- Email: [email protected]
Fractal tilings based on kite- and dart-shaped prototiles
Computers & Graphics 25 (2001) 323}331
Chaos and Graphics
Fractal tilings based on kite- and dart-shaped prototiles Robert W. Fathauer Tessellations Company, 688 W. 1st St., Ste. 5, Tempe, AZ 85281, USA
Abstract A family of `fractal tilingsa, or `f-tilingsa is presented in which the curves de"ned by the boundaries are fractals. There are two distinct branches, one with three f-tilings based on kite-shaped prototiles, and the other with three f-tilings based on dart-shaped prototiles. This appears to be the "rst report of a systematic exploration of tiling-like constructs with fractal character. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Tiling; Fractal; Self-similar; Self-similarity; Fractal tiling; Prototile
Fractals and tilings are two "elds in mathematics with strong visual appeal. In spite of this, there has been virtually no work on combining these two topics. The Dutch graphic artist M.C. Escher executed several prints in which tiles are reduced in size at either the center or boundary of the design [1]. In some of the these, the tiles are more-or-less similar in Euclidean space, while in others hyperbolic geometry is used to depict an in"nite tiling in a print of "nite area. While these designs possess self-similarity, none of the curves de"ned by the boundaries of the tilings are fractal. In GruK nbaum and Shephard's book Tilings and Patterns [2], a tiling is de"ned as a countable family of closed sets (tiles) which cover the plane without gaps or overlaps. The constructs described in this paper do not cover the entire Euclidean plane; however, they do obey the restrictions on gaps and overlaps. To avoid confusion with the standard de"nition of a tiling, these constructs will be referred to as `f-tilingsa, for fractal tilings. The tiles used here are `well behaveda by the criterion of GruK nbaum and Sheppard; namely, each tile is a (closed) topological disk. These f-tilings are edge to
E-mail address: [email protected] (R.W. Fathauer).
edge; i.e., the corners and sides of the tiles coincide with the vertices and edges of the tilings. However, they are not `well behaveda by the criteria of normal tilings in one particular; namely, they contain singular points, de"ned as follows. Every circular disk, however small, centered at a singular point meets an in"nite number of tiles. Since any f-tiling of the general sort described here will contain singular points, this will be not be considered a property that prevents an f-tiling from being described as `wellbehaveda. The "rst example of an f-tiling with a fractal boundary of which the author is aware was executed as an Escheresque print by Peter Raedschelders in 1985, though none of his subsequent prints have fractal boundaries [3]. Around 1990, Chaim Goodman-Strauss constructed some tilings with fractal boundaries in which the prototiles were Koch snow#akes and related tiles [4]. The next known example was executed by the author in 1993 [5], who was unaware at the time of either Raedschelders' or Goodman-Strauss' work. Other work has been reported in which the tiles themselves are fractal objects, but these are used to form tilings of in"nite extent
His prints can be seen on the World Wide Web at http://www.planetinternet.be/&praedsch/index.htm/. Additional f-tilings can be seen on the World Wide Web at http://www.tessellations.com/encyclopedia/html.
0097-8493/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 1 3 4 - 5
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R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
in the Euclidean plane in a regular, non-fractal manner [6]. While there is unique and interesting mathematical content in these f-tilings, the author's primary interest is esthetics. Fractals as visual objects are quite fascinating, and these f-tilings are fractal objects with a di!erent look than other types of fractals. In this paper, a family of f-tilings is presented in which the curves de"ned by the boundaries are fractals. These are based on kite- and dart-shaped polygonal prototiles. These f-tilings share the following properties: 1. Each f-tiling is constructed from a single prototile, to which all the tiles are similar. 2. The prototiles have edges of two lengths, denoted `longa and `shorta. 3. The f-tilings are by design edge to edge. As a consequence of this choice, from a given generation of tiles to the next smaller generation, the tiles are scaled by the ratio of the short to long edges of the prototile. 4. The f-tilings are bounded in the Euclidean plane. There are two distinct branches of this family of ftilings, each with its own matching rules. In the "rst, the prototiles are dart shaped, while in the second they are kite shaped. There appear to be only three members of each branch that allow well-behaved f-tilings; i.e., ones in which the tiles neither overlap nor leave gaps. The six prototiles for these f-tilings are shown in Fig. 1. While the author has not constructed a proof showing that the number of such f-tilings using kite- and dart-shaped prototiles is limited to six, he has convinced himself through extensive trial and error that this is the case. Typically, an f-tiling with n-fold rotational symmetry needs to be carried through n generations to determine if
Fig. 1. Six kite and dart prototiles that allow well-behaved f-tilings.
overlaps will occur. Note that there are innumerable f-tilings based on other types of prototiles [5]. The f-tilings are constructed as follows. The long edges of the largest ("rst) generation of tiles must match up, which is most readily achieved by arranging them as shown in Fig. 2. The second generation of tiles must then be arranged around these according to some simple set of rules. If the f-tilings are to be edge to edge, the only choices are to place the vertex between the two long edges of a second generation tile at either the vertex between the two short edges of a "rst generation tile or at the vertex where short edges of two "rst generation tiles meet. It turns out that the "rst choice only works for some kite-shaped prototiles and the second only works for some dart-shaped prototiles, as shown in Fig. 2.
Fig. 2. Arrangement of the "rst (lighter gray) and second (darker gray) generations of kite and dart tiles.
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
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Fig. 3. Three well-behaved kite f-tilings. Each blow up shows a region encompassing singular points within the boundary and a portion of the boundary.
These same matching rules are used for all subsequent generations; i.e., they completely de"ne the f-tilings obtained after an in"nite number of iterations. Note that some second generation tiles meet in a way that "rst generation tiles do not. For example, the two tiles labeled `aa and `ba in Fig. 2 meet along short edges, which does not happen for "rst generation tiles. These di!erences are what create a fractal boundary in many cases for such
tilings, rather than a polygonal one. In this particular case, this meeting of two tiles along short edges creates a low point (closer to the center) in the expanding boundary. There are more and more such low points for each subsequent generation, resulting in the fractal boundary shown at the top left of Fig. 3. Six full f-tilings are shown in Figs. 3 and 4. Note that all of these contain singular points within the boundary
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Fig. 4. Three well-behaved dart f-tilings. Each blow up shows a region encompassing singular points within the boundary and a portion of the boundary.
of the f-tiling, not just at the boundary. The dart f-tilings in fact contain two distinct types of singular points, a "rst type with local two-fold rotational symmetry and a second type with a di!erent local rotational symmetry for each f-tiling. A region containing these singular points and a portion of the boundary for each f-tiling is shown as a blow up in the "gures. Also note that these f-tilings
have n-fold global rotational symmetry, where n"6, 8, and 12 from top to bottom for each "gure. It is easily shown that the scaling factor from one generation of tiles to the next smaller generation is simply tan(n/) for all six f-tilings. The local groups of tiles in the vicinity of a singular point (of the second type for the dart f-tilings) have six-, four-, and three-fold rotational symmetry,
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
327
Fig. 5. The six-fold dart f-tiling, carried through nine generations, with a di!erent color assigned to each generation.
respectively. The "rst dart-based f-tiling is shown in more detail and in color in Fig. 5 in order to better reveal its complex structure. In addition to rotational symmetry, these f-tilings possess mirror symmetry. Each f-tiling possesses two distinct types of lines of mirror symmetry, one passing through the center of the f-tiling and along long edges of the largest generation of tiles, and another passing through the center of the f-tiling and bisecting two opposing tiles of the largest generation. There are n/2 lines of each type for each f-tiling. In addition to rotation and simple re#ection, which is a special case of glide re#ection, the other type of symmetry commonly used to describe tilings is translation [2]. These f-tilings do not possess translational symmetry in the usual sense. However, they do possess another type of symmetry commonly used to describe fractals, namely self-similarity. This self-similarity is local rather than global and can be linked to the other symmetries, including translation. If selected portions of one of the f-tilings are scaled and translated by an appropriate vector, then
those portions would perfectly overlap the tiles already there. For example, in "rst dart f-tiling, the local group of tiles around each of the singular points of the "rst type are the same except for di!erences in scale or angular orientation. The "gures shown here are all generated using Version 7.0 of the commercial drawing program FreeHand, on a Macintosh computer. The di!erent generations of f-tilings are constructed one by one using standard functions of the program, including cloning, rotating, grouping, and scaling. This process has been carried through a su$cient number of generations that di!erences between the "gure shown and the in"nite f-tilings are very minor for a "gure of the size presented here. In practice, this means that between four and ten generations are shown. While this construction process sounds laborious, it can be done relatively quickly by intelligent use of the program's features. Obviously, a computer program or plug in for an existing program could be written to generate such f-tilings automatically given a prototile and set of matching rules.
Fig. 6. Tilings obtained by perforating the "rst kite f-tiling once and twice. In each perforation step, the largest two generations of tiles are removed.
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
329
Fig. 7. An f-tiling obtained by perforating the eight-fold kite f-tilings four times. In each perforation step, the largest generation of tiles are removed.
Increased complexity and visual richness can be built into these constructs by `perforatinga them. This refers to removal of the one or two largest generations of tiles, followed by completing the groups (matching up the exposed long edges) of the largest remaining generation, and then building inward with the same matching rules used to generate the original f-tiling. This technique works particularly well esthetically with the kite f-tilings, as shown in Figs. 6 and 7. The f-tiling of Fig. 7 is similar in many ways to the Sierpinski carpet [7]. There is a large center hole, surrounded by eight next generation holes, etc. This progression of holes could be made in"nite by carrying out an in"nite number of perforation steps. The resulting object would be more complex than the Sierpinski carpet, because the holes in the f-tiling are fractal shapes as compared to simple squares for the Sierpinski carpet. Another prototile that allows f-tilings can be constructed by combining the 6-fold prototile of each type, as shown in Fig. 8. There are two basic choices for placement of the second generation of these tiles. In addition, because the tile does not possess bilateral symmetry, the
Fig. 8. A prototile constructed by combining the six-fold kite and six-fold dart prototiles.
tiles can be mirrored between successive generations. These four possibilities result in four di!erent wellbehaved f-tilings, with three very di!erent boundaries, as shown in Fig. 9. Combining the eight- and twelve-fold kite and dart prototiles results in prototiles that fail to generate well-behaved f-tilings. It is also possible, of course, to make an Escheresque variation of any of these tiles. An example is shown in Fig. 10. The basis is the f-tiling shown in Fig. 9a, but it has been rearranged to form a periodic f-tiling suggesting waves. An f-tiling is thus used to depict
330
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Fig. 9. Three di!erent f-tilings constructed with the prototile shown in Fig. 8. (a) An f-tiling obtained by one choice of matching rules; (b) An f-tiling obtained by a second choice of matching rules; (c) An f-tiling obtained by the "rst choice of matching rules, but where the tiles are mirrored between successive generations. The fourth possible f-tiling, with the rules used for tiling (b), but with the tiles mirrored between successive generations, is similar in overall appearance to f-tiling (b).
natural objects, waves that have fractal character. A tile motif, "sh, is chosen that is sympathetic to the overall wave motif. This lends the design a dual interpretation. Is the viewer looking at water, or at a school of "sh? In conclusion, a family of constructs has been presented in which the curves de"ned by the boundaries are fractals. These `f-tilingsa are constructed
from quadrilateral prototiles according to a simple set of matching rules, yet the resultant f-tilings possess rich visual complexity. Modi"cations of the kite f-tilings have been presented that increase the complexity of the f-tilings. F-tilings have also been presented with very di!erent boundaries based on a single prototile created by combining a kite tile and a dart tile.
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
331
References [1] Bruno Ernst, The Magic Mirror of M.C. Escher. New York: Balantine, 1976. [2] GruK nbaum B, Shephard GC. Tilings and Patterns. New York: WH Freeman, 1987. [3] P. Raedschelders, Private communication. [4] Private communication from Chaim Goodman-Strauss; also Pentakoch, Ptolemy mathcard no. 5, Ptolemy mathcard Co. 2000. [5] Robert W. Fathauer, Proceedings of the M.C. Escher Centennial Congress, Rome and Ravello, Italy, June 24}28, 1998. Germany: Springer Verlag, 2001. In press. [6] Vince A. Replicating Tessellations. SIAM Journal of Discrete Mathematics 1993;6:501}21. [7] Peitgen H-O, JuK rgens H, Saupe D. Fractals for the Classroom. New York: Springer-Verlag, 1992.
Fig. 10. A design based on an Escheresque variation of the f-tiling of Fig. 9a.
Chaos and Graphics
Fractal tilings based on kite- and dart-shaped prototiles Robert W. Fathauer Tessellations Company, 688 W. 1st St., Ste. 5, Tempe, AZ 85281, USA
Abstract A family of `fractal tilingsa, or `f-tilingsa is presented in which the curves de"ned by the boundaries are fractals. There are two distinct branches, one with three f-tilings based on kite-shaped prototiles, and the other with three f-tilings based on dart-shaped prototiles. This appears to be the "rst report of a systematic exploration of tiling-like constructs with fractal character. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Tiling; Fractal; Self-similar; Self-similarity; Fractal tiling; Prototile
Fractals and tilings are two "elds in mathematics with strong visual appeal. In spite of this, there has been virtually no work on combining these two topics. The Dutch graphic artist M.C. Escher executed several prints in which tiles are reduced in size at either the center or boundary of the design [1]. In some of the these, the tiles are more-or-less similar in Euclidean space, while in others hyperbolic geometry is used to depict an in"nite tiling in a print of "nite area. While these designs possess self-similarity, none of the curves de"ned by the boundaries of the tilings are fractal. In GruK nbaum and Shephard's book Tilings and Patterns [2], a tiling is de"ned as a countable family of closed sets (tiles) which cover the plane without gaps or overlaps. The constructs described in this paper do not cover the entire Euclidean plane; however, they do obey the restrictions on gaps and overlaps. To avoid confusion with the standard de"nition of a tiling, these constructs will be referred to as `f-tilingsa, for fractal tilings. The tiles used here are `well behaveda by the criterion of GruK nbaum and Sheppard; namely, each tile is a (closed) topological disk. These f-tilings are edge to
E-mail address: [email protected] (R.W. Fathauer).
edge; i.e., the corners and sides of the tiles coincide with the vertices and edges of the tilings. However, they are not `well behaveda by the criteria of normal tilings in one particular; namely, they contain singular points, de"ned as follows. Every circular disk, however small, centered at a singular point meets an in"nite number of tiles. Since any f-tiling of the general sort described here will contain singular points, this will be not be considered a property that prevents an f-tiling from being described as `wellbehaveda. The "rst example of an f-tiling with a fractal boundary of which the author is aware was executed as an Escheresque print by Peter Raedschelders in 1985, though none of his subsequent prints have fractal boundaries [3]. Around 1990, Chaim Goodman-Strauss constructed some tilings with fractal boundaries in which the prototiles were Koch snow#akes and related tiles [4]. The next known example was executed by the author in 1993 [5], who was unaware at the time of either Raedschelders' or Goodman-Strauss' work. Other work has been reported in which the tiles themselves are fractal objects, but these are used to form tilings of in"nite extent
His prints can be seen on the World Wide Web at http://www.planetinternet.be/&praedsch/index.htm/. Additional f-tilings can be seen on the World Wide Web at http://www.tessellations.com/encyclopedia/html.
0097-8493/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 1 3 4 - 5
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R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
in the Euclidean plane in a regular, non-fractal manner [6]. While there is unique and interesting mathematical content in these f-tilings, the author's primary interest is esthetics. Fractals as visual objects are quite fascinating, and these f-tilings are fractal objects with a di!erent look than other types of fractals. In this paper, a family of f-tilings is presented in which the curves de"ned by the boundaries are fractals. These are based on kite- and dart-shaped polygonal prototiles. These f-tilings share the following properties: 1. Each f-tiling is constructed from a single prototile, to which all the tiles are similar. 2. The prototiles have edges of two lengths, denoted `longa and `shorta. 3. The f-tilings are by design edge to edge. As a consequence of this choice, from a given generation of tiles to the next smaller generation, the tiles are scaled by the ratio of the short to long edges of the prototile. 4. The f-tilings are bounded in the Euclidean plane. There are two distinct branches of this family of ftilings, each with its own matching rules. In the "rst, the prototiles are dart shaped, while in the second they are kite shaped. There appear to be only three members of each branch that allow well-behaved f-tilings; i.e., ones in which the tiles neither overlap nor leave gaps. The six prototiles for these f-tilings are shown in Fig. 1. While the author has not constructed a proof showing that the number of such f-tilings using kite- and dart-shaped prototiles is limited to six, he has convinced himself through extensive trial and error that this is the case. Typically, an f-tiling with n-fold rotational symmetry needs to be carried through n generations to determine if
Fig. 1. Six kite and dart prototiles that allow well-behaved f-tilings.
overlaps will occur. Note that there are innumerable f-tilings based on other types of prototiles [5]. The f-tilings are constructed as follows. The long edges of the largest ("rst) generation of tiles must match up, which is most readily achieved by arranging them as shown in Fig. 2. The second generation of tiles must then be arranged around these according to some simple set of rules. If the f-tilings are to be edge to edge, the only choices are to place the vertex between the two long edges of a second generation tile at either the vertex between the two short edges of a "rst generation tile or at the vertex where short edges of two "rst generation tiles meet. It turns out that the "rst choice only works for some kite-shaped prototiles and the second only works for some dart-shaped prototiles, as shown in Fig. 2.
Fig. 2. Arrangement of the "rst (lighter gray) and second (darker gray) generations of kite and dart tiles.
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
325
Fig. 3. Three well-behaved kite f-tilings. Each blow up shows a region encompassing singular points within the boundary and a portion of the boundary.
These same matching rules are used for all subsequent generations; i.e., they completely de"ne the f-tilings obtained after an in"nite number of iterations. Note that some second generation tiles meet in a way that "rst generation tiles do not. For example, the two tiles labeled `aa and `ba in Fig. 2 meet along short edges, which does not happen for "rst generation tiles. These di!erences are what create a fractal boundary in many cases for such
tilings, rather than a polygonal one. In this particular case, this meeting of two tiles along short edges creates a low point (closer to the center) in the expanding boundary. There are more and more such low points for each subsequent generation, resulting in the fractal boundary shown at the top left of Fig. 3. Six full f-tilings are shown in Figs. 3 and 4. Note that all of these contain singular points within the boundary
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Fig. 4. Three well-behaved dart f-tilings. Each blow up shows a region encompassing singular points within the boundary and a portion of the boundary.
of the f-tiling, not just at the boundary. The dart f-tilings in fact contain two distinct types of singular points, a "rst type with local two-fold rotational symmetry and a second type with a di!erent local rotational symmetry for each f-tiling. A region containing these singular points and a portion of the boundary for each f-tiling is shown as a blow up in the "gures. Also note that these f-tilings
have n-fold global rotational symmetry, where n"6, 8, and 12 from top to bottom for each "gure. It is easily shown that the scaling factor from one generation of tiles to the next smaller generation is simply tan(n/) for all six f-tilings. The local groups of tiles in the vicinity of a singular point (of the second type for the dart f-tilings) have six-, four-, and three-fold rotational symmetry,
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
327
Fig. 5. The six-fold dart f-tiling, carried through nine generations, with a di!erent color assigned to each generation.
respectively. The "rst dart-based f-tiling is shown in more detail and in color in Fig. 5 in order to better reveal its complex structure. In addition to rotational symmetry, these f-tilings possess mirror symmetry. Each f-tiling possesses two distinct types of lines of mirror symmetry, one passing through the center of the f-tiling and along long edges of the largest generation of tiles, and another passing through the center of the f-tiling and bisecting two opposing tiles of the largest generation. There are n/2 lines of each type for each f-tiling. In addition to rotation and simple re#ection, which is a special case of glide re#ection, the other type of symmetry commonly used to describe tilings is translation [2]. These f-tilings do not possess translational symmetry in the usual sense. However, they do possess another type of symmetry commonly used to describe fractals, namely self-similarity. This self-similarity is local rather than global and can be linked to the other symmetries, including translation. If selected portions of one of the f-tilings are scaled and translated by an appropriate vector, then
those portions would perfectly overlap the tiles already there. For example, in "rst dart f-tiling, the local group of tiles around each of the singular points of the "rst type are the same except for di!erences in scale or angular orientation. The "gures shown here are all generated using Version 7.0 of the commercial drawing program FreeHand, on a Macintosh computer. The di!erent generations of f-tilings are constructed one by one using standard functions of the program, including cloning, rotating, grouping, and scaling. This process has been carried through a su$cient number of generations that di!erences between the "gure shown and the in"nite f-tilings are very minor for a "gure of the size presented here. In practice, this means that between four and ten generations are shown. While this construction process sounds laborious, it can be done relatively quickly by intelligent use of the program's features. Obviously, a computer program or plug in for an existing program could be written to generate such f-tilings automatically given a prototile and set of matching rules.
Fig. 6. Tilings obtained by perforating the "rst kite f-tiling once and twice. In each perforation step, the largest two generations of tiles are removed.
R.W. Fathauer / Computers & Graphics 25 (2001) 323}331
329
Fig. 7. An f-tiling obtained by perforating the eight-fold kite f-tilings four times. In each perforation step, the largest generation of tiles are removed.
Increased complexity and visual richness can be built into these constructs by `perforatinga them. This refers to removal of the one or two largest generations of tiles, followed by completing the groups (matching up the exposed long edges) of the largest remaining generation, and then building inward with the same matching rules used to generate the original f-tiling. This technique works particularly well esthetically with the kite f-tilings, as shown in Figs. 6 and 7. The f-tiling of Fig. 7 is similar in many ways to the Sierpinski carpet [7]. There is a large center hole, surrounded by eight next generation holes, etc. This progression of holes could be made in"nite by carrying out an in"nite number of perforation steps. The resulting object would be more complex than the Sierpinski carpet, because the holes in the f-tiling are fractal shapes as compared to simple squares for the Sierpinski carpet. Another prototile that allows f-tilings can be constructed by combining the 6-fold prototile of each type, as shown in Fig. 8. There are two basic choices for placement of the second generation of these tiles. In addition, because the tile does not possess bilateral symmetry, the
Fig. 8. A prototile constructed by combining the six-fold kite and six-fold dart prototiles.
tiles can be mirrored between successive generations. These four possibilities result in four di!erent wellbehaved f-tilings, with three very di!erent boundaries, as shown in Fig. 9. Combining the eight- and twelve-fold kite and dart prototiles results in prototiles that fail to generate well-behaved f-tilings. It is also possible, of course, to make an Escheresque variation of any of these tiles. An example is shown in Fig. 10. The basis is the f-tiling shown in Fig. 9a, but it has been rearranged to form a periodic f-tiling suggesting waves. An f-tiling is thus used to depict
330
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Fig. 9. Three di!erent f-tilings constructed with the prototile shown in Fig. 8. (a) An f-tiling obtained by one choice of matching rules; (b) An f-tiling obtained by a second choice of matching rules; (c) An f-tiling obtained by the "rst choice of matching rules, but where the tiles are mirrored between successive generations. The fourth possible f-tiling, with the rules used for tiling (b), but with the tiles mirrored between successive generations, is similar in overall appearance to f-tiling (b).
natural objects, waves that have fractal character. A tile motif, "sh, is chosen that is sympathetic to the overall wave motif. This lends the design a dual interpretation. Is the viewer looking at water, or at a school of "sh? In conclusion, a family of constructs has been presented in which the curves de"ned by the boundaries are fractals. These `f-tilingsa are constructed
from quadrilateral prototiles according to a simple set of matching rules, yet the resultant f-tilings possess rich visual complexity. Modi"cations of the kite f-tilings have been presented that increase the complexity of the f-tilings. F-tilings have also been presented with very di!erent boundaries based on a single prototile created by combining a kite tile and a dart tile.
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331
References [1] Bruno Ernst, The Magic Mirror of M.C. Escher. New York: Balantine, 1976. [2] GruK nbaum B, Shephard GC. Tilings and Patterns. New York: WH Freeman, 1987. [3] P. Raedschelders, Private communication. [4] Private communication from Chaim Goodman-Strauss; also Pentakoch, Ptolemy mathcard no. 5, Ptolemy mathcard Co. 2000. [5] Robert W. Fathauer, Proceedings of the M.C. Escher Centennial Congress, Rome and Ravello, Italy, June 24}28, 1998. Germany: Springer Verlag, 2001. In press. [6] Vince A. Replicating Tessellations. SIAM Journal of Discrete Mathematics 1993;6:501}21. [7] Peitgen H-O, JuK rgens H, Saupe D. Fractals for the Classroom. New York: Springer-Verlag, 1992.
Fig. 10. A design based on an Escheresque variation of the f-tiling of Fig. 9a.