Tiling

Some mathematical problems are simple to describe, however very difficult to solve. Integer factorization (e.g. 28=2x2x7) is one of them; apparently simple, it is exploited in the widely used RSA cryptography because there is no known method for a fast solution on a classical computer.
This project utilizes mathematical hard-to-solve properties related to integer factorization in order to add another level of unpredictability to the generative artwork.
Basically, it transforms any number (n) into an image made of exactly n tiles, attempting to capture the uncertainty and complexity embedded in the natural number sequence. Each integer is decomposed into its prime factors and the resulting structure is used for both an iterative fragmentation process and for coloring the resulting tessellation (see artwork example). By mathematical construction, the created image series has infinite unique variations, while preserving a common style.
The artwork also attempts to match shapes and colors in order to highlight the “numbers’ character”, by connecting color hue and intensity to specific integer properties (e.g. abundancy index). The result is that different types of integers, such as primes, stand out with their identity from the overall succession.
In addition to the variations produced by each integer within the sequence, several parameters can be tuned to generate many different series, which is typical of Generative Art.

This document presents an evolved Girih tile set beyond the five Girih Tiles popularized by Lu and Steinhardt. Additional tiles have been presented in multiple papers before, but here, a wider set is officially defined. I refer to this as the evolved tile set or evolved Girih tile set. It includes the adapter tiles, which also have been presented before, but not defined as a group or part of a tile set. Adapter tiles differ from the five Girih Tiles as the latter only have equilateral sides. This paper presents one category of adapters containing tiles having at least one side with the length of Φ. The aim is to establish non-equilateral sides as a necessary concept to enable tiling of a greater number of existing Islamic geometric patterns. This opens up for future introductions of other categories of non-equilateral tiles.

Islamic geometric patterns can often be decomposed into a set of modular equilateral tiles, each decorated with a small motif, which assemble to form patterns. As the floodgate for tiles with additional side lengths was opened in the previous Bridges paper (Adapter Tiles Evolves the Girih Tile Set) in which the Φ-category was defined, it is time to examine another non-equilateral category-the Short category, which include a new edge length. While many historical 5-fold patterns can be tiled in two ways, only one way has been used to replicate them. With the introduction of Short tiles, the other way is now possible. I argue that the Short category provides a better way to replicate some of the historical patterns. Together with categories not yet published, the Short tiles play a crucial role when creating patterns, especially historical patterns.

This chapter examines the relationship between mathematics and textile knot practice, i.e., how mathematics may be adopted to characterize knotted textiles and to generate new knot designs. Two key mathematical concepts discussed are knot theory and tiling theory. First, knot theory and its connected mathematical concept, braid theory, are used to examine the mathematical properties of knotted textile structures and explore possibilities of facilitating the conceptualization, design, and production of knotted textiles. Through the application of knot diagrams, several novel two-tone knotted patterns and a new material structure can be created. Second, mathematical tiling methods, in particular the Wang tiling and the Rhombille tiling, are applied to further explore the design possibilities of new textile knot structures. Based on tiling notations generated, several two- and three-dimensional structures are created. The relationship between textile knot practice and mathematics illuminates an objective and detailed way of designing knotted textiles and communicating their creative processes. Mathematical diagrams and notations not only reveal the nature of craft knots but also stimulate new ideas, which may not have occurred otherwise.

I was curious what would happen if I applied the double crossing on the thin (Dia) and thick (Rhomb) rhombus from Penrose P3 tile set.

Then I split the crossings into separate tiles and came up with two non-midpoint sets. At the time I didn't know I later would define these as a new category - the Naro category (a paper on this will be published in the future).

I moved on to split the crossings into having only one line each and thus breaking the edge rule of the polygon in contact method. I arrived at four sets where one already have been covered by Castera in his research about X-tiles.

This chapter examines the relationship between mathematics and textile knot practice, i.e., how mathematics may be adopted to characterize knotted textiles and to generate new knot designs. Two key mathematical concepts discussed are knot theory and tiling theory. First, knot theory and its connected mathematical concept, braid theory, are used to examine the mathematical properties of knotted textile structures and explore possibilities of facilitating the conceptualization, design, and production of knotted textiles. Through the application of knot diagrams, several novel two-tone knotted patterns and a new material structure can be created. Second, mathematical tiling methods, in particular the Wang tiling and the Rhombille tiling, are applied to further explore the design possibilities of new textile knot structures. Based on tiling notations generated, several two- and three-dimensional structures are created. The relationship between textile knot practice and mathematics illuminates an objective and detailed way of designing knotted textiles and communicating their creative processes. Mathematical diagrams and notations not only reveal the nature of craft knots but also stimulate new ideas, which may not have occurred otherwise.

Evolutionary studies usually assume that the genetic mutations are independent of each other. However, that does not imply that the observed mutations are independent of each other because it is possible that when a nucleotide is mutated, then it may be biologically beneficial if an adjacent nucleotide mutates too. With a number of decoded genes currently available in various genome libraries and online databases, it is now possible to have a large-scale computer-based study to test whether the independence assumption holds for pairs of adjacent amino acids. Hence the independence question also arises for pairs of adjacent amino acids within proteins. The independence question can be tested by considering the evolution of proteins within a closely related sets of proteins, which are called protein families. In this thesis, we test the independence hypothesis for three protein families from the PFAM library, which is a publicly available online database that records a growing number of protein families. For each protein family, we construct a hypothetical common ancestor, or consensus sequence. We compare the hypothetical common ancestor of a protein family with each of the descendant protein sequences in the family to test where the mutations occurred during evolution. The comparison yields actual probabilities for each pair of amino acids changing into another pair of amino acids. By comparing the actual probabilities with the theoretical probabilities under the independence assumption, we identify anomalies that indicate that the independence assumption does not hold for many pairs of amino acids.

Empirical program optimizers estimate the values of key optimization parameters by generating different program versions and running them on the actual hardware to determine which values give the best performance. In contrast, conventional compilers use models of programs and machines to choose these parameters. It is widely believed that model-driven optimization does not compete with empirical optimization, but few quantitative comparisons have been done to date. To make such a comparison, we replaced the empirical optimization engine in ATLAS (a system for generating a dense numerical linear algebra library called the BLAS) with a model-driven optimization engine that used detailed models to estimate values for optimization parameters, and then measured the relative performance of the two systems on three different hardware platforms. Our experiments show that model-driven optimization can be surprisingly effective, and can generate code whose performance is comparable to that of...

This paper proposes a family of origami tessellations called extruded Miura-Ori, whose folded state lies between two parallel planes with some faces on the planes, potentially useful for folded core materials because of face bonding. An extruded Miura-Ori is obtained by cutting Miura-Ori apart along the edges and face diagonals before inserting the extrusion of the section edges. We compute the extrusion direction to obtain a valid extruded Miura-Ori. The extruded Miura-Ori usually has three valid states. We analyse the third state (final folded state) in depth to show that a continuous family of parameters can produce origami tessellations that can completely tile the top and bottom planes.

This is my third paper submitted to the Bridges Conference. Unfortunately it was rejected, and the reasons given by the reviewers was perfectly valid. The case being that this topic is too big to fit into the page limitation of eight pages. This gives me an opportunity to tell the story as it was intended in the beginning, and then publish in another channel that allows more space.
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The subject is about pattern creation, using a technique that includes modular tiles within 5-fold patterns. The tiles have a motif that cross over each side and interacts with the motif in the adjacent tiles. What is new in this paper is that these crossings are not at mid-point. Previous papers focused on non-equilateral tiles, but all tiles had midpoint crossings. Here I introduce the first non-midpoint category, called the Flat category. The Flat category is the missing link between two types of historical patterns, the Starry and the Floral pattern types (terms coined by Castera). As Flat tiles link these two worlds together, they enable a broader spectrum of historical patterns to be tiled, and gives promise to new interesting pattern where the two different pattern types are more distinctly shown than in historical patterns.

We survey existing methods for counting two-cell tilings of finite planar lattice regions. We provide the only proof of the Pfaffian-squared theorem the author is aware of which is simultaneously modern, elementary, and fully correct. Detailed proofs of Kasteleyn's Rule and Theorem are provided. Theorems of Percus and Gessel-Viennot are proved in the standard fashions, hopefully with some advantage in clarity. Results of Kuperberg's are broken down and proved, again with significantly greater detail than we were able to find in a review of existing literature. We aim to provide a cohesive and compact, if bounded, examination of the structures and techniques in question.

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

Editorial introduction to a special double issue of the Journal of Mathematics and the Arts 12/2-3 (2018). Reza Sarhangi (1952-2016), as a mathematician, educator, and artist, had a profound impact on many interdisciplinary approaches to the study and teaching of mathematics and the arts. His vision and enthusiasm embraced and energized an international community of individuals engaged in explorations of intersections of art and mathematics. His legacy lives on. He is best known for the success of the establishment of the Bridges Conference (Mathematical Connections in Art, Music, and Science), which has been transformative in linking the fields of mathematics and art. From a conference of lectures focused on intersections, Bridges expanded to include workshops, musical performances, art exhibitions, poetry night, family day, short film festival, and theatrical programmes, all with mathematical content. The editorial goes on to explore the contents of this special issue, focused on tilings in the plane, and perspectives on the sphere.

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

Children of different classes of elementary schools in Portugal have participated in a mathematics pro ject which aims at increasing the interest for Mathematics in Portugal together with developing skills for day-to -day problem solving and inter-relating Mathematics with other S ciences and Arts. In this talk we shall present som e of the results obtained while exploring several topics rel ated to symmetry. The examples given relate to tili ng, kaleidoscopes, animal drawings and cooking.

Laddove i progressi nel campo della tecnologia dell'immagine rendono disponibili monitor, schermi e touch-screen con risoluzione sempre maggiore-nei quali la misura di ogni singolo pixel sembra ridursi sino alla soglia della effettiva visibilità-la cultura visuale contemporanea, soprattutto nel cam-po ludico e videoludico, conserva ancora oggi uno spiccato interesse per tipologie di raffigurazione a bassa risoluzione in cui la riconoscibilità del singolo picture element costituisce il fondamento di ogni cifra estetica. La storia della formazione delle immagini digitali-le cui origini si giovano di acquisizioni tardo otto-centesche nel campo dell'ottica e della pittura-si dipana tra gli anni Sessanta e Novanta del secolo scorso attraverso continui rimandi e contaminazioni tra le arti figurative e le più avanzate ricerche nel campo dell'informatica. Queste immagini di recente concezione, organizzate su fitte matrici di ele-menti quadrati, a ben vedere trovano addentellati nelle antiche tradizioni artigianali del mosaico ma ancor di più delle differenti tecnologie tessili le quali mostrano una strettissima analogia concettuale con le tecniche informatiche di supporto all'immagine. Una riflessione che utilizzi il disegno come strumento per cogliere i legami tra queste immagini di natura diversa può rendere evidente come i più moderni low res media mostrino una connessione diretta con strutture visuali maturate attraverso la pratica manifatturiera. Parole chiave pixel, tessitura, mosaico, immagini digitali.

We propose a software architecture for picture processing with three objectives. Images are represented as a set of tiles that are computed on demand, i.e. when needed as arguments to some operator. An image is indeed an operator instance that includes a call back function to respond to tile demands. Tiling avoids to store whole images. Demand driven evaluation shortens the time intermediate results must be kept in memory before they are used in further processing. This allows efficient memory management when many operators are applied to large images. Second, this architecture is a path towards automated parallelization. The kernel layer can spread the computation of image tiles to several processing threads. At last, application development should not be made more complex in comparison to classical techniques. Applications are written as functional programs that describe relationships between images. Writing a new operator class can mostly be done by copying other operator classes...

This chapter examines the relationship between mathematics and textile knot practice, i.e., how mathematics may be adopted to characterize knotted textiles and to generate new knot designs. Two key mathematical concepts discussed are knot theory and tiling theory. First, knot theory and its connected mathematical concept, braid theory, are used to examine the mathematical properties of knotted textile structures and explore possibilities of facilitating the conceptualization, design, and production of knotted textiles. Through the application of knot diagrams, several novel two-tone knotted patterns and a new material structure can be created. Second, mathematical tiling methods, in particular the Wang tiling and the Rhombille tiling, are applied to further explore the design possibilities of new textile knot structures. Based on tiling notations generated, several two- and three-dimensional structures are created. The relationship between textile knot practice and mathematics illuminates an objective and detailed way of designing knotted textiles and communicating their creative processes. Mathematical diagrams and notations not only reveal the nature of craft knots but also stimulate new ideas, which may not have occurred otherwise.

By sharing processor resources among threads at a very fine granularity, a simultaneous multithreading processor (SMT) renders thread-level parallelism (TLP) and instruction-level parallelism (ILP) operationally equivalent. Under what circumstances are they performance equivalent? In this paper, we show that operational equivalence does not imply performance equivalence. Rather, for some codes they perform equally well, for others ILP outperforms TLP, and for yet others, the opposite is true. In this paper, we...

this paper discusses the selection of an optimal supernode shape of a supernode transformation (also known as tiling). We assume that the communication cost is dominated by the startup penalty and therefore, can be approximated by a constant. We identify three parameters of a supernode transformation: supernode size, relative side lengths, and cutting hyperplane directions. For algorithms with perfectly nested loops and uniform dependencies, we give a closed form expression for an optimal linear schedule vector, and a necessary and sufficient condition for optimal relative side lengths. We prove that the total running time is minimized by cutting hyperplane direction matrix whose rows are from the surface of the polar cone of the cone spanned by dependence vectors, also known as tiling cone. The results are derived in continuous space and should for that reason be considered approximate.

Pasting Scheme is a generating model for Tiling and Tessellation pattern. Truchet
tiles are special type of tiles with continuous arc. In this paper the pasting scheme
model is used to study the construction of triangle shaped Truchet tiling and diamond
shaped Truchet tiling using the Truchet tiles.

This paper proposes a family of origami tessellations called extruded Miura-Ori, whose folded state lies between two parallel planes with some faces on the planes, potentially useful for folded core materials because of face bonding. An extruded Miura-Ori is obtained by cutting Miura-Ori apart along the edges and face diagonals before inserting the extrusion of the section edges. We compute the extrusion direction to obtain a valid extruded Miura-Ori. The extruded Miura-Ori usually has three valid states. We analyse the third state (final folded state) in depth to show that a continuous family of parameters can produce origami tessellations that can completely tile the top and bottom planes.

This paper explores patterns of particular variations of Truchet tiles that consist of arcs combined with squares and regular hexagons. With the help of matrixes with boolean values and observation of patterns in random tessellations the practically usable basic patterns have been systematized, which helped to construct also a few novel but simple patterns. The resulting patterns are characterized by smooth joints and organic look, therefore they can find numerous applications in design and architecture due to the high aesthetic value.

— We present a new basis for analysis of music. The basis vectors are sampled real waveforms of fixed frequency inside a Gaussian envelope. Their frequencies and time localizations are induced by a tiling of the time-frequency plane well adapted to music. Through a careful investigation of their properties they are subsequently slightly modified in order to give a stable system, without losing their time-frequency localization. Our new basis discriminates semitones, detects the overtones as well as the attack of notes, and gives a sparse representation of the signal. It will enhance the performance of all kinds of digital audio processors, and provide a useful tool for numerous multimedia applications.

Tiling improves the artistic appeal of many rooms’ floors and walls. However, it is necessary to take precautions to prevent the tiles from deteriorating over time. This is especially in wet or moisture-prone areas like the bathroom. Water and moisture that seep into the wall or floor can increase the probability of molting. This can then damage the construction over time. Before tiling, both wall and floor substrate must be waterproofed.

It is advisable to consult an expert tiling contractor, such as the Itile bathrooms NT team, to evaluate the issue or damage to waterproofing tiles. They will be able to identify if the issue is similar to failed waterproofing or cracked grout. If the waterproofing system fails, it is recommended to remove the tiles and begin the waterproofing and tiling procedure over.

Is it possible to waterproof over the bathroom tile?

Definitely, the answer to that is yes. It is possible to waterproof and tile over existing tiles. However, the existing tiles must be in excellent condition and securely attached to the substrate. Ensure that the tiles are not cracked, loose, or hollow-sounding. Moreover, make sure that the current floor level allows a height increase, particularly near door entrances.

How does bathroom waterproofing work?

Waterproof the substrate with a waterproofing compound like Tradies Choice WPM, a Grey Class III fibre reinforced water based polyurethane membrane that has been designed for a range of waterproofing applications for under Tile.

Step 1: Priming

The substrate must be in good shape, clean, and dry. After checking the substrate use Tradies Choice WPM. This product is appropriate for wet areas, balconies, and deck areas that will be tiled and not left exposed. Outstanding adhesion to concrete, masonry, renders, internal cement sheeting, and plasterboard sheeting. Then apply Tradies Choice Master Primer, a fast-drying water-based acrylic primer, which should be used on rough, porous surfaces.

Step 2: Application

For waterproofing, use Tradies Choice WPM this is a fast drying two component waterproof membrane that is suitable for a wide range of waterproofing applications.

Tradies Choice WPM should be tested on a small section of tile and grout to assess any potential discoloration.
The application should begin at the top of the structure and work its way down. Tradies Choice WPM should be carefully applied under tiles and grout. Finish by installing the bathroom base. Tradies Choice WPM should be properly applied to every surface.
Give at least 15 minutes for its first application to take effect before repeating the procedure throughout the entire area.
Require at least 15 minutes for the second application to penetrate before wiping away any excess on the tile surface with a dry cloth.
Allow Tradies Choice WPM to dry for 24 hours before showering.
Step 3: Drying Time

After the second application, let the Tradies Choice WPM to dry for at least 15 minutes before polishing off the surface with a clean soft cloth. Allow at least 24 hours before using it.

Guaranteed Bathroom Workmanship

Waterproofing over old tiles is a specialised job. We do not recommend to have it as a DIY project. To have the advantage of a guarantee, we highly recommend hiring Itile Bathrooms’ professional team.

We have over 20 years of tiling and waterproofing expertise and have been serving Darwin, Palmerston, and Humpty Doo. Itile Bathrooms NT is a licensed tiler and waterproofer of walls and floors.

Our team can handle full bathroom renovations, including tile demolition, waterproofing, tiling, and plumbing, as well as remedial waterproofing. You can be confident that our years of experience will meet all of your remedial waterproofing needs, including leaky balconies, showers, and tile-free bathrooms. Contact us today and let us know how we can help you.

We all know how frustrating it is to see leaking tiles in the bathroom. The feeling of wanting to retain the bathroom tile, but wanting to repair it poses a dilemma. In most cases, due to lack of ideas on how to address the tiling issue and also to avoid the repair costs, we just let the tiles leak or leave them as is. But the more we delay resolving the problem tile, the more it gets worse. Good thing that we now have Clear Waterproofing Membrane.

Clear Waterproofing Membrane

It is a two part clear, hybrid water based, aliphatic polyurethane waterproofing membrane or coating that treats to form a clear, tough, durable, flexible, trafficable, abrasion, and anti-scratch resistant, waterproof membrane in clear gloss, satin, or matte finish. Simply put, this membrane can be applied on any surface where water is unavoidable.

The product can be installed on top of leaking tiled surfaces just like a shower or a deck. From this, it then forms a waterproof barrier while retaining the original tile appearance.

Product Formulation

The formulation of Clear Waterproofing Membrane is suitable for sunlight exposure. Moreover, it preserves the substrate’s original appearance, like tiles. This waterproofing membrane can be used:

1. To rectify a leaking tiled spot without having to strip the tiles
2. To serve as floor coating for timber
3. To waterproof porous materials

Is the affected area a non-porous surface? No need to worry. There are adhesion enhancers included in Clear Waterproofing Membrane so that it will stick with non-porous areas.

clear waterproofing membrane by itilent, darwin, au
Save Repair Cost

Back in time when Clear Waterproofing Membrane is not yet invented, the cost of repair can really hurt your budget. The money you will spend on actual repair is insignificant if you are to compare it with the cost of doing the water damage repair brought by the leak. With the innovative waterproofing membrane, you can save more and get the best results. You are not only addressing the leaking tiles, but also your concern on the repair costs.

Remedial Membrane Certified Installer

To ensure that Clear Waterproofing Membrane is correctly installed, hire a Remedial Membrane Certified installer like iTileNT. We have successfully completed a course on how to use and install Clear Waterproofing Membrane. This guarantees an expertly done project worth your spending and trust. Our team has more than 20 years in commendable bathroom renovation and tiling services that you can count on. Let’s help you renovate or repair your bathroom today. Request for a quote or call us at 04 0072 9303.

Children of different classes of elementary schools in Portugal have participated in a mathematics project which aims at increasing the interest for Mathematics in Portugal together with developing skills for day-today problem solving and interrelating Mathematics with other Sciences and Arts. In this talk we shall present some of the results obtained while exploring several topics related to symmetry. The examples given relate to tiling, kaleidoscopes, animal drawings and cooking.

Pasting Scheme is a generating model for Tiling and Tessellation pattern. Truchet tiles are special type of tiles with continuous arc. In this paper the pasting scheme model is used to study the construction of triangle shaped Truchet tiling and diamond shaped Truchet tiling using the Truchet tiles.

Pasting Scheme is a generating model for Tiling and Tessellation pattern. Truchet tiles are special type of tiles with continuous arc. In this paper the pasting scheme model is used to study the construction of triangle shaped Truchet tiling and diamond shaped Truchet tiling using the Truchet tiles.