Movies (Notes) … (a Mosaic Design)

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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 extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland.

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Other natural patterns occur in foams ; these are packed according to Plateau's laws , which require minimal surfaces. Such foams present a problem in how to pack cells as tightly as possible: In , Denis Weaire and Robert Phelan proposed the Weaire—Phelan structure , which uses less surface area to separate cells of equal volume than Kelvin's foam. Tessellations have given rise to many types of tiling puzzle , from traditional jigsaw puzzles with irregular pieces of wood or cardboard [89] and the tangram [90] to more modern puzzles which often have a mathematical basis.

For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles. For example, Dudeney invented the hinged dissection , [93] while Gardner wrote about the rep-tile , a shape that can be dissected into smaller copies of the same shape. Triangular tiling , one of the three regular tilings of the plane. Snub hexagonal tiling , a semiregular tiling of the plane. Floret pentagonal tiling , dual to a semiregular tiling and one of 15 monohedral pentagon tilings.

The Voderberg tiling , a spiral, monohedral tiling made of enneagons. Alternated octagonal or tritetragonal tiling is a uniform tiling of the hyperbolic plane. From Wikipedia, the free encyclopedia. For the song by Alt-J, see Tessellate song. For the computer graphics technique, see Tessellation computer graphics. Euclidean tilings of regular polygons , Uniform tiling , and List of convex uniform tilings. Aperiodic tiling and List of aperiodic sets of tiles. Tiling puzzle and recreational mathematics. A honeycomb is a natural tessellated structure.

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Mosaics of the Greek and Roman world. Retrieved 26 May Geometric Patterns from Roman Mosaics.

Harmonices Mundi [ Harmony of the Worlds ]. Los Alamos National Laboratory. Retrieved 6 April Petersburg Mineralogical Society], series 2 in Russian. The Symmetries of Things. Regular Polytopes 3rd ed. Mathematical Models 2nd ed. Basilica di San Marco. Retrieved 26 April Try the Conway Criterion!

Tessellation

Geometric Symmetry in Patterns and Tilings. Journal of Combinatorial Theory, Series A. Retrieved 29 April Notices of the American Mathematical Society. Retrieved 29 May Bell System Technical Journal. Memoirs of the American Mathematical Society.

The Grammar of Ornament folio ed. Delaunay Triangulation and Meshing: Application to Finite Elements. Lectures on Random Voronoi Tessellations. Firefly Guide to Gems. Minerals and Rocks in Colour 2nd ed. Arthur; Canadian Mathematical Society Selected Writings of H. Retrieved 27 May Regular honeycombs in hyperbolic space. The Beauty of Geometry: National University of Singapore. Retrieved 17 May Sensational Designs From Interlocking Patterns. Hutchinson "Crack patterns in thin films" J.

Solids 48, Retrieved 7 November Applications of Undergraduate Mathematics in Engineering. Random patterns in two dimensions". Aspects of Australian sandstone landscapes. American Jigsaw Puzzle Society. Archived from the original on 11 February Retrieved 28 May The Tao of Tangram. A guide to puzzles and problems in tiling. Mathematical Association of America. A Two Volume Collection: Archived from the original PDF on Polykleitos Canon Vitruvius De architectura.

Journal of Mathematics and the Arts. Instead, the atypical movement of the image to the left expresses her dominating power to move and rearrange anything at a distance.

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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: There is no masking over the shot as in the first example from The Thomas Crown Affair. The pieces were clearly moved away to amplify the feeling of isolation and loneliness. An image of the bus on the road is situated at the bottom right and is considerably bigger than the other three. That alone makes it prominent, but this image is given even more emphasis because it is rotated to the right — a rotation that we follow. This more dynamic positioning combined with the movement of the bus creates a greater disparity between this image and the stillness of the fragments on the top left.

At times, the film combines multiple insets forming a saturated and dispersed image, cubist in its effect — encompassing multiplied viewpoints, simple quadrilateral forms, interlocking planes, and a sense of collage, obviously connected with the idea of mosaic. Billy Gallo sits on a bench right after his release from jail and an image of him in the shower is placed at the centre of the screen. One by one, several images of his recent past fill the screen until the initial shot of the protagonist on a bench disappears under them. These concise analyses prove that this formal and technical practice exists and has a set of distinctive characteristics.

Even so, this is not the same as arguing for the appropriateness of the term, which may be unsuitable to designate such a technique.

Craig Knowles recuperates the concept of mosaic as an art form that can be explored in filmmaking. According to him, multiple-imaging in motion pictures is part of a history of decorative and artistic mosaic forms whose origin can be traced back to the Greeks , 9. For Knowles, mosaic forms maintain a connection with this source: It is not surprising then that he simply gathers modes of presenting several images on screen: Legitimately for his goals, the author not only makes no attempt to distinguish between multiple-image practices, but also mentions the listed techniques as simply creating image mosaics — and hence their effects seems similar and their distinctions become vague.

As stated in the description of one of these services, interactive video mosaics4 can display as few as two and as many as twelve video streams. These video windows can be scaled and the mosaic also includes a channel list and a menu for additional details on selected video streams or television channels. What is fundamental to our discussion is that the variety of information to be displayed begs for an arrangement that is conceptually similar to the mosaic-screen technique: The phosphor screen of television sets is not a surface for projection; it is part of a device that generates the image.

This does not mean that the word is, in itself, problematic. This is an entrenched and comprehensible use of the word. Examining the two words combined leads to other observations. The verb indicates that the split screen emerges from the form and boundaries of the screen; it entails the action of dividing the screen into parts. The noun denotes that the mosaic-screen is arranged over the screen; it identifies a type of composition of images on screen.

The foremost evidence that this new term is needed is how the images produced with this technique have observable and significant differences from the ones created employing split-screen imaging. Behind this assertion is the idea that specific terms can be established and refined — and sometimes established through a process of refinement. As such, the differentiation between the split screen and the mosaic-screen is as critical, and as subtle or apparent, as the one between a matched cut and a jump cut.

The clear distinction drawn between split-screen and mosaic-screen in the first sections demonstrated the necessity and usefulness of this new term by showing its descriptive value and practical application.

Tessellation - Wikipedia

One of the advantages of what is proposed here is that it does not require or call for the substitution of one term for another. This is of special significance given that terms become ingrained by their usage — which means that if this essay were proposing such a replacement, even if for a more precise expression, it would encounter resistance, at the least, or indifference, most likely.

It has been made clear that split-screen and mosaic-screen are two distinct multi-image techniques and this is the chief reason why we should have two terms to designate them. The split screen divides the screen into two or more parts. The mosaic-screen arranges one or more detached images on screen. These two general definitions are open. They do not tell us if the split-screen parts have varying or equal dimensions, or if the mosaic-screen images have similar or different characteristics.

Usually, split-screen parts are equivalent and mosaic-images are heterogeneous, but that is not always the case. Let us allow some room for the kind of formal diversity celebrated throughout this text regarding these frames within frames, these images within images. This paper has been narrow in its purpose, but it aimed at being penetrating.

It identified a form and distinguished it from another form through observation. It also proposed an adequate and defensible name to give to this new form. In brief, this has been mainly an analytical and theoretical effort. Now that the mosaic-screen is clearly identifiable there is much work to conduct. The histories of this stylistic device and of the split screen are not coincident — even though there is a common genealogy that, for instance, links the multiple screen projections of Napoleon Abel Gance, or Chelsea Girls Paul Morrissey and Andy Warhol, with these two forms of multi-image composition.

As we have seen, the split screen was first achieved optically in The effect of a mosaic-screen could have not been attained in the same way, but a mask placed in the camera could have solved the problem.

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