In Latin, tessella is a small cubical piece of clay , stone or glass used to make mosaics. It corresponds to the everyday term tiling , which refers to applications of tessellations, often made of glazed clay. Tessellation in two dimensions, also 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, and that no corner of one tile can lie along the edge of another. Among those that do, a regular tessellation has both identical [a] regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
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.
Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. 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 polygons or any other shapes. 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: Mathematically, tessellations can be extended to spaces other than the Euclidean plane.
These are the analogues to polygons and polyhedra in spaces with more dimensions. Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the vertex configuration , which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4. The tiling of regular hexagons is noted 6.
Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles; it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling is a tiling where every vertex point is identical; that is, the arrangement of polygons about each vertex is the same.
The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tiling is any polygonal tessellation where adjacent tiles only share one full side, i. In an edge-to-edge tiling, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks. A normal tiling is a tessellation for which every tile is topologically equivalent to a disk , the intersection of any two tiles is a single connected set or the empty set , and all tiles are uniformly bounded.
This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling; the condition disallows tiles that are pathologically long or thin. A monohedral tiling is a tessellation in which all tiles are congruent ; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in ; the Voderberg tiling has a unit tile that is a nonconvex enneagon.
Hunt in , is a pentagon tiling using irregular pentagons: An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling. A regular tessellation is a highly symmetric , edge-to-edge tiling made up of regular polygons , all of the same shape. There are only three regular tessellations: All three of these tilings are isogonal and monohedral.
A semi-regular or Archimedean tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings or nine if the mirror-image pair of tilings counts as two. Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups , of which 17 exist. Though this is disputed, [33] the variety and sophistication of the Alhambra tilings have surprised modern researchers.
Mosaic (web browser) - Wikipedia
Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns. Penrose tilings , which use two different quadrilateral prototiles, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings , which use tiles that cannot tessellate periodically. The recursive process of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is the rep-tiles ; these tilings have surprising self-replicating properties.
Aperiodic tilings, while lacking in translational symmetry , do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches. Wang tiles are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wang dominoes. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any Turing machine can be represented as a set of Wang dominoes that tile the plane if and only if the Turing machine does not halt.
Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable. These can tile the plane either periodically or randomly. Sometimes the colour of a tile is understood as part of the tiling; at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity one needs to specify whether the colours are part of the tiling or just part of its illustration. This affects whether tiles with the same shape but different colours are considered identical, which in turn affects questions of symmetry.
The four colour theorem states that for every tessellation of a normal Euclidean plane , with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The colouring guaranteed by the four colour theorem does not generally respect the symmetries of the tessellation. To produce a colouring which does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as in the picture at right. Next to the various tilings by regular polygons , tilings by other polygons have also been studied.
Any triangle or quadrilateral even non-convex can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre.
We can divide this by one diagonal, and take one half a triangle as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting. If only one shape of tile is allowed, tilings exists with convex N -gons for N equal to 3, 4, 5 and 6. Voronoi or Dirichlet tilings are tessellations where each tile is defined as the set of points closest to one of the points in a discrete set of defining points. Think of geographical regions where each region is defined as all the points closest to a given city or post office.
The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges. Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill or tile three-dimensional space, including the cube the only Platonic polyhedron to do so , the rhombic dodecahedron , the truncated octahedron , and triangular, quadrilateral, and hexagonal prisms , among others.
A Schwarz triangle is a spherical triangle that can be used to tile a sphere.
Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular [c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions. The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically. It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry.
A uniform tiling in the hyperbolic plane which may be regular, quasiregular or semiregular is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces ; these are vertex-transitive transitive on its vertices , and isogonal there is an isometry mapping any vertex onto any other.
A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs , generated as Wythoff constructions , and represented by permutations of rings of the Coxeter diagrams for each family. In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns.
Some of the most decorative were the Moorish wall tilings of Islamic architecture , using Girih and Zellige tiles in buildings such as the Alhambra [66] and La Mezquita. Tessellations frequently appeared in the graphic art of M. Escher ; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts. Tessellations are also a main genre in origami paper folding , where pleats are used to connect molecules such as twist folds together in a repeating fashion.
Tessellation is used in manufacturing industry to reduce the wastage of material yield losses such as sheet metal when cutting out shapes for objects like car doors or drinks cans. Tessellation is apparent in the mudcrack -like cracking of thin films [77] [78] — with a degree of self-organisation being observed using micro and nanotechnologies. The honeycomb provides a well-known example of tessellation in nature with its hexagonal cells. In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit.
Flowers including the fritillary [81] and some species of Colchicum are characteristically tessellate. Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations , [83] also known as random crack networks. The model, named after Edgar Gilbert , allows cracks to form starting from randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons.
The same needs to be done regarding the new term. These analyses will draw a contrast between the two expressions and what they designate. To differentiate the mosaic-screen from the split screen, it is imperative to revisit the latter and the functions it typically plays.
Tessellation
Timecode Mike Figgis, is a recent work that uses this technique. The screen is divided into four equal sections; each occupied by continuous takes that are coincident in time. This video experiment transferred to film elucidates something essential about the split screen: A glance at the history of this practice confirms this. In its simplest form, the split screen bisects the screen. As the narrative splits into two, so does the screen, each with its own half, each with its own shots.
In this sense, the act of splitting is inseparable from the act of separating what was one and the same. The characters are filmed in two frontal close-ups during the conversation. The physical touch followed by eye contact make them closer — and it is this new closeness that stands out against their separation on screen. Accordingly, the camera rotates and the two shots merge into one shot.
Although eventually giving way to unity, the split screen here remains associated foremost with division. Conversations with Other Women Technical decisions and choices of screen format can be important factors when employing the split screen. Much like Timecode , Conversations with Other Women Hans Canosa, divides the screen from the start, but in two parts instead of four. The film tells a love story between a man Aaron Eckhart and a woman Helena Bonham Carter who remain nameless until the end.
The wide-screen format of the movie emulates the aspect ratio of anamorphic cinematographic processes like Panavision 2. This allows the two associated shots to contain more visual information and detail. It consequently enables the split screen to fulfil various purposes throughout the motion picture. A shot and a counter-shot of the pair in their first exchange of words can be presented at the same time.
One frame is able to accommodate the two of them or just one while displaying a flashback on the left or showing a flash-forward on the right. The ending accomplishes something similar to the sequence from The Rules of Attraction. It joins two shots, one of him, one of her, both in the back seat of two taxicabs, into one shot.
Not through a camera movement, as in the previous case, but simply by way of a gradual matching of the two shots, facilitated by digital video technology. Bye Bye Birdie The partition of the screen may take other more inventive and complex forms — namely, it may be in more than two parts, the partitioning may be uneven, or the dividing lines may not be straight. Two examples suffice to illustrate this diversity. Bye Bye Birdie George Sidney, belongs to a cycle of popular wide-screen teenage comedies produced in the s.
Mosaic (web browser)
Sketchy lines split the screen during a musical number about phone gossip, mirroring the carefree life of six adolescents. The Laramie Project shows multiple views of the same event. Bye Bye Birdie displays views of different, subsequent events. The two recall that the split screen is frequently linked with simultaneity and causality. The screen is subdivided into three areas. Billie and Joy are at the top — each one on the phone —, and Earl and Randy at the bottom — listening in together.
To save agriculture from climate change, we need better weather forecasting
This causal connection explored by the split screen has been commonly allied with phone conversations: The split screen is generally connected with simultaneity as well as division — the technique is often used as a division that allows simultaneity. That is why it is regularly employed in television news and live sportscasts. Brian De Palma has adopted the split screen almost as a signature technique, amongst others like slow motion. His filmography contains many different uses of the split screen, from Sisters to Femme Fatale — in the first, for example, suspense is intensified when the screen is split into two crossing points of view, from an apartment where criminals are cleaning the scene of a murder, and from a flat across the street where a witness awaits the police.
Spatial relations are maintained as the screen is split in two: Carrie is on the right, directing three gazes toward the left that result in the shutting of all exit doors, as shown on the left. Her image then moves to the left to preserve the spatial coordinates of the scene on screen: The consistency of spatial relationships could have been maintained by an immediate swap of images. Instead, the atypical movement of the image to the left expresses her dominating power to move and rearrange anything at a distance.
This succinct survey of the usage and functions of the split screen reveals the importance of technical decisions like screen ratio and the prevalence of relationships like causality and simultaneity. It also explored examples of more ingenious uses of the technique that sometimes highlight authorship. Obviously, some of these aspects may also be related with the new term — but not screen division, the first mentioned trait. The mosaic-screen does not divide the screen.
The mosaic-screen presents fragments on screen. It may be used to produce similar effects to the split screen, but it allows for other ways of achieving them. Let us return to 24 to investigate these ideas further. Recent style analyses of the series invariably look at the use of multi-frame imagery. As Peacock attentively notices,. The full-screen image of Bauer only gives way when he recognizes the voice and tone of the caller.
However, the writer overlooks how their physical distance is emphasised exactly because the screen is not simply divided. The two distinct images are presented over a black background. The full close-up of Bauer gives weight to his calm assessment of the situation and his attempt to find a way to help. The partial close-up of Walsh stresses his feeling of entrapment, the way he feels cornered. Nevertheless, the possibility of a loss of contact between the characters is made visible by the introduction of the mosaic-screen.
This arrangement of images on screen conveys, not quite an anticipation of disconnection, but the prospect of a disconnection, establishing it as something that can happen at any instant. This creates a permanent tension and, more interestingly, an uncertainty about what is going to follow. It therefore constantly brings to mind the fact that some events are not shown — and consequently that those that are shown were selected and are fragments of an unravelling sequence of simultaneous events. As in the mosaic-screen analysed in the introduction, the preponderance of the limits of the screen has vanished.
The spectator does not look at the screen as a whole but concentrates on one of its parts like in the split screen. The isolated images and intervening spaces of the mosaic-screen ask us to choose between images as if we were selecting from various smaller screens, each one with its own narrative — unlike Timecode , which does not formally disperse its images.
What occurs at the end of this sequence relates to this notion of fragmentation: The mosaic-screen arranges diverse images normally with distinct aesthetic properties: In another paper on 24 , Michael Allen is sensitive to these differences between mosaic-screen and split-screen. His stimulating account of the history of multi-panel forms spans from medieval paintings to comic books and covers an ampler view of multiple-image techniques. The scholar views the way multiple images are organised in the series as different from screen splitting and describes it as image composition on screen — even if he accepts the inapplicable expression split-screen.
In this way, 24 utilises one the major features of the comic book layout aesthetic to reveal and substantiate narrative and psychological detail developed on other layers of the text. In this passage, Allen is responsive to how the mosaic-screen composes attention — which differs strikingly from the way the split screen divides it. If the split screen draws attention to points of division both along and within the screen edges, the mosaic-screen draws it to the relationships of the detached images set out on a customarily black background. The split screen is routinely used to connect images whereas the mosaic-screen is habitually used to disconnect them.
There are, of course, cases in which the mosaic-screen explores situations that have become usual in the split screen — phone conversations, as evidenced, are regular in 24 —, but here, the space around and in between the frames, more easily conveys degrees of disconnection, prompting the above interpretation. The Thomas Crown Affair However, it is a mistake to think that large gutters are enough to define a mosaic-screen. The Thomas Crown Affair Norman Jewison, is notable for its use of multiple-frame imagery, split screens as well as mosaic-screens.
In an exemplary moment, the film presents two shots on screen. A close-up of Thomas Crown Steve McQueen , a millionaire thief, occupies the whole screen, but is split into six by an added grid. A medium shot of one of his accomplices in the bank robbery then fills one of the resultant parts.
This moment can be fruitfully contrasted with the last instants of the opening credit sequence. Two pictures on the left get smaller: A narrow vertical figure of Vicki is expanded from left to right; the final height is more than three times that of the first pictures. The scale and colour differences contribute to a sense of balance: A moving picture of a man walking in a hallway is inserted and is at variance with the first three still pictures — but it appears at the centre of the screen, prolonging the equilibrium, the inaugural stability of this often tense thriller.
In the first moment, the split screen is used as a means to unevenly fracture a shot and incorporate another. In the initial credits, the mosaic-screen is employed to achieve a vivid sense of adjusted balance. A mosaic is produced from arranged pieces. The use of the mosaic-screen is at times striking — as when the year-old is travelling at the back of an empty city bus, naked, and covered by a torn shower curtain.
On the left, there are fragments of the same shot: