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Tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional spaces, higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and ''semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An ''aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern. A ''tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Tilings
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wallpaper Groups
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is in a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations and tiles as well as wallpaper. What this page calls pattern Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‑to‑edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the boun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathematician John Conway called it a hextille. The internal angle of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. Applications The hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is optimal. However, the less regular Weaire–Phel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular Tiling
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Euclidean tilings Euclidean tilings are usually named after Cundy & Rollett’s notation. This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 36; 36; 34.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 36; 36 (both of different transitivity class), or (36)2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 34.6, 4 more contiguous equilateral triangles and a single regular hexagon. However, this notation has two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Islamic Geometric Patterns
Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create a representation of an important Islamic figure according to many holy scriptures. The geometric designs in Islamic art are often built on combinations of repeated squares and circles, which may be overlapped and interlaced, as can arabesques (with which they are often combined), to form intricate and complex patterns, including a wide variety of tessellations. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, or may retreat into the background around other motifs. The complexity and variety of patterns used evolved from simple stars and lozenges in the ninth century, through a variety of 6- to 13-point patterns by the 13th century, and finally to include also 14- and 16-point stars in the sixteenth century. Geometric patterns occur in a variety of forms in Islamic a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zellige
''Zellij'' ( ar, الزليج, translit=zillīj; also spelled zillij or zellige) is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various patterns on the basis of tessellations, most notably elaborate Islamic geometric motifs such as radiating star patterns. This form of Islamic art is one of the main characteristics of architecture in the western Islamic world. It is found in the architecture of Morocco, the architecture of Algeria, early Islamic sites in Tunisia, and in the historic monuments of al-Andalus (in the Iberian Peninsula). From the 14th century onwards, ''zellij'' became a standard decorative element along lower walls, in fountains and pools, on minarets, and for the paving of floors. After the 15th century the traditional mosaic ''zellij'' fell out of fashion in most countries except for Morocco, where it continues to be produced today. ''Zellij'' is found in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patterns
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design. Any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, rarely exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation Of Space
In geometry, a honeycomb is a ''space filling'' or ''close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dimensions. Its dimension can be clarified as ''n''-honeycomb for a honeycomb of ''n''-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. Classification There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or ''slabs'' of prisms based on some tessellations of the plane. In particula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patterns In Nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time. In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Islamic Art
Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslim populations. Referring to characteristic traditions across a wide range of lands, periods, and genres, Islamic art is a concept used first by Western art historians since the late 19th century. Public Islamic art is traditionally non- representational, except for the widespread use of plant forms, usually in varieties of the spiralling arabesque. These are often combined with Islamic calligraphy, geometric patterns in styles that are typically found in a wide variety of media, from small objects in ceramic or metalwork to large decorative schemes in tiling on the outside and inside of large buildings, including mosques. Other forms of Islamic art include Islamic miniature painting, artefacts like Islamic glass or pottery, and textile arts, such as carpets and embroidery. The early developments of Isla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alhambra
The Alhambra (, ; ar, الْحَمْرَاء, Al-Ḥamrāʾ, , ) is a palace and fortress complex located in Granada, Andalusia, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of the historic Islamic world, in addition to containing notable examples of Spanish Renaissance architecture. The complex was begun in 1238 by Muhammad I Ibn al-Ahmar, the first Nasrid emir and founder of the Emirate of Granada, the last Muslim state of Al-Andalus. It was built on the Sabika hill, an outcrop of the Sierra Nevada which had been the site of earlier fortresses and of the 11th-century palace of Samuel ibn Naghrillah. Later Nasrid rulers continuously modified the site. The most significant construction campaigns, which gave the royal palaces much of their definitive character, took place in the 14th century during the reigns of Yusuf I and Muhammad V. After the conclusion of the Christian Reconquista in 1492, the site became th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quilting
Quilting is the term given to the process of joining a minimum of three layers of fabric together either through stitching manually using a needle and thread, or mechanically with a sewing machine or specialised longarm quilting system. An array of stitches is passed through all layers of the fabric to create a three dimensional padded surface. The three layers are typically referred to as the top fabric or quilt top, batting or insulating material and the backing. Quilting varies from a purely functional fabric joinery technique to highly elaborate, decorative three dimensional surface treatments. A wide variety of textile products are traditionally associated with quilting that includes bed coverings, soft home furnishings, garments and costumes, wall hangings, artistic objects and cultural artefacts. A wide range of effects can be employed by the quilter that contribute to the final surface quality and utility of the quilted material. The quilter controls these effect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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