Geometry, Aperiodic tiling, Mathematical symmetry.

Transcription

1 Conference of the International Journal of Arts & Sciences, CD-ROM. ISSN: :: 07(03): (2014) Copyright c 2014 by UniversityPublications.net Tiling Theory studies how one might cover the plane with various shapes. Aperiodic tilings are nonperiodic patterns that cannot be rearranged to form periodic tilings. Examples of this aperiodic symmetry are found in a variety of places from medieval Islamic architecture to 20 th Century Penrose tilesand also characterize newly discovered quasicrystals. Construction techniques, other than the usual methods for producing periodic tilings, are required to successfully generate these aperiodic patterns. Historic manuscripts like the Topkapi Scrollprovide insight into how these intricate designs might have been produced in the past. This paper focuses on the discoveries of aperiodic symmetry with a brief historical perspective and examines the mathematics of dissection and iterative techniques for generating some well-known patterns. Geometry, Aperiodic tiling, Mathematical symmetry. Throughout history, virtually all cultures around the world have created simple and intricate symmetric designs and ornamentation to express aesthetic feelings and to decorate their dwellings, workplaces, highways, and places of worship. As the patterns developed from simple ornaments centered around a point to more intricate repetitive patterns, the methods for constructing these patterns evolved as well. From ancient times straightedge and compass constructions allowed for precise patterns. Numerous historical sites exist such as the Alhambra [6], in Granada, Spain which shows the mastery of skilled craftsmen in producing intricate precise ornamentation that has survived to the present day. In the modern world, recursive computer techniques allow for these patterns to be studied and produced with new tools. Students equipped with Geometer s Sketchpad [5] can create modern versions of historical designs. Recent discoveries have found intriguing connections between modern day aperiodic tilings and medieval Islamic architecture as well as to the real-world research into quasicrystals. Mathematical symmetry provides tools for measuring the symmetry of ornamental patterns and thereby allows for these patterns to be classified. It is well know that for two-dimensional patterns, symmetry is expressed in terms of the four possible planar transformations for a pattern, rotations, reflections, translations, and glide reflections. If all the symmetries of a pattern are given the additional structure of multiplication, they form a group referred to as a symmetry group.the simplest finite ornamental designs 343

2 344 Dissection Methods for Aperiodic Tilings: From Medieval Islamic Architecture to Quasicrystals formed around a central point have symmetry which may include rotations and reflections and the resulting finite symmetry groups are either cyclic in the case of a pattern with all rotational symmetry or dihedral in the case of a pattern with both rotational and reflective symmetry. Tiling theory describes methods for constructing and classifying ornamental patterns in the plane. A tiling is defined as a covering of the plane by finite shapes so that there are no gaps or overlaps between tiles. The simplest tilings are the three regular tilings which involve covering the plane with congruent copies of one regular polygon. The three regular tilings are formed by equilateral triangles, squares, and regular hexagons. The eight tilings by two or more regular polygons with the condition that each vertex is surrounded by polygons in the same way are called semiregular. The regular and semiregular tilings have been known since antiquity. If the condition that the tiles be regular is relaxed, there is a wide variety of possibilities for tiling the plane. Group theory provides tools for measuring the symmetry of ornamental tilings or patterns and thereby allows for these patterns to be described and classified. For two-dimensional patterns, symmetry is expressed in terms of the four possible planar transformations that may leave a pattern unchanged, namely rotations, reflections, translations, and glide reflections. If all the symmetries of a pattern are given the additional structure of multiplication of transformations, they form a group referred to as the symmetry group of the pattern. The simplest finite ornamental patterns formed around a central point have symmetry which may include rotations and reflections and the resulting finite symmetry groups are either cyclic in the case of a pattern with all rotational symmetry or dihedral in the case of a pattern with both rotational and reflective symmetry. Patterns that can be repeated by translations in two linearly independent directions to cover the entire plane are called periodic and may also contain rotational, reflective, and glide symmetry. The symmetry groups for these infinite periodic patterns form the seventeen two-dimensional crystallographic groups [4]. Tilings of the plane that cannot be translated in two linearly independent directions are called nonperiodic. The simplest nonperiodic tilings involve modifying one of the tiles to be unique so that no translation of the tiling is possible. Other nonperiodic tilings involve patterns that are generated from a central point and progress radially outward orthat spiral around one or more points. Another technique involves combing congruent copies of one tile to form a larger version of the tile and then continuing this iterative step to cover the plane. A nonperiodic tiling which cannot be rearranged to form a periodic tiling along with the additional property that the tiling does not contain arbitrarily large periodic patchesis called aperiodic [7]. In 1964 Hao Wang conjectured that no such set of aperiodic tiles exists. In 1966 Wang s Conjecture was proved false and shortly thereafter Robert Berger found a set of 20,426 Wang Dominoes that tile the plane aperiodically. This was improved on when aperiodic tiles were found by Donald Knuth (92 tiles) in 1968, Raphael Robinson (6 tiles) in 1971, and Roger Penrose (2 tiles kite and dart) in In 2008, Peter Lu [1] discovered a connection between aperiodic tilings and medieval Islamic architecture in Iran, Uzbekistan and other historic locations (Figure 1a). The 15 th Century Topkapi Scroll contains instructions for matching rules which involve five girih tiles (decagon, hexagon, bowtie, pentagon, and rhombus. The enhanced lines red lines on Drawing 28 of the Topkapi (Figure 1b) show how the tiling is dissected and the dotted lines show the girih tiles, [2].

3 Raymond Tennant 345 Portal of the Darb-i Imam Shrine, Isfahan, Iran, 15 th Century CE. Topkapi Scroll Late 15 th Century, C Whether a single tile (monotile) tiles the plane aperiodically in a similar fashion has been under investigation for a number of years. In 2010 Joan Taylor in Tasmania developed a single hexagonal prototile [3] which contains matching rules on both sides of the tile allowing it to be flipped (Figure 2a). To construct the tiling, black lines must connect and the purple one-sided arrow heads must align.for this tile to be modified so that it can tile by one shape alone without being flipped requires that the tile not be one contiguous piece (Figure 2b). For this tiling, all the components of one color are considered a tile. Therefore, the question remains open as to whether there is a single contiguous monotile that tiles the plane in an aperiodic manner. Two-sided Hexagonal Tile. One-sided Non Contiguous Tiles. All tilings can be constructed by giving an algorithm to follow in building the pattern. In the case of periodic tilings (Figure 3) this process is straightforward and generally involves producing a fundamental finite region and then completing the tiling by translations. In the case of nonperiodic tilings, the algorithm for generating the pattern has matching rules for connecting tiles to one another for constructing the pattern outward toward infinity (Figure 4).

4 346 Dissection Methods for Aperiodic Tilings: From Medieval Islamic Architecture to Quasicrystals Periodic Tiling by Hexagons Periodic Tilings. Periodic Tiling by Octagons and Squares Above Penrose Tiling with Kites and Darts Below Penrose Tiling with Matching Rules Periodic and Aperiodic Tilings. Above Portal from the Darb-I Imam Shrine Below Overlay of a Girih Tiling Constructions of both periodic and nonperiodic tilings can be described by the method of dissection where each prototile is dissected into smaller versions of all the prototiles that make up the overall

5 Raymond Tennant 347 pattern. The information on numbers of smaller tiles involved in this process can be used to form a matrix of the dissection. As examples, the periodic tiling by regular hexagons, each tile can be dissectedd into four smaller regular hexagon while the semiregular tiling by regular octagons and squares can be dissected in a similar fashion. The aperiodic tilings by Penrose tiles and by girih tiles are dissected according to their respective matching rules. Table 1 below shows the prototiles with matrices and eigenvalues is given for the four tilings described above. The non-zero eigenvalues for the two aperiodic tilings are multiples of powers of the golden ratio which reinforces that if the dissection matrix has irrational eigenvalues then it follows that the tiling is nonperiodic. Matrices of Dissection and Eigenvalues Tilings Prototiles Dissection Matrix Eigenvalues Hexagons 4 4 Octagons and Squares , and 64 Penrose Kites and Darts (Roger Penrose) and Kites and Darts Girih Tiles (Peter Lu) and 4 Bowties, Hexagons and Decagons The field of aperiodic tilings remains an interesting crossroads of mathematical symmetry, art, cultural ornamentation and quasicrystals with the open question being the search for a contiguous monotile that tiles the plane aperiodically.

6 348 Dissection Methods for Aperiodic Tilings: From Medieval Islamic Architecture to Quasicrystals Peter J. Lu and Paul J. Steinhardt,, Science. vol. 315, pp , Gülru Necipolu, The Topkapi Scroll Geometry and Ornament in Islamic Architecture (Sketchbooks & Albums), Getty Center Publications, Joshua E.S. Socolar, Joan M. Taylor,, Journal of Combinatorial Theory, Series A 118, pp , Raymond F. Tennant,, Visual Mathematics: Art and Science Electronic Journal, vol. 4, no. 4, December, Raymond F. Tennant,, BRIDGES/ISAMA International Conference Proceedings, pp , Raymond F. Tennant,, Teachers, Learners and Curriculum, Vol. 2, pp , Raymond F. Tennant,, Symmetry: Culture andscience, vol. 19, nos. 2-3, pp , 2009.