Sumbangan MC Ascher

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Sumbangan MC Ascher

M.C. Escher Interlocking Tiling

M.C. Escher has uncommon visions and intuitions. Many of Escher's drawingscontain mathematical ideas never seen elsewhere. One of the things that he did

touched me deeply - the interlocking shapes of people, birds and fish which repeatover a flat surface with no gaps. This group of drawings are not his best contributionto mankind, but it is art for me because it stirred my imagination immediately. This issomething I can understand as an engineer and I think I can develop these ideasfurther with computer graphics. The mathematics behind the interlocking shapes ona flat surface was established by Heesch and Kienzle in 1963. The followingdiscussion is limited to regular pattern in a flat surface.

The best way to understand Escher's interlocking shapes is to dissect the figures.Take a piece of tracing paper and copy the outline of a shape (in the case of thehorseman, the combined outline of the horse and the ri der). This silhouette allowsyou to focus on the line properties between the shapes. Look in the original periodic

drawing and locate the vertices: points where 3 (or more) lines joint together. Thevertex is also a place where 3 or more neighboring shapes touch. There are at least3 vertices and up to 6 vertices on the silhouette on the tracing paper. The linesegments between the vertices holds the secrets of the mathematics. Look forrotation symmetry at these vertices. Let's discuss two famous Escher drawings toillustrate the structure: "Reptiles" where the lizards claw out of a book, and "Cycle"where a short person runs down the stairs and tumble into a hexagon form. In thesefigures, there are 6 vertices and 6 line segments. Upon closer examination, there areonly 3 distinct line segments and 2 copies of each of these three lines. Suchsimplicity ! Now, the secret becomes lucent, half of the silhouette are made out ofcopies from the other half. Three of the 6 vertices have rotation symmetry C3, whichmeans the line segments attached to these vertices can pivot about this point andradiate out 3 copies like the propeller of an electric fan. The two copies of identicalline segments are connected at the rotation vertices C3. The other 3 vertices are thejunctions of the dissimilar line segments. The line segments repeat across the pageperiodically and hook up with other line segments at the vertices to create therepetitive pattern. In all the drawings made by Escher there are 6 different kinds ofrepetition method for the line segments: translation T, half -turn C2, one-third turn C3,quarter turn C4, one-sixth turn C6 and gliding mirror reflection G. Only 6 differentcongruent lines and you can do everything ! The lines are equivalent to the sub -atomic particles in physics. The atoms are recognized forms which have aninteresting meaning to our eyeballs: people, birds, fish, horse, animals. These atomicunits may or may not nest on a surface. The molecules are the smallest unit that can

interlock - what I call interlocking shapes. The molecules may be a silhouette of ahorseman, which interlocks perfectly, and consists of two atoms: the horse and therider. The atom has meaning to our eyeballs, but no mathematical interpretation. Thesilhouette, is formed by groups of congruent lines just like a molecule is build withshared orbits of electrons. The method of repetition of the interlocking shapes isrooted in the congruent movements of the lines. Thus, a silhouette that is build withC6 lines also repeat in 60 degree rotations all over the flat surface.


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Crystallographers illustrate the molecular packing with regular drawings. Thesediagrams consists of geometric arrangement with no color or grayscale contrast. TheJapanese quilts and Chinese lattice window s are also lacking in color (betweenadjacent shapes). The Islamic tiles have wonderful colors, which gives the picturedepth and foreground/background. All the above still lack another characteristic:recognizable animal form and meaningful objects. The challenge Escher overcome

is to build the shapes with familiar animals and things. One explanation for the purelyabstract shapes in Islamic tile is that the religion forbid the use of familiarrecognizable animals and human images. Escher sees the beauty in structure andinfinity. He forced the idea of meaningful lines into the mathematical framework ofregular plane division. He likes to challenge the logic of seeing. You can see thewhite birds and regard the black as background. Or you can see the black fish andregard the white as background. When the mind jumps back and forth, that is whenyou are seeing two things existing together as interlocking shapes.

Escher said that he felt irresistible joy with putting multiple copies of an image on adrawing and making them fit together in a structure. Over a period of 40 years, hegradually learn about the mathematical rules and about techniques used by other

cultures, and the science of packing molecules. Within each interlocking shape, halfof the boundary is a clone of the other half. These identical sections of the boundarymake it possible for neighboring shapes to mate together without gaps. This pre -condition often prevents an arbitrary shape from mating together, or it makes a goodfitting shape look meaningless (absurd silhouette). Here is another rule, theboundary of a silhouette cannot cross itself. The vertices of Escher's lizard drawingform a regular hexagon. Three of the vertices are rotation points for the 3 pairs ofline segments. When you replace the straight edges of the hexagon with thesecurved lines, the lines wind in and out of the hexagon. It is difficult to let the linescome to close proximity (to form the legs of the lizard) and yet never intersectanother boundary line. Benoit Mandelbrot stated this as "self avoiding curves" in hisdiscussion of Koch Island. Through 40 year of practice, Escher found a way tomediate the need for the eyeballs to find meaning and beauty, and the need for themathematical rules to be followed. This is a tension which haunts every student whotries to use these computer programs to draw interlocking shapes. You lack thefreedom to draw the clone half of the silhouette. Escher has a wonderful descrip tionof this thinking process in his book "Exploring the Infinite". Here is a direct quote: "Acontour line between two interlocking figures has a double function, and the act oftracing such a line therefore presents a special difficulty. On either side o f it, a figuretakes shape simultaneously. But, as the human mind can't be busy with two things atthe same moment, there must be a quick and continuous jumping from one side tothe other. The desire to overcome this fascinating difficulty is perhaps the v eryreason for my continuing activity in this field".

Escher's repeating lizards have 4 properties: (1)The drawing is periodic and canextend to infinity. (2) The lizards can rotate 1/3 turn and superimpose on anotherlizard of a different color. (3) There are no gaps between the adjacent lizards, so thetiles interlock. (4)There is only one tile with 3 different colors. The first 2 aspects aresymmetry properties of the aggregate. The 3rd and 4th are properties of the tileboundary (line segments of the tile). The study of symmetry in crystallography giverise to the 17 wallpaper groups. There is an international notation (IUC) whichdescribes pattern properties such as m(mirror), g(gliding mirror), p1(translation).


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Many people said the mathematics of symme try describes the properties of Escherdrawings. But the 3rd and 4th properties are missing from the "Wallpaper groups"which is based on aggregates and not tiles. A tile is an enclosure of finite area.There is a close relationship between the tile symmetry and the aggregatesymmetry. Robert Fathauer remarked to me that all tilings are patterns, but apatterns is not necessary a tile. A pattern(aggregate) may not have boundary or

enclosure or finite size. Most people interested in Escher -like drawings want todesign tiles by a step-by-step cookbook method. The mathematics guides andrestricts design, but the mathematics will not replace vision. This is why so fewinterlocking shapes have ever been designed successfully. Heesch studied tileproperties and defined 28 basic tile formation based on line segments. Here we havemoved one level down and talk about movements of the line segments. The tile itselfis an assembly of 3 to 6 line segments which always come in pairs. Computerprograms are available that speed up the design of such tiles and patterns using themathematics of symmetry.

Now let us look at how the pattern repeats to infinity. Each regular pattern has tileproperties and cluster properties. The tile, which is similar to a molecule, may repeat

in rotation, translation and gliding mirror. We can build a cluster of molecules, similarto a small crystal that repeats only in translation. If you take the Escher example ofthe lizard and the tumbling man, a cluster consist of one of each color lizard (3 in thecluster). This cluster can shed all rotation and reproduce by translation. Lineartranslation is one of the easiest things for the computer to do: increment the x and yco-ordinates by the period. These cluster contains 1 to 6 molecules. There are atleast 2 translation axes and sometimes 3 translation axes. Doing tessellation withJAVA program means finding a cluster of lines or polygons and the shift vectors forthis group. Usually, a x-vector is used to repeat the shape from left to right. Thesecond vector may be at 60 degrees to x-vector when the next row has an offset inthe x-direction. This offset is typically half the regular x displacement. To make awebpage wallpaper, the two vectors must be orthogonal (perpendicular). The x and yvectors must have units equal to the number of pixels in the width and height of theGIF or JPEG file. This is done by regrouping the cluster, cut some off a shape andpatch that onto the other side. As Hop David said "If you want the tiles to appearseamless, take care that the left side matches the right side and that the top matchesthe bottom." The wallpaper tiles are straightly rectangular. Once you have the tile, toactivate this tile on the webpage is easy, because HTML has a build -in tiling functionin BODY statement with the parameter "BACKGROUND=catcat3.gif". It is interestingto look for the cluster in the JAVA graphics examples.

Let's summarize the mathematics. The lines are congruent and composed of anarray of pixels. The forms make sense to the eye but has no mathematical

significance. The tile is the interlocking shape and may be composed of several sub -divisions which are recognizable forms. Clusters of 1-6 tiles can form a translationalrepeating unit. For this translational unit to be suitable for wallpaper on a webpage,make sure that the tile is rectangular and the two axes of repetition are same as thewidth and height of the tile. Engineers have used interlocking shapes to conservematerials in stamping . The links and references at the bottom of this page will showmany uses of tessellation in different art and science professions.


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I developed a set of computer programs at the University of Illinois using the abovemathematics. In 1978, the programs were used for students to drawing Escher -likedesigns. People interested in this kind of work can check the technicalpapers."Automatic Generation of Interlocking Shapes" Computer Graphics andImage Processing April 1979, Volume 9, Number 4, pp. 333 -353."InterlockingShapes in Art and Engineering" Computer Aided Design, January 1980, Volume 12

Number 1, pp.29-34. These journals can be found in engineering libraries at majoruniversities.

Wikipedia

Escher's first print of an impossible reality was Still Life and Street, 1937. His artistic

expression was created from images in his mind, rather than directly from

observations and travels to other countries. Well known examples of his work also

include Drawing Hands, a work in which two hands are shown, each drawing the

other; Sky and Water, in which light plays on shadow to morph the water background

behind fish figures into bird figures on a sk y background; andAscending and

Descending, in which lines of people ascend and descend stairs in an infinite loop,

on a construction which is impossible to build and possible to draw only by taking

advantage ofquirks of perception and perspective.

He worked primarily in the media of lithographs and woodcuts, though the

few mezzotints he made are considered to be masterpieces of the technique. In his

graphic art, he portrayed mathematical relationships among shapes, figures and

space. Additionally, he explored interlocking figures using black and white toenhance different dimensions. Integrated into his prints were mirror images of cones,

spheres, cubes, rings and spirals.

In addition to sketching landscape and nature in his early years, he also sketched

insects, which frequently appeared in his later work. His first artistic work, completed

in 1922, featured eight human heads divided in different planes. Later around 1924,

he lost interest in "regular division" of planes, and turned to sketching landscapes

in Italy with irregular perspectives that are impossible in natural form.


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superpositionof perboli pl neon fi ed di ensional plane, and t e

incorporationof t ree-di ensional objectssuchasspheres, columnsandcubes into

his orks. orexample, inaprint called eptiles", hecombined t oand three-

dimensional images. Inoneofhispapers, Escheremphasi ed the importanceof

dimensionalit anddescribedhimselfas"irritated"by flat shapes: "I make themcomeout of theplane."

Waterfall,

Sculptureof thesmall stellateddodecahedronthat appears in Escher'sGravitation. It

canbe found in front of the"Mesa+"buildingon the ampusof the ente

ni ersity.

Escheralsostudied themathematical conceptsof topology. e learnedadditional

concepts inmathematics from the Britishmathematician ogerPenrose. rom this

knowledgehecreatedWaterfallandUp andown, featuring irregularperspecti es

similarto theconcept of theMbiusstrip.


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Escher printed Meta or

osis Iin 1937, which was a beginning part of a series of

designs that told a story through the use of pictures. These works demonstrated a

culmination of Escher's skills to incorporate mathematics into art. In Meta

or

osis

I, he transformedconvex polygons into regular patterns in a plane to form a human

motif. This effect symbolizes his change of interest from landscape and nature toregular division of a plane.

One of his most notable works is the piece Meta or

osis III, which is wide enough

to cover all the walls in a room, and then loop back onto itself.

After 1953, Escher became a lecturer at many organizations. A planned series of

lectures in North America in 1962 was cancelled due to an illness, but the

illustrations and text for the lectures, wr itten out in full by Escher, were later

published as part of the book Esc er on Esc er. In July 1969 he finished his last

work, a woodcut called Snakes, in which snakes wind through a pattern of linked

rings which fade to infinity toward both the center and the edge of a circle.

Over his life, the Dutch graphic artist M.C. Escher created over a hundred

ingenious tesselations in the plane. Some were simple and geometric, used as

prototypes for more complex endeavors. But in most the tiles were recognizable

animal forms such as birds, fish and reptiles. The definitive reference on Escher's

divisions of the plane is Doris Schattschneider's Visions of Symmetry, now in a

second edition. A good place to view some of Escher's work online is the World of

Escher. Be sure also to check out these examples from Escher's

notebook: E25, E70, and E72.

Escher was able to discover such tilings through a combination of natural ability

and sheer determination. Can we automate the discovery of tilings by recognizable

motifs? More formally, we pose theEscherization problem :

Given a shape S, find a new shape Tsuch that:

1. Tis as close as possible to S; and

2. Copies ofTfit together to form a tiling of the plane.

We have developed an algorithm that can produce reasonable solutions to the

Escherization problem. It is based on three large components:

A parameterized space of tilings. We develop a parameterization of the

space of isohedraltilings, a family of tilings that can express the designs

Escher created. Every possible tile shape boils down to a sequence of real

numbers. An interactive tool lets us modify these numbers in a natural way

(corresponding to Escher manipulating drawings by hand). More


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importantly, we can direct the computer to search through possible

sequences of numbers, trying to find good tilings by brute force.

A comparison function for shapes. We can dial up a collection of tile

shapes using the parameterization above. We need to know how well each

of these tiles approximates the goal shape provided by the user. We use a

metric from the computer vision literature that efficiently finds theL2distance between two polygons. Our objective then becomes to locate the

particular tile shape that compares most favourably to the goal shape.

An optimization algorithm. We need a meaningful strategy for sorting

through the large shape of possible tile shapes for the best one (the one that's

closest to the goal shape). We use a simulated annealing algorithm

from Numerical Recipes. When suitably tuned, it tends to find good tile

shapes without getting stuck in objectionable local minima.

Dihedral Escherization

Escher also created a number of dihedral tilings: designs featuring twodifferent shapes. He was fond of these designs, since with more than one

shape it was possible to tell a story, to have the shapes complement each

other somehow.

The Escherization problem can be adapted in an obvious way to two goal

shapes. We can also extend the space of tilings by including in the

parameterization a path that splits an isohedral tile shape into two pieces.

We compare the two pieces to the goal shapes and run the optimization as

before, attempting to minimize the maximum of the two comparisons.

Dihedral Escherization can also be exploited to create designs in the style of

Escher's Sky and Water.

We need to restrict the search to a narrower classof possible tiling types (what Dress calls Heaven and Hell Patterns). Once

the desired tiling has been discovered the rest of the process is fairly easy,

since we already have an association between a realistic, user-supplied goal

shape and an abstract, more geometric tile shape. We just need to morph

between those two extremes. Here's one example, based on Rembrandt

and Mrs. van Rijn above.

Aperiodic Escherization

The Penrose tilings P2 (kites and darts) and P3 (thin and thick rhombs) can

also be parameterized and fed to the Escherization system. It's much harder

to discover satisfying results, because of the idiosyncratic shapes of the

Penrose tiles. It's exciting to be able to produce these aperiodic pictures

Penrose and Escher were friends, but sadly Escher passed away before

Penrose discovered P2 and P3. Penrose has asked what sorts of designs

Escher would have been able to create from these tilings.


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Non-Euclidean Escherization

Escher made four wonderful Circle Limitpatterns, designs based on non-

Euclidean geometry. Escher didn't have the mathematical background tomanipulate hyperbolic patterns symbolically, but he certainly had the

intuition necessary to create pictures of them.

Our Escherization algorithm cannot easily be translated into non-Euclidean

geometry, for deep reasons having to do with the notion of shape, as used in

the shape comparison metric. However, for some isohedral tiling types it is

possible to transfer the outputof Escherization from the Euclidean plane to

the hyperbolic plane or to the surface of a sphere. The process is a kind of

non-linear warp of a piece of the Escherized tiling.

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