- Email: [email protected]
Fractal tilings based on v-shaped prototiles
Computers & Graphics 26 (2002) 635–643
Chaos and Graphics
Fractal tilings based on v-shaped prototiles Robert W. Fathauer Tessellations Company, 3913 E. Bronco Trail, Phoenix, AZ 85044, USA
Abstract We present a family of ‘‘fractal tilings’’ or ‘‘f-tilings’’ in which the curves defined by the boundaries are fractals. All of the tiles in any given tiling are similar to a single prototile that may be generally described as v-shaped. We carry out an analysis of the various allowed edge-to-edge f-tilings based on a bilaterally symmetric hexagonal v-shaped prototile. We also present examples of f-tilings in which these conditions are relaxed. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Tiling; Fractal; Self-similar; Self-similarity; Fractal tiling; Hexagon; Prototile
Fractals and tiling are two fields in mathematics that are generally thought of as separate. However, they can be combined to form a variety of visually appealing constructs that possess fractal character and at the same time obey many of the properties of tilings. Recently, we described families of fractal tilings based on segments of regular polygons and on kite- and dart-shaped quadrilateral prototiles [1,2]. These papers appear to be the first attempts at a systematic treatment of this topic. The first example of such a tiling of which the author is aware is Escher’s ‘‘Square Limit’’ print of 1964 [3]. His four ‘‘Circle Limit’’ prints are of a different nature, namely hyperbolic tilings using the Poincar!e disk [3]. In Grunbaum . and Shephard’s book Tilings and Patterns [4], a tiling is defined as a countable family of closed sets (tiles) that 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 definition of a tiling, these constructs will be referred to as ‘‘f-tilings’’, for fractal tilings. The tiles used here are ‘‘well behaved’’ by the criterion of Grunbaum . and Sheppard; namely, each tile is a (closed) topological disk. The f-tilings are also edge to edge unless otherwise noted; i.e., the corners and sides of the tiles coincide with the vertices and edges of the tilings. However, they are not ‘‘well behaved’’ by the
E-mail address: [email protected] (R.W. Fathauer).
criteria of normal tilings in one particular; namely, they contain singular points, defined as follows. Every circular disk, however small, centered at a singular point meets an infinite 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 ‘‘wellbehaved’’. This paper is written on a recreational mathematics level, rather than a research mathematics level. The author’s primary interest in these objects is recreational and esthetic. The fractal constructs described here
Fig. 1. Hexagonal prototile analyzed for allowed well-behaved f-tilings.
0097-8493/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 0 2 ) 0 0 0 9 7 - 3
636
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 2. The two different choices for arranging second generation v-shaped prototiles around first generation tiles.
Fig. 3. The first four members of an infinite family of Type-A f-tilings with prototiles of the form (n; 1, n22).
are of interest because they are distinct from other fractal constructs, though they are similar in some aspects. A great variety of prototiles that can generally be described as v-shaped admit well-behaved f-tilings. To quantify and systematize the topic to some degree,
the simplest sort of v-shaped prototile that allows f-tilings will be considered in some depth. This sort of tile is hexagonal, with two long edges of the same length and four short edges of the same length. The tile is further restricted to possessing bilateral symmetry, as shown in Fig. 1.
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643 Table 1 Candidate v-shaped prototiles for well-behaved f-tilings obeying the Type A matching rule, through n ¼ 8 n
c¼1
c¼2
c¼3
3
i ¼ 1|
—
—
4
i ¼ 1 i ¼ 2|
—
—
5
i ¼ 1| i ¼ 2 i ¼ 3|
i ¼ 2| i ¼ 3 i ¼ 4
—
6
i ¼ 1 i ¼ 2| i ¼ 3 i ¼ 4|
i ¼ 2 i ¼ 3| i ¼ 4 i ¼ 5
—
7
i ¼ 1 i ¼ 2 i ¼ 3| i ¼ 4 i ¼ 5|
i ¼ 2| i ¼ 3 i ¼ 4| i ¼ 5 i ¼ 6
i ¼ 3 i ¼ 4 i ¼ 5 i ¼ 6
8
i ¼ 1 i ¼ 2| i ¼ 3 i ¼ 4| i ¼ 5 i ¼ 6|
i ¼ 2 i ¼ 3| i ¼ 4 i ¼ 5| i ¼ 6 i ¼ 7
i ¼ 3 i ¼ 4| i ¼ 5 i ¼ 6 i ¼ 7
A check mark indicates a well-behaved f-tiling, while an indicates an f-tiling that either contains overlaps or gaps.
The f-tilings are constructed by first matching the long edges of identical prototiles, then cloning the protile, reducing its size, and fitting multiple copies of these smaller tiles around the first generation of tiles according to a matching rule, and finally iterating this process an infinite number of times. In practice, the trend becomes clear after a few to several generations. The appearance of the overall tiling changes little after 5–10 generations, since the tiles become extremely small relative to the first-generation tiles. The figures shown here were all generated using Version 7.0 of the commercial drawing program FreeHand, on a Macintosh computer. The different generations of tiles in the f-tilings were constructed one by one using standard functions of the program, including cloning, rotating, grouping, and scaling. While this construction process sounds laborious, it can be carried out relatively quickly through intelligent use of the program’s features. The angle a in the prototile of Fig. 1 must be of the form 2p=n; where n is an integer, if a group of first generation tiles is to meet without gaps or overlaps. Any f-tiling constructed from such a prototile and following a single matching rule will have n-fold rotational symmetry. There are two choices for the matching rule,
637
or arranging the second generation of tiles around the first, as illustrated in Fig. 2. In the first of these, denoted ‘‘Type A’’, the points joining the two long edges of each second generation tile mate to one of the points a in Fig. 1. In the second, denoted ‘‘Type B’’, these points mate to the points b: It will be shown that there are infinitely many well-behaved f-tilings of Type A, while there appears to be only one well-behaved f-tiling of Type B. In Type-A f-tilings, tiles fit down in the crotch of nextlarger generation tiles. The number of tiles that fits in the crotch of a single tile in a given f-tiling will be denoted ‘‘c’’; note that c must be on=2 for a v-shaped prototile. The shape of a prototile is specified by three integers: n; c; and i; using the notation (n; c; i). This last integer describes the interior angle b at the wing tips of the prototile, and is related to it by i ¼ nb=p: For a v-shaped prototile, the angles b; g and d must all be op: Under these restrictions, we list the possible allowed combinations of n; c; and i in Table 1, through n ¼ 8: We have constructed Type-A f-tilings for each of the prototiles described in Table 1 through a sufficient number of generations to determine whether or not a well-behaved f-tiling results. Carrying out this process, it quickly becomes clear that the angle g must be of the form m=np; where m is an integer, if overlaps are to be avoided. This is the reason that every other prototile in any given n, c block in Table 1 fails to allow a wellbehaved f-tiling. Every prototile of the form (n; 1; n 2) for n ¼ 328 is observed to form a well-behaved f-tiling, and it is clear from Fig. 3 that this pattern will continue for all n. In this case, all of the tiles within a given 2p=n wedge defined by a first-generation tile are oriented the same, with a scaling factor from one generation of tile to the next-smaller generation of 0.5. Furthermore, each of these f-tilings has a boundary that is a regular n-gon. This subset of allowed Type-A f-tilings is therefore an infinite family, though these f-tilings are not particularly interesting. Another family of Type-A f-tilings that appears to be infinite is that with prototiles of the form (n; 1; n 4). It is not so obvious that this results in well-behaved f-tilings for all n; but it seems likely from inspection of Fig. 4 to be the case. This is a much more interesting family of f-tilings, with fractal boundaries. Note that the overall n ¼ 6 f-tiling forms the classical fractal known as the Koch snowflake, or Koch island [5]. The other f-tilings in this family have boundaries that are analogs to the Koch snowflake for other values of n; i.e., a curve of this shape could be constructed using an analogous algorithm to that used for construction of the Koch snowflake, but with different angles, related to n in a consistent manner. The overall n ¼ 6 and 8 f-tilings will tile with similar copies of themselves; in the n ¼ 8 case,
638
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 4. The first four members of an apparently infinite family of Type-A f-tilings with prototiles of the form (n; 1, n24).
the number of choices for arranging these meta-tiles is almost bewildering [6]. For the f-tilings of Fig. 4, the scaling factor s from one generation of tiles to the next smaller generation can be shown with a little trigonometry and algebra to be given by 1=ð4cos2 ðp=nÞ). The fractal dimension of the perimeter of these f-tilings can be computed using the selfsimilarity dimension D¼ logðaÞ=logð1=sÞ; where a is the minimum number of identical pieces into which a portion of the perimeter can be divided [5]. Each of these is s times the length of the larger piece. For each of the f-tilings of Fig. 4, a is 4, so that D¼ logð4Þ=logð4 cos2 ðp=nÞÞ: This gives a value of 1.2619y for n ¼ 6; as expected for the Koch curve. For n ¼ 4 (not shown in Fig. 4), the prototile has no area, and the f-tiling is a sort of space filling curve. In this case, D ¼ 2; as expected for such a curve. For n ¼ 5; D ¼ 1:4404y As n gets large, D approaches 1. Intuitively, from inspection of Fig. 4, one would expect
the perimeter to approach a circle for large n; in which case D would indeed be 1. We show two other Type-A f-tilings not included in the above families that are esthetically pleasing in Fig. 5. For Type-B f-tilings, we will argue, though not rigorously prove, that there is exactly one well-behaved f-tiling. First, since the wing tips of two prototiles will meet in the crotch, dX2b: By definition, d ¼ c2p=n; and b ¼ ip=n: The condition on d can thus be rewritten as c2p=nX2ip=n; or cXi: If c > i; the prototile would not be v shaped, so c ¼ i for any Type-B f-tiling. Second, the prototile can be bisected into two similar quadrilaterals, for which the sums of the four interior angles must be 2p: This condition can be written as a=2 þ ðp2d=2Þ þ g þ b ¼ 2p: Third, for second-generation tiles clustered around the points b without gaps or overlaps, the angle g must be of the form m2p=n; where m is an integer. Combining the second and third conditions and doing
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
639
Fig. 5. Two Type-A f-tilings deemed esthetically pleasing, constructed from (a) the prototile specified by (8, 1, 2), and (b) the prototile specified by (12, 1, 2). Both the full f-tilings and blow-ups of specific regions are shown.
Fig. 6. A 2p=n wedge showing the first two or three generations of tiles for Type-B f-tilings formed by prototiles specified by (3, 1, 1), (5, 2, 2), and (7, 3, 3).
640
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 7. The sole well-behaved Type-B f-tiling.
some simple algebra leads to the requirement that m ¼ ðn þ 1Þ=2: If n is even, m is not an integer, so even n are not allowed. At this point in the argument, the only possibilities are the prototiles described by (3, 1, 1), (5, 1, 1), (5, 2, 2), (7, 1, 1), (7, 2, 2), (7, 3, 3), (9, 1, 1), y Fourth, note that when two second-generation tiles fit in the crotch of a first-generation tile, the upper wing tips are not allowed to overlap. This means that the angle dX2e; where e is the angle defined by a line connecting the outer points a: Since the three interior angles of any triangle sum to p; 2e þ a ¼ p; which means that dXðn22Þp=n: Applying this condition to the list of possible prototiles above, the list is reduced to prototiles described by (3, 1, 1), (5, 2, 2), (7, 3, 3), (9, 4, 4), y We show the first three of these in Fig. 6. By inspection, it is clear that the sole prototile that will not result in
Fig. 8. Two f-tilings based on a v-shaped prototile that has been skewed so that the four short edges are of two different lengths. The starting point was the bilaterally symmetric prototile specified by (6, 1, 2), the f-tiling for which is shown in Fig. 4.
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
641
Fig. 9. A non-edge-to-edge f-tiling based on a bilaterally symmetric decagonal v-shaped prototile.
overlaps is (3, 1, 1). For all others, the gap above the first-generation crotch left by the second generation tiles is not sufficiently wide to allow third generation tiles without overlaps. We show the full (3, 1, 1) Type-B ftiling in Fig. 7. In the remainder of the paper, we present a variety of f-tilings with v-shaped prototiles, for which some of the conditions imposed above have been relaxed. No attempt is made to systematically describe all of the possibilities, as the task would be enormous due to the large number of possibilities. The f-tilings shown are chosen for variety and esthetic value. In Fig. 8, we show two f-tilings for which the prototile is a variation of that used for n ¼ 6 in Fig. 4. The protile has been skewed so that the four short edges are of two lengths, rather than all of the same length. Edge-to-edge matching is maintained by mirroring of adjacent tiles, reducing the overall rotational
symmetry from 6- to 3-fold. Such a skewing of a prototile does not in general allow a well-behaved edgeto-edge f-tiling. In Fig. 9, we show an f-tiling that is not edge to edge, and for which the prototile is a bilaterally symmetric decagon. In Fig. 10, the prototile is not bilaterally symmetric, nor is the f-tiling edge to edge. This is also the case for Fig. 11; however, in this case, the tiles are reflected between successive generations. Finally, Fig. 12 shows an f-tiling based on an octagonal prototile that does not possess bilateral symmetry, but is edge to edge. Interesting fractal ‘‘inlets’’ are formed in the boundary of this f-tiling, as shown in a blow up. In conclusion, we have demonstrated the existence of an infinite number of well-behaved f -tilings based on v-shaped prototiles. These are distinct from fractal objects described previously, and many of them possess considerable visual appeal.
642
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 10. A non-edge-to-edge f-tiling based on a prototile that does not possess bilateral symmetry.
Fig. 11. A non-edge-to-edge f-tiling based on a prototile that does not possess bilateral symmetry, and in which the prototiles have been reflected between successive generations.
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
643
Fig. 12. An edge-to-edge f-tiling based on an octagonal v-shaped prototile that does not possess bilateral symmetry, with a blow-up of a particular region shown in an inset.
References [1] Fathauer RW. Fractal tilings based on kite- and dart-shaped prototiles. Computers & Graphics 2001;25(2):323–31. [2] Fathauer RW. Self-similar tilings based on prototiles constructed from segments of regular polygons. In: Winfield KS, editors. Proceedings of the Bridges 2000 Conference. 2000. p. 285–92.
[3] Ernst B. The magic mirror of M.C. Escher. New York: Ballantine Books, 1976. . [4] Grunbaum B, Shephard GC. Tilings and patterns. New York: WH Freeman, 1987. [5] Peitgen H-O, Jurgens . H, Saupe D. Fractals for the classroom, part one. New York: Springer, 1992. [6] Goodman-Strauss C. Private communication.
Chaos and Graphics
Fractal tilings based on v-shaped prototiles Robert W. Fathauer Tessellations Company, 3913 E. Bronco Trail, Phoenix, AZ 85044, USA
Abstract We present a family of ‘‘fractal tilings’’ or ‘‘f-tilings’’ in which the curves defined by the boundaries are fractals. All of the tiles in any given tiling are similar to a single prototile that may be generally described as v-shaped. We carry out an analysis of the various allowed edge-to-edge f-tilings based on a bilaterally symmetric hexagonal v-shaped prototile. We also present examples of f-tilings in which these conditions are relaxed. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Tiling; Fractal; Self-similar; Self-similarity; Fractal tiling; Hexagon; Prototile
Fractals and tiling are two fields in mathematics that are generally thought of as separate. However, they can be combined to form a variety of visually appealing constructs that possess fractal character and at the same time obey many of the properties of tilings. Recently, we described families of fractal tilings based on segments of regular polygons and on kite- and dart-shaped quadrilateral prototiles [1,2]. These papers appear to be the first attempts at a systematic treatment of this topic. The first example of such a tiling of which the author is aware is Escher’s ‘‘Square Limit’’ print of 1964 [3]. His four ‘‘Circle Limit’’ prints are of a different nature, namely hyperbolic tilings using the Poincar!e disk [3]. In Grunbaum . and Shephard’s book Tilings and Patterns [4], a tiling is defined as a countable family of closed sets (tiles) that 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 definition of a tiling, these constructs will be referred to as ‘‘f-tilings’’, for fractal tilings. The tiles used here are ‘‘well behaved’’ by the criterion of Grunbaum . and Sheppard; namely, each tile is a (closed) topological disk. The f-tilings are also edge to edge unless otherwise noted; i.e., the corners and sides of the tiles coincide with the vertices and edges of the tilings. However, they are not ‘‘well behaved’’ by the
E-mail address: [email protected] (R.W. Fathauer).
criteria of normal tilings in one particular; namely, they contain singular points, defined as follows. Every circular disk, however small, centered at a singular point meets an infinite 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 ‘‘wellbehaved’’. This paper is written on a recreational mathematics level, rather than a research mathematics level. The author’s primary interest in these objects is recreational and esthetic. The fractal constructs described here
Fig. 1. Hexagonal prototile analyzed for allowed well-behaved f-tilings.
0097-8493/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 0 2 ) 0 0 0 9 7 - 3
636
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 2. The two different choices for arranging second generation v-shaped prototiles around first generation tiles.
Fig. 3. The first four members of an infinite family of Type-A f-tilings with prototiles of the form (n; 1, n22).
are of interest because they are distinct from other fractal constructs, though they are similar in some aspects. A great variety of prototiles that can generally be described as v-shaped admit well-behaved f-tilings. To quantify and systematize the topic to some degree,
the simplest sort of v-shaped prototile that allows f-tilings will be considered in some depth. This sort of tile is hexagonal, with two long edges of the same length and four short edges of the same length. The tile is further restricted to possessing bilateral symmetry, as shown in Fig. 1.
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643 Table 1 Candidate v-shaped prototiles for well-behaved f-tilings obeying the Type A matching rule, through n ¼ 8 n
c¼1
c¼2
c¼3
3
i ¼ 1|
—
—
4
i ¼ 1 i ¼ 2|
—
—
5
i ¼ 1| i ¼ 2 i ¼ 3|
i ¼ 2| i ¼ 3 i ¼ 4
—
6
i ¼ 1 i ¼ 2| i ¼ 3 i ¼ 4|
i ¼ 2 i ¼ 3| i ¼ 4 i ¼ 5
—
7
i ¼ 1 i ¼ 2 i ¼ 3| i ¼ 4 i ¼ 5|
i ¼ 2| i ¼ 3 i ¼ 4| i ¼ 5 i ¼ 6
i ¼ 3 i ¼ 4 i ¼ 5 i ¼ 6
8
i ¼ 1 i ¼ 2| i ¼ 3 i ¼ 4| i ¼ 5 i ¼ 6|
i ¼ 2 i ¼ 3| i ¼ 4 i ¼ 5| i ¼ 6 i ¼ 7
i ¼ 3 i ¼ 4| i ¼ 5 i ¼ 6 i ¼ 7
A check mark indicates a well-behaved f-tiling, while an indicates an f-tiling that either contains overlaps or gaps.
The f-tilings are constructed by first matching the long edges of identical prototiles, then cloning the protile, reducing its size, and fitting multiple copies of these smaller tiles around the first generation of tiles according to a matching rule, and finally iterating this process an infinite number of times. In practice, the trend becomes clear after a few to several generations. The appearance of the overall tiling changes little after 5–10 generations, since the tiles become extremely small relative to the first-generation tiles. The figures shown here were all generated using Version 7.0 of the commercial drawing program FreeHand, on a Macintosh computer. The different generations of tiles in the f-tilings were constructed one by one using standard functions of the program, including cloning, rotating, grouping, and scaling. While this construction process sounds laborious, it can be carried out relatively quickly through intelligent use of the program’s features. The angle a in the prototile of Fig. 1 must be of the form 2p=n; where n is an integer, if a group of first generation tiles is to meet without gaps or overlaps. Any f-tiling constructed from such a prototile and following a single matching rule will have n-fold rotational symmetry. There are two choices for the matching rule,
637
or arranging the second generation of tiles around the first, as illustrated in Fig. 2. In the first of these, denoted ‘‘Type A’’, the points joining the two long edges of each second generation tile mate to one of the points a in Fig. 1. In the second, denoted ‘‘Type B’’, these points mate to the points b: It will be shown that there are infinitely many well-behaved f-tilings of Type A, while there appears to be only one well-behaved f-tiling of Type B. In Type-A f-tilings, tiles fit down in the crotch of nextlarger generation tiles. The number of tiles that fits in the crotch of a single tile in a given f-tiling will be denoted ‘‘c’’; note that c must be on=2 for a v-shaped prototile. The shape of a prototile is specified by three integers: n; c; and i; using the notation (n; c; i). This last integer describes the interior angle b at the wing tips of the prototile, and is related to it by i ¼ nb=p: For a v-shaped prototile, the angles b; g and d must all be op: Under these restrictions, we list the possible allowed combinations of n; c; and i in Table 1, through n ¼ 8: We have constructed Type-A f-tilings for each of the prototiles described in Table 1 through a sufficient number of generations to determine whether or not a well-behaved f-tiling results. Carrying out this process, it quickly becomes clear that the angle g must be of the form m=np; where m is an integer, if overlaps are to be avoided. This is the reason that every other prototile in any given n, c block in Table 1 fails to allow a wellbehaved f-tiling. Every prototile of the form (n; 1; n 2) for n ¼ 328 is observed to form a well-behaved f-tiling, and it is clear from Fig. 3 that this pattern will continue for all n. In this case, all of the tiles within a given 2p=n wedge defined by a first-generation tile are oriented the same, with a scaling factor from one generation of tile to the next-smaller generation of 0.5. Furthermore, each of these f-tilings has a boundary that is a regular n-gon. This subset of allowed Type-A f-tilings is therefore an infinite family, though these f-tilings are not particularly interesting. Another family of Type-A f-tilings that appears to be infinite is that with prototiles of the form (n; 1; n 4). It is not so obvious that this results in well-behaved f-tilings for all n; but it seems likely from inspection of Fig. 4 to be the case. This is a much more interesting family of f-tilings, with fractal boundaries. Note that the overall n ¼ 6 f-tiling forms the classical fractal known as the Koch snowflake, or Koch island [5]. The other f-tilings in this family have boundaries that are analogs to the Koch snowflake for other values of n; i.e., a curve of this shape could be constructed using an analogous algorithm to that used for construction of the Koch snowflake, but with different angles, related to n in a consistent manner. The overall n ¼ 6 and 8 f-tilings will tile with similar copies of themselves; in the n ¼ 8 case,
638
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 4. The first four members of an apparently infinite family of Type-A f-tilings with prototiles of the form (n; 1, n24).
the number of choices for arranging these meta-tiles is almost bewildering [6]. For the f-tilings of Fig. 4, the scaling factor s from one generation of tiles to the next smaller generation can be shown with a little trigonometry and algebra to be given by 1=ð4cos2 ðp=nÞ). The fractal dimension of the perimeter of these f-tilings can be computed using the selfsimilarity dimension D¼ logðaÞ=logð1=sÞ; where a is the minimum number of identical pieces into which a portion of the perimeter can be divided [5]. Each of these is s times the length of the larger piece. For each of the f-tilings of Fig. 4, a is 4, so that D¼ logð4Þ=logð4 cos2 ðp=nÞÞ: This gives a value of 1.2619y for n ¼ 6; as expected for the Koch curve. For n ¼ 4 (not shown in Fig. 4), the prototile has no area, and the f-tiling is a sort of space filling curve. In this case, D ¼ 2; as expected for such a curve. For n ¼ 5; D ¼ 1:4404y As n gets large, D approaches 1. Intuitively, from inspection of Fig. 4, one would expect
the perimeter to approach a circle for large n; in which case D would indeed be 1. We show two other Type-A f-tilings not included in the above families that are esthetically pleasing in Fig. 5. For Type-B f-tilings, we will argue, though not rigorously prove, that there is exactly one well-behaved f-tiling. First, since the wing tips of two prototiles will meet in the crotch, dX2b: By definition, d ¼ c2p=n; and b ¼ ip=n: The condition on d can thus be rewritten as c2p=nX2ip=n; or cXi: If c > i; the prototile would not be v shaped, so c ¼ i for any Type-B f-tiling. Second, the prototile can be bisected into two similar quadrilaterals, for which the sums of the four interior angles must be 2p: This condition can be written as a=2 þ ðp2d=2Þ þ g þ b ¼ 2p: Third, for second-generation tiles clustered around the points b without gaps or overlaps, the angle g must be of the form m2p=n; where m is an integer. Combining the second and third conditions and doing
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
639
Fig. 5. Two Type-A f-tilings deemed esthetically pleasing, constructed from (a) the prototile specified by (8, 1, 2), and (b) the prototile specified by (12, 1, 2). Both the full f-tilings and blow-ups of specific regions are shown.
Fig. 6. A 2p=n wedge showing the first two or three generations of tiles for Type-B f-tilings formed by prototiles specified by (3, 1, 1), (5, 2, 2), and (7, 3, 3).
640
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 7. The sole well-behaved Type-B f-tiling.
some simple algebra leads to the requirement that m ¼ ðn þ 1Þ=2: If n is even, m is not an integer, so even n are not allowed. At this point in the argument, the only possibilities are the prototiles described by (3, 1, 1), (5, 1, 1), (5, 2, 2), (7, 1, 1), (7, 2, 2), (7, 3, 3), (9, 1, 1), y Fourth, note that when two second-generation tiles fit in the crotch of a first-generation tile, the upper wing tips are not allowed to overlap. This means that the angle dX2e; where e is the angle defined by a line connecting the outer points a: Since the three interior angles of any triangle sum to p; 2e þ a ¼ p; which means that dXðn22Þp=n: Applying this condition to the list of possible prototiles above, the list is reduced to prototiles described by (3, 1, 1), (5, 2, 2), (7, 3, 3), (9, 4, 4), y We show the first three of these in Fig. 6. By inspection, it is clear that the sole prototile that will not result in
Fig. 8. Two f-tilings based on a v-shaped prototile that has been skewed so that the four short edges are of two different lengths. The starting point was the bilaterally symmetric prototile specified by (6, 1, 2), the f-tiling for which is shown in Fig. 4.
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
641
Fig. 9. A non-edge-to-edge f-tiling based on a bilaterally symmetric decagonal v-shaped prototile.
overlaps is (3, 1, 1). For all others, the gap above the first-generation crotch left by the second generation tiles is not sufficiently wide to allow third generation tiles without overlaps. We show the full (3, 1, 1) Type-B ftiling in Fig. 7. In the remainder of the paper, we present a variety of f-tilings with v-shaped prototiles, for which some of the conditions imposed above have been relaxed. No attempt is made to systematically describe all of the possibilities, as the task would be enormous due to the large number of possibilities. The f-tilings shown are chosen for variety and esthetic value. In Fig. 8, we show two f-tilings for which the prototile is a variation of that used for n ¼ 6 in Fig. 4. The protile has been skewed so that the four short edges are of two lengths, rather than all of the same length. Edge-to-edge matching is maintained by mirroring of adjacent tiles, reducing the overall rotational
symmetry from 6- to 3-fold. Such a skewing of a prototile does not in general allow a well-behaved edgeto-edge f-tiling. In Fig. 9, we show an f-tiling that is not edge to edge, and for which the prototile is a bilaterally symmetric decagon. In Fig. 10, the prototile is not bilaterally symmetric, nor is the f-tiling edge to edge. This is also the case for Fig. 11; however, in this case, the tiles are reflected between successive generations. Finally, Fig. 12 shows an f-tiling based on an octagonal prototile that does not possess bilateral symmetry, but is edge to edge. Interesting fractal ‘‘inlets’’ are formed in the boundary of this f-tiling, as shown in a blow up. In conclusion, we have demonstrated the existence of an infinite number of well-behaved f -tilings based on v-shaped prototiles. These are distinct from fractal objects described previously, and many of them possess considerable visual appeal.
642
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
Fig. 10. A non-edge-to-edge f-tiling based on a prototile that does not possess bilateral symmetry.
Fig. 11. A non-edge-to-edge f-tiling based on a prototile that does not possess bilateral symmetry, and in which the prototiles have been reflected between successive generations.
R.W. Fathauer / Computers & Graphics 26 (2002) 635–643
643
Fig. 12. An edge-to-edge f-tiling based on an octagonal v-shaped prototile that does not possess bilateral symmetry, with a blow-up of a particular region shown in an inset.
References [1] Fathauer RW. Fractal tilings based on kite- and dart-shaped prototiles. Computers & Graphics 2001;25(2):323–31. [2] Fathauer RW. Self-similar tilings based on prototiles constructed from segments of regular polygons. In: Winfield KS, editors. Proceedings of the Bridges 2000 Conference. 2000. p. 285–92.
[3] Ernst B. The magic mirror of M.C. Escher. New York: Ballantine Books, 1976. . [4] Grunbaum B, Shephard GC. Tilings and patterns. New York: WH Freeman, 1987. [5] Peitgen H-O, Jurgens . H, Saupe D. Fractals for the classroom, part one. New York: Springer, 1992. [6] Goodman-Strauss C. Private communication.