Abstract Combining Art and Mathematics A Closer Look at the Automation of Escher’s Combina

Combining Art and MathematicsA Closer Look at the Automation of Escher’s Combinatorial PatternsMark OmranDecember 7, 2000Stetson UniversityDeLand, FLAbstract:An Internet-based application that will allow users to explore and createunique tiling patterns will be implemented. Database driven, this softwarewill give the users the ability to choose an assortment of pre-designed motifs and arrange/color them specific to their personal taste. Finally the userswill be able to view the created tile in a mach-three dimensional modeled room. The ideas and inspiration behind this is a study motivated from M.C. Escher’s mathematical motifs and tile art pieces.I.)Basis of ProposalBeginning with M.C. Escher, forms of tiling have often intrigued many persons. The diverse way that Escher arranged different motifs(square-like stylized designs), depending on certain polygons andhyperbolic planes, gave way to a very algorithmic way of thinking.Dr. Rick Mabry has expounded on this programmatic thought process toautomate Escher’s handwork using the software Mathematica. Mabry’sideas will be observed and utilized in designing a database drivenapplication for textile companies.The proposed project will utilize fundamentals of mathematical andcombinatorial algorithms to interpret the artistic ideas of M.C.Escher’s tiles. An Internet-based application designed to allowusers to independently create multiple tilings (discussed later),will mimic the automation sequence proposed by Mabry [2]. Databasedriven, this software will give the users the ability to choose anassortment of pre-designed motifs and arrange/color them to suittheir personal taste. Subsequent features will allow the user tovisualize the tiling design in a modeled three-dimensional room. Byusing client/server architecture with real-time interaction, the user will be able to view their decisions immediately. Furthermore, thisproject lends its entirety to the combining of mathematics and art,by means of a computer gateway.II.)Brief Historical Introduction on Escher IdeaEscher has always seemed to be interested in the mathematics behindhis art. The laws that were to fascinate Escher most were those ofthe regular division of the plane.‘For me it remains an open question whether [thiswork] pertains to the realm of mathematics or tothat of art.’ [1]Despite Escher’s artistic stardom in 1913, it was not until 1936 that Escher began to study Moorish tilings in Alhambra.‘The fitting together of congruent figures whose shapesevoke in the observer an association with an object or aliving creature intrigued me increasingly after that firstSpanish visit in 1922. And although at the time I wasmainly interested in free graphic art, I periodicallyreturned to the mental gymnastics of my puzzles. In about1924 1 first printed a piece of fabric with a wood block ofa single animal motif which is repeated according to aparticular system, always bearing in mind the principlethat there may not be any "empty spaces…I exhibited thispiece of printed fabric together with my other work, but itwas not successful. This is partly the reason why it wasnot until 1936, after I had visited the Alhambra a secondtime, that I spent a large part of my time puzzling withanimal shapes.’ [6]Escher began experimenting with the idea of plane-filling techniques, shapes and transformations. After settling on a specific design for a motif, many more thoughts intrigued Escher. To create more interesting and analytical art, Escher began exploring methods of creating a 2x2 tile by taking four copies of this motif and rotating each through some number of counterclockwise 90 degree turns [2]. He would then use this tile in his signature plane-filling art. The result was a highly mathematical art piece whose foundation rested upon a sole design. This lucrative concept has brought about numerous mathematical and computational studies. One of thesestudies will be examined thoroughly.III.)Studied WorksThe research of Dr. Rick Mabry will be examined to provide afoundation for the proposed application [2]. Mabry conclusivelyproved how Mathematica is used in automating Escher’s originalinvestigations of repeating patterns to resemble a more familiarEuclidean plane. Escher’s rudimentary idea is simple; begin with adecorated motif and compose a 2X2 tile-design (tile). The resultingtile is achieved through counterclockwise-90 degree rotations of thebeginning motif [2].Figure 1Utilizing the Escher’s original rotation, each unique tile willcorrespond with a signature [Schattschneider 1997]. The integers 1, 2, 3, and 4 are used to denote counterclockwise turns of 0°, 90°,180°, and 270°. Thus the signature, in matrix format, yields thepattern on the right in Figure 1.To automate Escher’s process multiple functions must be comprised. A majority of Mabry’s work rests on a computational function,EscherTiling. Before an analysis can be completed on this function,a few fundamental methods should be observed. Rotation andtranslation of an object are the first of these methods. Thesimplest approach to achieve either of these ideas is to replace each point in the object by its translated or rotated image.Note: For all purposes of Mathematica 3, a restriction holds true for the item being replaced: the item must be represented by a numericobject (NumericQ). Observe the following group of functions:The rotate method begins by validating the numerical value of the variables x and y. After subtracting {0.5, 0.5} from {x, y} (motifs current position), the rotation matrix is the applied, and finally {0.5, 0.5} is then added back to the resulting matrix. The addition and subtraction of {0.5, 0.5} occurs because the functions areworking with motifs in the unit square. Translating the motiffollows the same validation process of x and y and analyzes the numerical value of the variables being used to move the motif to the new position (u and v). After validation, the original values are added to u and v to result in the new desired position.Before moving the discussion to the conclusive function EscherTiling, geometry weaving and color instructions must be examined. Weaving varies the layers in which the polygons of a motif are reversed. Escher noticed that there were numerous motifs that could be created by allowing this varying layer order. Escher referred to wovenmotifs by appending the letter “a” to the integers in his signatures. The following code examined by Maybry takes into account coloring of the polygons (discussed later), as well as the order in which the geometry is displayed.MotifPieces, simply accepts a color list and applies it to theoriginal motif (using motif[]) and the reversed motif (usingmotifA[]). Notice the reverse command in motifA[]. This will apply the reversing of the geometry in the motif to yield the following diagrams:Original Motif and the Reversal (MotifA)IV.)Implementation•Database Driven Application•Three Dimensional ResultOnce a satisfactory design (either for floor implementation orwall implementation) has been chosen by the client, a threedimensional extrapolation will be created. The client will be agiven a choice of multiple mach-three dimensional rooms which arepre-fabricated awaiting only the created design from the customer.Utilizing the programming languages PovRay and/or Open GL, theinformation from the customer design will be imported and used ina pattern filling method to decorate the are desired.V.) RequirementsAdministrators: Rules and Requirements•Utilize all features of software through interface.•Must type in a name and password.•Be able to click and turn tile for different look•Expand matrix of tile signature for multiple tile designs•Be able to see signatures for each different combination•See image that displays the tile completely•Be able to resize tile (more smaller tile in a given space orfew large tiles in the same space)•Have some type of zoom tool•Be able to access all user information - Save Users, loadusers, load information•Associate favorite signature/color instructions with tiles•Type in signature by hand / changing signatureDatabase Information: Rules and Requirements•All Motifs•What customer chose what tile•All Designs (signatures)•Any Orders PlacedCustomer: Rules and Requirements•Options for Signatures(s) limited•Be able to adjust Signatures•Be able to keep track of what they have been interested in sofar•Change Color instructions•Have a status that depends on if they are new, have boughtanything, or a repeat customer•Depending on customer status, interface may be limited.•Will need a login name and password.•As status increases, so might the amount of functionality theysee.•Customer will never have as much control through the interfaceas Admin.Database Information: Rules and Requirements•Name•Unique ID•Login Name•Password•Signature(s) Chosen•Color Instruction(s) Chosen•Tiles they are interested in (spacing and type of tile)•Motifs they are interested•Any images that have been created for them (names of)•What the have bought•Address•Phone Number•Any Comments•Tiles: Rules and Requirements on Information stored•Name•Unique Identifier•Signature(s) - Could be created dynamically in code•Design Aspect•Color Instructions•Big Tile Size•Date EnteredVI.) Entity Relationship ModelVII.) Normalized Table LayoutVIII.)ConclusionIX.)References:[1] Mathematical Art of M.C. Escher,/platonic_realms/minitext/escher.html[2] Rick Mabry, Stan Wagon and Doris Schattschneider, “AutomatingEscher’s Combinatorial Patterns," Mathematica in Education andResearch, 5, no. 4 (1996), 38-52.[3] Dan Davis, On a tiling scheme from M. C. Escher, ElectronicJournal of Combinatorics 4 (no. 2) 1997, #R23.[4] Doris Schattschneider, Visions of Symmetry, Notebooks, PeriodicDrawings, and Related Work of M. C. Escher, W. H. Freeman, New York, 1990.[5] Doris Schattschneider, Escher’s combinatorial patterns,Electronic Journal of Combinatorics 4 (no. 2) 1997, #R17.[6] M.C. Escher, J.L. Locher, F.Bool, M.C. Escher : His Life andComplete Graphic Work。