis of the many aesthetically interesting patterns that can be created with the four possible orientations of such a tile (Fig. 1, top left). Although Truchet's analysis focussed on structured nonrandom designs, tilings formed by filling a square grid with random integers from {0, 1, 2, 3} and substituting the appropriate tile can still be aesthetically interesting (Fig. 1, bottom left). ...
... Although Truchet's analysis focussed on structured nonrandom designs, tilings formed by filling a square grid with random integers from {0, 1, 2, 3} and substituting the appropriate tile can still be aesthetically interesting (Fig. 1, bottom left). In his 1987 description of Truchet's tiles, metallurgist and historian Cyril Stanley Smith described the tile variation shown inFig. 1 (top right) consisting of two circular 901 arcs with radius equal to half the tile width and centred at opposite corners [1]. Random tilings of this modified tile in its two possible orientations form visually interesting results [2,3]. ...
Spanning tree contours, a special class of Truchet contour based upon a random spanning tree of a Truchet tiling's underlying graph, are presented. This spanning tree method is extended to three dimensions to define a Truchet surface with properties similar to its two-dimensional counterpart. Both contour and surface are smooth, have known minimum curvature and known maximum distance to interior points, and high ratios of perimeter to area and area to volume, respectively. Expressions for calculating contour length, contour area, surface area and surface volume directly from the spanning tree are given.
... Mais TRUCHET fut le premier à tenter l'analyse exhaustive d'un pavage non trivial, à travers la mise en oeuvre d'une méthode combinatoire. Sébastien TRUCHET est cité, mais en une seule ligne, dans le livre de référence consacré à la théorie des pavages [9], impressionnante oeuvre de synthèse de l'ensemble des connaissances théoriques accumulées dans ce domaine peu connu et difficile des mathématiques, mais c'est à Cyril SMITH [17] que revient le mérite d'avoir écrit la première (et seule...) étude récente consacrée aux recherches de TRUCHET sur les pavages , auxquelles on ne s'était plus intéressé depuis les travaux d'un contemporain de TRUCHET, le Père Dominique DOUAT, qui en avait publié le fruit en [ [17]). En effet, TRUCHET considère une seule tuile de deux couleurs séparée par une ligne diagonale, admettant quatre positions possibles, et il assemble ces tuiles pour couvrir une surface tout en faisant varier l'orientation de celles-ci selon certaines règles précises (figures 1 et 2). ...
... Mais TRUCHET fut le premier à tenter l'analyse exhaustive d'un pavage non trivial, à travers la mise en oeuvre d'une méthode combinatoire. Sébastien TRUCHET est cité, mais en une seule ligne, dans le livre de référence consacré à la théorie des pavages [9], impressionnante oeuvre de synthèse de l'ensemble des connaissances théoriques accumulées dans ce domaine peu connu et difficile des mathématiques, mais c'est à Cyril SMITH [17] que revient le mérite d'avoir écrit la première (et seule...) étude récente consacrée aux recherches de TRUCHET sur les pavages , auxquelles on ne s'était plus intéressé depuis les travaux d'un contemporain de TRUCHET, le Père Dominique DOUAT, qui en avait publié le fruit en [ [17]). En effet, TRUCHET considère une seule tuile de deux couleurs séparée par une ligne diagonale, admettant quatre positions possibles, et il assemble ces tuiles pour couvrir une surface tout en faisant varier l'orientation de celles-ci selon certaines règles précises (figures 1 et 2). ...
Résumé. Un problème algorithmique de pavage du plan a fait l'objet d'un concours il y a trois ans. Après avoir présenté ce problème, nous donnons les principales solutions qui nous furent proposées. Le vainqueur désigné fut Rouben TER MINASSIAN. Abstract. Three years ago, an algorithmic problem on tiling of a plane was set as a contest puzzle. After presentating various aspects to the puzzle, we give the main answers received. The winner was Rouben Ter Minassian.
... The sound absorption coefficients are then used as material-data inputs for computer simulations. The simulations use 3D modeled panels that were designed using a generative grammar we developed (Walter et al., 2023), based in the logic of the Truchet Tile (Truchet, 1704;Smith, 1987). ...
... A potential frame for such a TI assembly is given in 2(c). The planar TI assemblies can be classified by Truchet tiles (for example [17]) together with an assembly-rule as a combinatorial model. Each Truchet tile encodes a certain placement of the Versatile Block as given in the first four pictures of Figure [8] and one aperiodic assembly. ...
Topological interlocking is an abstract concept which requires that, given an assembly of blocks with a fixed frame as a constraint, no group of blocks can be removed. We introduce a construction kit based on a block, called Versatile Block, that leads to a wide range of possible topological interlocking assemblies. We show that the combinatorics of assembling copies of this block can be linked to Truchet tiles. We analyze the self assembly behaviour of the block by presenting experimental results based on 3D-printing with inserted magnets and computational experiments using the computer algebra system GAP. Furthermore, we present structural examples demonstrating the functionalities of each block in the construction kit. We investigate different design strategies and discuss possible applications in civil engineering design, considering possible manufacturing with different choices of material such as carbon reinforced concrete.
... We find that there are two such motifs, up to recolourings, that are related by ± 1 4 -turn rotations [20]. These kinds of motifs exhibit or possess a property that Dow has referred to as a 'universal visual matching rule' by analogy with Truchet tilings [20,24,46]. This seems to nicely reflect much about the role of visual reasoning in the aesthetic appeal about mathematical artwork. ...
We pursue an exploration into the interdisciplinarity given by amalgamations of the study of mathematical artwork and the study of automatic sequences.
... A Trouchet tiling is a set of identical square tiles arranged together in different rotations. Named after Trouchet who wrote about them in 1704, they were popularised more recently by [Smi87], especially the top left tile Figure 1. These tilings are widely used in generative art work and graphic design, for example [Kra11], [Car18], and also have been applied in computer graphics [Bro08]. ...
This article describes a new method of producing space filling fractal dragon curves based on a hinged tiling procedure. The fractals produced can be generated by a simple L-system. The construction as a hinged tiling has the advantage of automatically implying that the fractiles produced tessellate, and that the Heighway fractal dragon curve, and the other curves constructed by this method, do not cross themselves. This also gives a new limiting procedure to apply to certain Trouchet tilings. I include the computation of the fractal dimension of the boundary of one of the curves, and describe an algorithm for computing the sim value of the fractal boundary of these curves. The curves produced are well known. The hinged tiling approach is new, as is the algorithm for computing the sim value.
... In 1987, Cyril Smith analyzed the structure of Truchet's tiling and first abstracted them into simple diagonal lines and then into two arcs starting and ending at edge midpoints ( Figure 1B). 4 Smith wrote about the closures that were being formed-circles-and also showed a color-filled example to further highlight these positive and negative, concave and convex patterns 1 ( Figure 1C). ...
Repeating patterns in architecture are utilized in elements at a variety of scales, from a façade to perforated ceilings and wall reliefs to carpeting and tile stonework. The Truchet tiling concept is one means to develop a modular non-repeating pattern. This paper explores some of the basic concepts of Truchet tilings, variations developed, and some current examples of using these methods with digital generation and fabrication methods.
... Such structures are found in the fully-packed loop models of statistical mechanics, with connections to alternating sign matrices, percolation and the six-vertex model [6]. They also resemble Truchet tilings [18]; their classification lies in the field of enumerative combinatorics. ...
Through the hands-on creation of two sashiko pieces of work - a counted thread kogin bookmark and a single running stitched hitomezashi sampler - participants will explore not only the living cultural history of this traditional Japanese needlework but will also experience the mathematics of sashiko in a tangible form, and will take away with them items of simple beauty.
... The complete works of Sébastien Truchet is recapitulated in André and Denis (1999). Regarding recent works, Smith (1987) proposed alternatives for Truchet tiles with only two possible orientations instead of four, resulting to binary coding. Lord and Ranganathan (2006) gave three-dimensional generalizations. ...
This paper is interested in developing a new application in combinatorial analysis, which will be called "combinatorial arabesque". There are many disciplines, which have a relationship with analytic combinatorics such as "Graph theory", "combinatorial optimization" and "Probabilistic combinatorics". However, applying combinatorial analysis on geometric patterns is actually a new topic and there is no previous study on this subject. This topic seems to be a simple composition of geometrical patterns, but it really uses many branches of mathematics such as counting techniques, analysis, and structural algebra related to geometric transformations. The results of this paper is two tenors, the first is mathematical by determining all of the specific kind of motifs we called "Entirely symmetrical motifs". The second tenor is artistic and decorative by constructing beautiful motif from ugly one. Finally, a mathematical definition of beauty degree of a motif is given.
... Truchet showed how an infinity of patterns can arise by repeating a single tile oriented in different ways, providing us with a first representation of the principles of combinatorial theory and crystallographic symmetry [1]. His technique uses simple topological separation and junction rules to generate different patterns. ...
This paper introduces a method to combine structural efficiency and ornamental qualities in funicular shells through the application of a rib pattern based on the principles of Truchet tiles. The inherent algorithmic and iterative nature of form finding processes contributed to improve research in contemporary computational shell design research to the point that the Block Research Group at ETH developed a software for the design of funicular shells named RhinoVault. Furthermore, stereotomy, emerged in the gothic age, underwent a renewed interest in the relation between form and force as demonstrated by Pier Luigi Nervi's works on rib patterns. Starting from the RhinoVault simulation, the shell is first discretized into topologically identical elements based on the funicular polygon of the structure, which are then populated with a combination of a limited set of rib-patterned elements; structural data is used to modulate the pattern expression in each element in terms of local orientation (towards principal stress direction), pattern density and rib height. The combinatorial nature of such pattern interprets the structural data, which modulates its distribution across the shell, yet resisting the degeneration into a replica of pure structural trajectories or parameters expression diagram; this friction is a crucial condition in managing the interplay of force pattern distribution and autonomous architectural expression of ornamental and constructive qualities. The main goal is to provide a design strategy to manage a tessellation and stereotomy algorithmic process able to generate ornamental and structural patterns, challenging the classical structure/ornament dichotomy.
... To scale up the complexity and diversity of origami arrays, we present a strategy using random DNA tilings, which are as easy to construct as periodic arrays but with substantially more control over the complexity of the global patterns. The strategy is guided by stochastic Truchet tiling 29,30 . A Truchet tile has a rotationally asymmetric pattern that is designed to continue into neighbouring tiles. ...
Scaling up the complexity and diversity of synthetic molecular structures will require strategies that exploit the inherent stochasticity of molecular systems in a controlled fashion. Here we demonstrate a framework for programming random DNA tilings and show how to control the properties of global patterns through simple, local rules. We constructed three general forms of planar network-random loops, mazes and trees-on the surface of self-assembled DNA origami arrays on the micrometre scale with nanometre resolution. Using simple molecular building blocks and robust experimental conditions, we demonstrate control of a wide range of properties of the random networks, including the branching rules, the growth directions, the proximity between adjacent networks and the size distribution. Much as combinatorial approaches for generating random one-dimensional chains of polymers have been used to revolutionize chemical synthesis and the selection of functional nucleic acids, our strategy extends these principles to random two-dimensional networks of molecules and creates new opportunities for fabricating more complex molecular devices that are organized by DNA nanostructures.
... In 1704, Truchet investigated the patterns obtainable from a single square tile that was bisected along a diagonal into two uniquely colored isosceles right triangles [5]. In 1987, Smith [4] published an article containing a translation of Truchet's original paper with some commentary and new ideas including the use of a random tiling rather than a structured pattern. Smith also included a variant of the Truchet tile that replaced the triangular segmentation with two quarter-circle arcs, resulting in a tiling that is comprised of an aesthetically pleasing, meandering set of mostly closed curves. ...
A way to create text with Truchet tiles is presented where each letter is represented by a particular grouping of Truchet tiles. The resulting text is rendered in a surprisingly and visually appealing manner within a tiled region.
... A method that is closely related to Taubin's work is Truchet tiles, which was originally introduced by Sebastien Truchet as all possible patterns formed by tilings of right triangles oriented at the four corners of a square [20,21] in a square grid structure. Truchet's triangulations of a grid can be considered a special case of triangulation of quad meshes. ...
... Decorated tiles with simple motifs have been used to enhance the visual appeal of tilings by creating additional patterns. The commonly known Truchet tiles, square tiles decorated with quarter circle arcs, date to paper by Smith [2]. A more complete historical account of these tiles is given by Reimann [3] and Browne [4]. ...
Decorated tiles with simple motifs have been used to enhance the visual appeal of tilings by infusing the underlying tessellation with additional patterns. This paper explores a generalization of Truchet tiles by decorating tiles made from regular polygons with simple Bézier curves and considering more than one arc per side. Examples of the generalized tilings for each of the Archimedean tilings are presented using one and two arcs per side. The tension present between the global irregularity and both the local similarity and positional regularity of the generated curves provides excitement and movement not present in the underlying tessellations.
... The early LLG models include the static mirror (SM, [58]) and static rotator (SR, [42]) models on Z 2 . The former appears to be closely related to bond percolation [39] and the problem of random tiling of the plane with Truchet tiles, also known as hull percolation [29], [54], [55], [57], [59]. Its relation to the formation of polymers and smart kinetic walks has also been noticed by various authors (see e.g. ...
Ph.D. Leonid Bunimovich
... The motif or shape indicated at bottom left was inserted and rotated 45° with each occurrence of heads; the probability is and the distribution is largely homogeneous or even. Despite the simplicity of the generating algorithm, the complexity (information content) of the array exceeds that found in the intricate patterns of Truchet (Smith, 1987) or even the aperiodic tilings of Penrose (1978). Interestingly, the localized areas of the array are far from homogeneous (see outlined area). ...
Formal grammars in design are discussed in regard to such topics as stochastically generated forms, ascribed meaning to graphical data, and the intuitive nature of rule making or breaking in design. An original shape-production system is introduced based on the interplay of stochastic processes and shape-generating algorithms. Proposed is the notion that shape production systems, of one form or another, offer the best prospect for future computer-based aids to designers.
Acoustic materials are widely used for improving interior acoustics based on their sound absorptive or sound diffusive properties. However, common acoustic materials only offer limited options for customizable geometrical features, performance, and aesthetics. This paper focuses on the sound absorption performance of highly customizable 3D-printed Hybrid Acoustic Materials (HAMs) by means of parametric stepped thickness, which is used for sound absorption and diffusion. HAMs were parametrically designed and produced using computational design, 3D-printing technology, and feedstock material with adjustable porosity, allowing for the advanced control of acoustic performance through geometry-related sound absorbing/diffusing strategies. The proposed design methodology paves the way to a customizable large-scale cumulative acoustic performance by varying the parametric stepped thickness. The present study explores the challenges posed by the testing of the sound absorption performance of HAMs in an impedance tube. The representativeness of the test samples (i.e., cylindrical sections) with respect to the original (i.e., rectangular) panel samples is contextually limited by the respective impedance tube’s geometrical features (i.e., cylindrical cross-section) and dimensional requirements (i.e., diameter size). To this aim, an interlaboratory comparison was carried out by testing the normal incidence sound absorption of ten samples in two independent laboratories with two different impedance tubes. The results obtained demonstrate a good level of agreement, with HAMs performing better at lower frequencies than expected and behaving like Helmholtz absorbers, as well as demonstrating a frequency shift pattern related to superficial geometric features.
When tiles decorated to lower their symmetry are joined together, they can form aperiodic and labyrinthine patterns. Such Truchet tilings offer an efficient mechanism of visual data storage related to that used in barcodes and QR codes. We show that the crystalline metal-organic framework [OZn4][1,3-benzenedicarboxylate]3 (TRUMOF-1) is an atomic-scale realization of a complex three-dimensional Truchet tiling. Its crystal structure consists of a periodically arranged assembly of identical zinc-containing clusters connected uniformly in a well-defined but disordered fashion to give a topologically aperiodic microporous network. We suggest that this unusual structure emerges as a consequence of geometric frustration in the chemical building units from which it is assembled.
Several artists, neuroscientists, and art psychologists have investigated the existence of a relationship between perceived motion and beauty in figurative and abstract paintings. In our study, we created stimulus pictures by combining the same matrices, consisting of modular stochastic polygons, to obtain regular (translational symmetry) and irregular (non-symmetry) combinations. Some of these combinations consisted of many small matrices, making it difficult to read the ‘shapes’ of stochastic polygons. Our sample consisted of both art experts and non-art experts. We hypothesised that irregular combinations, with fewer and greater numbers of the same matrices, would have stimulated more perception of motion, complexity and beauty than regular compositions. Results showed that stochastic irregular combinations are generally dynamic, more complex, and more aesthetically pleasing than stochastic regular compositions. Perhaps the greater dynamism of irregular combinations influences beauty evaluation for compositions with stochastic matrices. Research has shown that specific artistic competence influences the assessment of irregular or asymmetrical stimuli as beautiful. Our study, on the other hand, shows that irregular stochastic combinations are more beautiful for both art experts and non-experts.
Symmetry and antisymmetry are resources used by humans since pre-historic times in their artistic creations. In this work we start by introducing antisymmetry and the 17 possible asso- ciated groups. We then present some counts of friezes, built with a particular module, the Truchet tile, which is itself antisymmetric. These counts take into consideration the frieze antisymmetry group and the number of tiles composing the unitary cell which originates the frieze by translation. These Truchet friezes led to the creation of a set of 5 paintings, named Crossings, which illustrate all the possible groups, in an artistic context.
Nel novembre del 2009 il Museo del Louvre decise di affidare a Umberto Eco la direzione di una serie di conferenze da tenere su un argomento di sua scelta. Eco scelse il tema della ‘lista’ conferendo a quella manifestazione il nome di Vertige de la liste a cui seguì un saggio dallo stesso titolo pubblicato da Bompiani. Questa breve nota prende in prestito la stessa tematica che, per via del suo carattere altamente eterogeneo, è in grado di coinvolgere diversi ambiti del sapere senza minimamente accennare ad esaurirsi. L’elenco, strettamente legato alla figura retorica dell’accumulazione, può infatti costituirsi di segni, parole, suoni e di qualsiasi altra forma di rappresentazione utile a instaurare dialoghi tra due o più soggetti. Nello specifico, cercando di impostare lo stesso testo come una ulteriore ‘lista vertiginosa’, si analizzano tre casi, vicini alla rappresentazione dell’architettura, utili ad accentuare la propensione della enumerazione quale complesso dispositivo semiotico mediato dall’esperienza personale che ne interpreta i ripetuti ‘eccetera’: il testo Tentative d’épuisement d’un lieu parisien dello scrittore George Perec, la tavola 4 dell’Architecture Civile dell’architetto-disegnatore Jean Jacques Lequeu e il trattato Methode pour faire une infinité de desseins differens del padre carmelitano Dominique Douat. Tutte liste ‘pratiche’ e al tempo stesso ‘poetiche’ in grado, si spera, di aprire ulteriori territori d’indagine, nonché di far riflettere sulla dilagante attitudine del ‘voler dire tutto’ tipica dei tempi odierni.
Periodic tilings can store information if individual tiles are decorated to lower their symmetry. Truchet tilings - the broad family of space-filling arrangements of such tiles - offer an efficient mechanism of visual data storage related to that used in barcodes and QR codes. Here, we show that the crystalline metal-organic framework [OZn$_4$][1,3-benzenedicarboxylate]$_3$ (TRUMOF-1) is an atomic-scale realisation of a complex three-dimensional Truchet tiling. Its crystal structure consists of a periodically-arranged assembly of identical zinc-containing clusters connected uniformly in a well-defined but disordered fashion to give a topologically aperiodic microporous network. We suggest that this unusual structure emerges as a consequence of geometric frustration in the chemical building units from which it is assembled.
In this paper, we introduce a new topic on geometric patterns issued from Truchet tile, that we agree to call combinatorial arabesque. The originally Truchet tile, by reference to the French scientist of 17th century Sébastien Truchet, is a square split along the diagonal into two triangles of contrasting colors. We define an equivalence relation on the set of all square tiling of same size, leading naturally to investigate the equivalence classes and their cardinality. Thanks to this class notion, it will be possible to measure the beauty degree of a Truchet square tiling by means of an appropriate algebraic group. Also, we define many specific arabesques such as entirely symmetric, magic and hyper-maximal arabesques. Mathematical characterizations of such arabesques are established facilitating thereby their enumeration and their algorithmic generating. Finally the notion of irreducibility is introduced on arabesques.
In 1704, the French priest Sébastien Truchet published a paper where he explored and counted patterns made up from a square divided by a diagonal line into two coloured parts, , now known as a Truchet tile. A few years later, Father Dominique Doüat continued Truchet's work and published a book in 1722 containing many more patterns and further counts of configurations. In this paper, we extend the work introduced by Truchet and Doüat by considering all possible rosettes made up of an m×n array of square or non-square Truchet tiles, or . We then classify the rosettes according to their symmetry group and count all the distinct rosettes in each group, for all possible sizes. The results are summarized in a separate section where we further analyse the asymptotic behaviour of the counts for square arrays. Finally, some applications are shown using two types of square flexagons.
Resumen. En este trabajo presentamos un procedimiento algorítmico para construir conjuntos de baldosas y, dado un entero n, asignar una baldosa a cada clase módulo n. Con esta asignación es posible construir mosaicos que, además de su valor estético, reflejan en su geometría algunas propiedades aritméticas del producto de enteros módulo n. La idea ha sido llevada a la práctica exitosamente con alumnos de bachillerato (16-18 años).
Abstract In this work we present an algorithmic method to construct sets of tiles and, for every integer n, assign a tile to each class modulo n. With this assignment it is possible to construct tilings which, in addition to their esthetical value, reproduce in their geometrical structure some arithmetical properties of the multiplication modulo n. This idea has been successfully put into practice with high-school students (16-18 years old).
Resumo Neste travalho apresentamos um método algorítmico para a construção de conjuntos de lajotas e, para cada inteiro n, atribuir uma lajota a cada classe módulo n. Com esta atribução, é possível construir mosaicos que, além de seu valor estético, refletem em sua estrutura geométrica algumas propriedades da multiplicação módulo n. A idéia foi posta em prática com sucesso com alunos do ensino médio (16-18 anos).
In comparison with the six chapters of the Programming volume [767] of the GuideBooks, this chapter is very long—almost one and a half times longer than the average chapter text, without the plots. The reason for this is threefold: The reader is now familiar with the structure of Mathematica expressions, and we can make full use of this knowledge to write larger programs. Often, one of the simplest ways to check whether a calculation produces the desired result is to look at it graphically for special values of parameters or/and limiting cases. Originally, it was not my intention to discuss a lot of graphical methods and examples. However, because of numerous requests by the students who listened to the original lectures that resulted in this book, I have now included, in my opinion, a sufficient amount of graphics. Apparently many students and colleagues wanted to use Mathematica not only for symbolic calculations, but also (sometimes, unfortunately, only) to produce plots of measured data or sets of data produced by other programs. Visualization of mathematical knowledge is very important for teaching and “understanding” mathematics [94], [835], [597], [228], [364], [438], [649], [361], [45], [446], [428], [269], [226].
We explore the use of Truchet-like tiles (tiles with half area shaded) for single-tone (monochrome) line-based rendering of maps. We borrow concepts from information theory to develop some qualitative and quantitative measures, and use them to discuss four rendering styles which use Truchet-like tiles. We first present a style based on the original Truchet tiles, then we review the tile-based approach used by Inglis and Kaplan for Op Art rendering of 2-color maps, and extend the concept to 3- and 4-color input maps. We present two more concepts capable of rendering maps with up to 6 colors. We highlight some relationships between the four rendering styles, and utilize these relationships to generate Op Art/labyrinth renderings from input maps of up to 6 colors.
Stereotomy, as a historical subject, shows how to build stone/wood vaulted architectural systems of hard spatial complexity, where specific geometrical rules and correspondences set the relationship between system and part. Such a difficult, but useful knowledge for the tailleur de pierre (stone cutter) can be made easily accessible by the experts, thanks to the evolution of the existing three-dimensional modelling software that allows the shape-planning and shape-building process to be checked unambiguously. This essay aims to show the capabilities of computer-modelling techniques when applied on classical stereotomy studies, and to analyze the topological transformation techniques applied on freestone architecture elements. Our working hypothesis is to compare two computer-modelling techniques: direct and indirect modelling, both used to study the stereotomic shape. In the first case the shape rises by consequential protrusion processes and Boolean works on two-dimensional shapes coming from trait géométriques; in the second case the shape rises by modelling processes based on topological principles, that means using volumetric transformation and deformation tools. Indirect modelling enables very complex stereometric shapes to be built and checked through hard conceptual work and easy practical operations. The geometrical conformation of every single ashlar, of a specific vaulted system, will be the result of an appropriate series of simple solids geometrical transformations, producing a topological correspondence.
We present a new approach for stippling by recursively dividing a grayscale image into rectangles with equal amount of ink, then we use the resulting structure to generate novel line-based halftoning techniques. We present four different rendering styles which share the same underlying structure, two of which bear some similarity to Bosch-Kaplan's TSP Art and Inoue-Urahama's MST Halftoning. The technique we present is fast enough for real time interaction, and at least one of the four rendering styles is well-suited for maze construction.
In 1704 Father Sébastien Truchet published an article, ‘Mémoire sur les combinaisons’, that describes his mathematical and artistic investigations into how a simple set of square tiles, each divided by a diagonal into a white half and a black half, can be arranged to form an infinity of pleasing designs. In this paper, we describe how to modify Truchet’s tiles so that a collection of them can be used for halftoning, the reproduction of user-supplied greyscale target images in pure black and white. We do this by allowing the diagonals of the tiles to ‘flex’ or bend at their midpoints in accordance with the brightness of an individual pixel, or a collection of pixels, from the target image. We also present hexagonal variations, a similar scheme for the Truchet-like tiles – each decorated with two quarter-circle arcs centred at opposite corners of the square – proposed by Cyril Stanley Smith in 1987, and an extension that can be applied to all regular and semiregular tilings.
This paper explores methods for colouring Truchet-like tiles, with an emphasis on the resulting visual patterns and designs. The methods are extended to non-square tilings that allow Truchet-like patterns of noticeably different character. Underlying parity issues are briefly discussed and solutions presented for parity problems that arise for tiles with odd numbers of sides. A new tile design called the arch tile is introduced and its artistic use demonstrated.
Tilings and patterns dominate our visual and material world. Cyril Stanley Smith was acutely sensitive to this in every aspect
of his work. In a diversion he rediscovered the Truchet tilings of 1704 and added to their richness and variety.
Formal grammars in design are discussed. A design tool is proposed, consisting of an original design production system based on the interplay of stochastic processes and shape generating algorithms. Graphic designs generated by the system are presented.
We describe a 2 parameter family of polygon exchange transformations
parameterized by points in a square. Whenever the two parameters are
irrational, the polygon exchange has periodic orbits of arbitrarily large
period. We show that for almost all parameters, the polygon exchange map has
the property that almost every point is periodic. However, there is a dense set
of irrational parameters for which this fails. By choosing parameters
carefully, the measure of non-periodic points can be made arbitrarily close to
full measure. These results are powered by a notion of renormalization which
holds in a more general setting. Namely, we consider a renormalization of
tilings arising from the Corner Percolation Model.
Today, characters glyphes are defined as surfaces. This is not new, for they were modelised with compasses and ruler since the Renaissance (e.g. Durer in Germany, Pacioli in Itlay or Tory in France) and with Truchet. Old models are exhibited, the approximation of O's outlines being used as example. Finaly, the today model using Bezier splines is explained. Aujourd'hui, les caractères d'imprimerie sont traités comme des surfaces géométriques. L'idée n'est pas nouvelle puisque la modélisation de leurs contours par le compas et la règle remonte à la Renaissance (avec notamment Pacioli en Italie, Dürer en Allemagne et Tory en France) et a été reprise sous Louis XIV (Truchet, à qui on doit aussi le concept de point typographique). Après avoir cité les principaux modèles anciens, nous montrons notamment comment ils permettaient d'approcher les contours d'un O par des arcs de cercle, ce que l'on fait aujourd'hui par des courbes de Bézier.
The Ruijgrok-Cohen (RC) mirror model (Phys. Lett. A 133, 415 (1988)) of a Lorentz lattice gas, in which particles are reflected by left and right diagonally oriented mirrors randomly placed on the sites of a square lattice, is further investigated. Extensive computer simulations of individual trajectories up to 2{sup 24} steps in length, on a lattice of 65 536{times}65 536 sites, are carried out. This model generates particle trajectories that are related to a variety of kinetic growth and smart'' (nontrapping) walks, and provides a kinetic interpretation of them. When all sites are covered with mirrors of both orientations with equal probability, the trajectories are equivalent to smart kinetic walks that effectively generate the hulls of bond percolation clusters at criticality. For this case, 10{sup 6} trajectories were generated, yielding with unprecedented accuracy an orbit size-distribution exponent of {tau}=2.1423{plus minus}0.0003 and a fractal dimension of {ital d}{sub {ital f}}=1.750 47{plus minus}0.000 24 (without correcting for finite-size effects), compared with theoretical predictions of 15/7=2.142 857. . . and 7/4, respectively. When the total concentration of mirrors {ital C} is less than unity, so that the trajectories can cross, the size distribution of the closed orbits does not follow a power law, but appears to be described by a logarithmic function.
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