In Greek mythology, the Labyrinth was an elaborate, confusing structure designed and built by the legendary artificer Daedalus for King Minos of Crete at Knossos. Its function was to hold the Minotaur, the monster killed by the hero Theseus. Daedalus had so cunningly made the Labyrinth that he could escape it after he built it. Although early Cretan coins exhibit branching patterns, the single-path seven-course "Classical" design without branching or dead ends became associated with the Labyrinth on coins as early as 430 BC, similar non-branching patterns became used as visual representations of the Labyrinth – though both logic and literary descriptions make it clear that the Minotaur was trapped in a complex branching maze; as the designs became more elaborate, visual depictions of the mythological Labyrinth from Roman times until the Renaissance are invariably unicursal. Branching mazes were reintroduced only. In English, the term labyrinth is synonymous with maze; as a result of the long history of unicursal representation of the mythological Labyrinth, many contemporary scholars and enthusiasts observe a distinction between the two.
In this specialized usage maze refers to a complex branching multicursal puzzle with choices of path and direction, while a unicursal labyrinth has only a single path to the center. A labyrinth in this sense has an unambiguous route to the center and back and presents no navigational challenge. Unicursal labyrinths appeared as designs on pottery or basketry, as body art, in etchings on walls of caves or churches; the Romans created many decorative unicursal designs on walls and floors in tile or mosaic. Many labyrinths set in floors or on the ground are large enough. Unicursal patterns have been used both in group ritual and for private meditation, are found for therapeutic use in hospitals and hospices. Labyrinth is a word of pre-Greek origin, which the Greeks associated with the palace of Knossos in Crete, excavated by Arthur Evans early in the 20th century; the word appears in a Linear B inscription as da-pu-ri-to. As early as 1892 Maximilian Mayer suggested that labyrinthos might derive from labrys, a Lydian word for "double-bladed axe".
Evans suggested that the palace at Knossos was the original labyrinth, since the double axe motif appears in the palace ruins, he asserted that labyrinth could be understood to mean "the house of the double axe". This designation may not have been limited to Knossos, since the same symbols were discovered in other palaces in Crete; however Nilsson observes that in Crete the "double axe" is not a weapon and always accompanies goddesses or women and not a male god. Beekes finds the relation with labrys speculative, suggests instead the relation with lavra, narrow street; the original Minoan word appears to refer to labyrinthine grottoes, such as seen at Gortyn. Pliny the Elder's four examples of labyrinths are all complex underground structures, this appears to have been the standard Classical understanding of the word, it is possible that the word labyrinth is derived from the Egyptian loperohunt, meaning palace or temple by the lake. The Egyptian labyrinth near Lake Moeris is described by Strabo.
By the 4th century BC, Greek vase painters represented the Labyrinth by the familiar "Greek key" patterns of endlessly running meanders. When the Bronze Age site at Knossos was excavated by explorer Arthur Evans, the complexity of the architecture prompted him to suggest that the palace had been the Labyrinth of Daedalus. Evans found various bull motifs, including an image of a man leaping over the horns of a bull, as well as depictions of a labrys carved into the walls. On the strength of a passage in the Iliad, it has been suggested that the palace was the site of a dancing-ground made for Ariadne by the craftsman Daedalus, where young men and women, of the age of those sent to Crete as prey for the Minotaur, would dance together. By extension, in popular legend the palace is associated with the myth of the Minotaur. In the 2000s, archaeologists explored other potential sites of the labyrinth. Oxford University geographer Nicholas Howarth believes that'Evans's hypothesis that the palace of Knossos is the Labyrinth must be treated sceptically.'
Howarth and his team conducted a search of an underground complex known as the Skotino cave but concluded that it was formed naturally. Another contender is a series of tunnels at Gortyn, accessed by a narrow crack but expanding into interlinking caverns. Unlike the Skotino cave, these caverns have smooth walls and columns, appear to have been at least man-made; this site corresponds to an unusual labyrinth symbol on a 16th-century map of Crete contained in a book of maps in the library of Christ Church, Oxford. A map of the caves themselves was produced by the French in 1821; the site was used by German soldiers to store ammunition during the Second World War. Howarth's investigation was shown on a documentary produced for the National Geographic Channel. More labyrinth might be applied to any complicated maze-like structure. Herodotus, in Book II of his Histories, describes as a "labyrinth" a building complex in Egypt, "near the place called the City of Crocodiles," that he considered to surpass the pyramids: It has twelve covered courts — six in a row facing north, six south — the gates of the one range fronting the gates of the other.
Inside, the building is of two storeys and contains three thousand rooms, of which half are underground, the other half directly above them. I was taken through the rooms in
Information visualization or information visualisation is the study of visual representations of abstract data to reinforce human cognition. The abstract data include both numerical and non-numerical data, such as text and geographic information. However, information visualization differs from scientific visualization: "it’s infovis when the spatial representation is chosen, it’s scivis when the spatial representation is given"; the field of information visualization has emerged "from research in human-computer interaction, computer science, visual design and business methods. It is applied as a critical component in scientific research, digital libraries, data mining, financial data analysis, market studies, manufacturing production control, drug discovery". Information visualization presumes that "visual representations and interaction techniques take advantage of the human eye’s broad bandwidth pathway into the mind to allow users to see and understand large amounts of information at once.
Information visualization focused on the creation of approaches for conveying abstract information in intuitive ways."Data analysis is an indispensable part of all applied research and problem solving in industry. The most fundamental data analysis approaches are visualization, data mining, machine learning methods. Among these approaches, information visualization, or visual data analysis, is the most reliant on the cognitive skills of human analysts, allows the discovery of unstructured actionable insights that are limited only by human imagination and creativity; the analyst does not have to learn any sophisticated methods to be able to interpret the visualizations of the data. Information visualization is a hypothesis generation scheme, which can be, is followed by more analytical or formal analysis, such as statistical hypothesis testing; the modern study of visualization started with computer graphics, which "has from its beginning been used to study scientific problems. However, in its early days the lack of graphics power limited its usefulness.
The recent emphasis on visualization started in 1987 with the special issue of Computer Graphics on Visualization in Scientific Computing. Since there have been several conferences and workshops, co-sponsored by the IEEE Computer Society and ACM SIGGRAPH", they have been devoted to the general topics of data visualisation, information visualization and scientific visualisation, more specific areas such as volume visualization. In 1786, William Playfair published the first presentation graphics. Cartogram Cladogram Concept Mapping Dendrogram Information visualization reference model Graph drawing Heatmap HyperbolicTree Multidimensional scaling Parallel coordinates Problem solving environment Treemapping Information visualization insights are being applied in areas such as: Scientific research Digital libraries Data mining Information graphics Financial data analysis Health care Market studies Manufacturing production control Crime mapping eGovernance and Policy Modeling Notable academic and industry laboratories in the field are: Adobe Research IBM Research Google Research Microsoft Research Panopticon Software Scientific Computing and Imaging Institute Tableau Software University of Maryland Human-Computer Interaction Lab VviConferences in this field, ranked by significance in data visualization research, are: IEEE Visualization: An annual international conference on scientific visualization, information visualization, visual analytics.
Conference is held in October. ACM SIGGRAPH: An annual international conference on computer graphics, convened by the ACM SIGGRAPH organization. Conference dates vary. EuroVis: An annual Europe-wide conference on data visualization, organized by the Eurographics Working Group on Data Visualization and supported by the IEEE Visualization and Graphics Technical Committee. Conference is held in June. Conference on Human Factors in Computing Systems: An annual international conference on human-computer interaction, hosted by ACM SIGCHI. Conference is held in April or May. Eurographics: An annual Europe-wide computer graphics conference, held by the European Association for Computer Graphics. Conference is held in April or May. PacificVis: An annual visualization symposium held in the Asia-Pacific region, sponsored by the IEEE Visualization and Graphics Technical Committee. Conference is held in March or April. For further examples, see: Category:Computer graphics organizations Computational visualistics Data art Data Presentation Architecture Data visualization Geovisualization Infographics Patent visualisation Software visualization Visual analytics List of information graphics software List of countries by economic complexity, example of Treemapping Ben Bederson and Ben Shneiderman.
The Craft of Information Visualization: Readings and Reflections. Morgan Kaufmann. Stuart K. Card, Jock D. Mackinlay and Ben Shneiderman. Readings in Information Visualization: Using Vision to Think, Morgan Kaufmann Publishers. Jeffrey Heer, Stuart K. Card, James Landay. "Prefuse: a toolkit for interactive information visualization". In: ACM Human Factors in Computing Systems CHI 2005. Andreas Kerren, John T. Stasko, Jean-Daniel Fekete, Chris North. Information Visualization – Human-Centered Issues and Perspectives. Volume 4950 of LNCS State-of-the-Art Survey, Springer. Riccardo Mazza. Introduction to Information Visualization, Springe
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to 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, 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. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons; such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings.
Tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.
Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules; these rules can be varied. Common ones are that there must be no gaps between tiles, that no corner of one tile can lie along the edge of another; the tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
There are only three shapes that can form such regular tessellations: the equilateral triangle and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can be made from other shapes such as pentagons, polyominoes and in fact any kind of geometric shape; the artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, these can be used to decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries.
These tiles may be any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane; the Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; the Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions, he further defined the Schläfli symbol notation to make it easy to describe polytopes.
For example, the Schläfli symbol for an equilateral triangle is. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is. Other methods exist for describing polygonal tilings; when the tessellation
Tiling with rectangles
A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 side ratio; the tilings with straight polyominoes of shapes such as 1 × 3, 1 × 4 and tilings with polyominoes of shapes such as 2 × 3 fall into this category. Some tiling of rectangles include: The smallest square that can be cut into rectangles, such that all m and n are different integers, is the 11 × 11 square, the tiling uses five rectangles; the smallest rectangle that can be cut into rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, the tiling uses five rectangles. Squaring the square Tessellation Tiling puzzle
Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of catoptric tessellation; the cubille is the only Platonic tessellation of 3-space, is self-dual. There are other uniform honeycombs constructed as prismatic stacks which are excluded from these categories; the pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed; these four symmetry groups are labeled as: Crystallography of Quasicrystals: Concepts and Structures by Walter Steurer, Sofia Deloudi, p.54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry Conway, John H..
"21. Naming Archimedean and Catalan Polyhedra and Tilings"; the Symmetries of Things. A K Peters, Ltd. pp. 292–298. ISBN 978-1-56881-220-5. Inchbald, Guy. "The Archimedean honeycomb duals". The Mathematical Gazette. Leicester: The Mathematical Association. 81: 213–219. JSTOR 3619198. Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4, 49 - 56. Norman Johnson Uniform Polytopes, Manuscript A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative, Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF George Olshevsky, Uniform Panoploid Tetracombs, Manuscript PDF Pearce, Peter. Structure in Nature is a Strategy for Design; the MIT Press. Pp. 41–47. ISBN 9780262660457. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter and Semi-Regular Polytopes III, See p318
In geometry, the rhombille tiling known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 120 ° angles. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles; the rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling, it can be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals of each rhomb are in the ratio 1:√3; this is the dual tiling of the trihexagonal kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, in the face configuration for monohedral tilings it is denoted, it is one of 56 possible isohedral tilings by quadrilaterals, one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, more such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube; the rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion. In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. In another of his works, Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements and an upstairs patio tiled with the rhombille tiling.
A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so. These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more and includes a depiction of the reversible cubes illusion on a flag within the scene; the rhombille tiling is used as a design for parquetry and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. It appears in ancient Greek floor mosaics from Delos and from Italian floor tilings from the 11th century, although the tiles with this pattern in Siena Cathedral are of a more recent vintage. In quilting, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation; as a quilting pattern it has many other names including cubework, heavenly stairs, Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape.
See Quilts of the Underground Railroad. In these decorative applications, the rhombi may appear in multiple colors, but are given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms; the rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field.
The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers. In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice, it is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals, it has been studied in percolation theory. The rhombille tiling has *632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a *333 symmetry; the rhombille tiling is the dual of the trihexagonal tiling, as such is part of a set of uniform dual tilings. It is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry, starting from the cube, which can be seen as a rhombic hexahedron where the rhombi are squares; the nth element in this sequence has a face configuration of V3.n.3.n. The rhombille tiling is one of many different ways of tiling the plane by congruent rhombi.
Others include a diagonally flattened variation of the square tiling, the tiling used by the Miura-ori folding pattern, the Penrose tiling which
A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s; the aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right. Penrose tiling is non-periodic, it is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through "inflation" and every finite patch from the tiling occurs infinitely many times, it is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order. Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules and project schemes and coverings.
Penrose tilings are simple examples of aperiodic tilings of the plane. A tiling is a covering of the plane by tiles with no gaps; the most familiar tilings are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period of the tiling. If a tiling has no periods it is said to be non-periodic. A set of prototiles is said to be aperiodic if it tiles the plane but every such tiling is non-periodic; the subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles, posed the "Domino Problem": to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges, he observed that if this problem were undecidable there would have to exist an aperiodic set of Wang dominoes.
At the time, this seemed implausible, so Wang conjectured no such set could exist. Wang's student Robert Berger proved that the Domino Problem was undecidable in his 1964 thesis, obtained an aperiodic set of 20426 Wang dominoes, he described a reduction to 104 such prototiles. The color matching required in a tiling by Wang dominoes can be achieved by modifying the edges of the tiles like jigsaw puzzle pieces so that they can fit together only as prescribed by the edge colorings. Raphael Robinson, in a 1971 paper which simplified Berger's techniques and undecidability proof, used this technique to obtain an aperiodic set of just six prototiles; the first Penrose tiling is an aperiodic set of six prototiles, introduced by Roger Penrose in a 1974 paper, but it is based on pentagons rather than squares. Any attempt to tile the plane with regular pentagons leaves gaps, but Johannes Kepler showed, in his 1619 work Harmonices Mundi, that these gaps can be filled using pentagrams and related shapes.
Traces of these ideas can be found in the work of Albrecht Dürer. Acknowledging inspiration from Kepler, Penrose found matching rules for these shapes, obtaining an aperiodic set, his tiling can be viewed as a completion of Kepler's finite Aa pattern. Penrose subsequently reduced the number of prototiles to two, discovering the kite and dart tiling and the rhombus tiling; the rhombus tiling was independently discovered by Robert Ammann in 1976. Penrose and John H. Conway investigated the properties of Penrose tilings, discovered that a substitution property explained their hierarchical nature. In 1981, De Bruijn explained a method to construct Penrose tilings from five families of parallel lines as well as a "cut and project method", in which Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure. In this approach, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges.
The three types of Penrose tiling, P1–P3, are described individually below. They have many common features: in each case, the tiles are constructed from shapes related to the pentagon, but the basic tile shapes need to be supplemented by matching rules in order to tile aperiodically. Penrose's first tiling uses pentagons and three other shapes: a five-pointed "star", a "boat" and a "diamond". To ensure that all tilings are non-periodic, there are matching rules that specify how tiles may meet each other, there are three different types of matching rule for the pentagonal tiles, it is co