A Method to Generate Polyominoes and ... - Semantic Scholar
A Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry Hiroshi Fukuda1 , Nobuaki Mutoh1 , Gisaku Nakamura2 , Doris Schattschneider3 1 2 3
School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka, 4228526 JAPAN Research Institute of Education, Tokai University, 2-28-4 Tomigaya, Shibuya-ku Tokio, JAPAN Mathematics Dept. PPHAC Moravian College, 1200 Main St. Bethlehem, PA 18018-6650
Abstract. We show a simple method to generate polyominoes and polyiamonds that produce isohedral tilings with p3, p4 or p6 rotational symmetry by using n line segments between lattice points on a regular hexagonal, square and triangular lattice, respectively. We exhibit all possible tiles generated by this algorithm up to n = 9 for p3, n = 8 for p4, and n = 13 for p6. In particular, we determine for n ≤ 8 all n-ominoes that are fundamental domains for p4 isohedral tilings. Key words. Polyominoes, Polyiamonds, Isohedral Tilings, Rotational Symmetry
1. Polyominoes, polyiamonds, and isohedral tilings Polyominoes and polyiamonds are among the simplest shapes for tiles and are easily produced by computer or by hand. A polyomino (or n-omino) is a tile made up of n congruent squares joined at their edges; the name is a natural extension of “domino”, which is two squares joined at an edge. A polyiamond (or n-iamond) is a tile made up of n congruent triangles joined at an edge; this name is an extension of “diamond”, which is two congruent triangles joined at an edge. There is a rich store of problems (both solved and unsolved) concerning these tiles. For example, how many n-ominoes and how many n-iamonds are there for a given n? The answers are known for at least n ≤ 28, but not known for large n, nor is there any formula for the general case. [6] Most of the problems about these tiles are about their tiling properties – for example, the different ways in which a complete set of n-ominoes for a given n can fill a rectangle or other shape, or which of the polyominoes or polyiamonds can tile the plane. [3, 6] In this article, we focus on isohedral tilings of the plane by these tiles in which the tilings have rotation symmetry – for polyominoes, the tilings will have 4-fold rotation symmetry, and for polyiamonds, the tilings will have 3-fold or 6-fold rotation symmetry. An isohedral tiling of the plane is one in which congruent copies of a single tile fill the plane without gaps or overlaps, and the symmetry group of the tiling acts transitively on the tiles. Send offprint requests to:
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We describe an algorithm (which we have implemented by computer) that can systematically produce polyominoes and polyiamonds that will tile the plane isohedrally, and the tilings have the desired rotation symmetry. In our discussion, we assume knowledge of the lattice of rotation centers of the symmetry groups p3, p4, and p6. References [1, 8] give details on these and the other 14 symmetry groups of plane tilings. Figure 1 shows examples of such tilings with symmetry groups p4 and p3.
Fig. 1. An isohedral p4 tiling by an 8-omino and an isohedral p3 tiling by an 18-iamond.
2. Polyominoes for tilings with 4-fold rotation symmetry We first consider the case of producing polyominoes that can tile the plane by repeated 4-fold rotations. The symmetry group of such a tiling will be of type p4 or it will contain a p4 subgroup. In Fig. 2 (a), symmetry elements of a p4 pattern are shown. Filled
L
(a)
(b)
(c)
Fig. 2. (a) Symmetry elements of a p4 pattern. (b) Arrangement of tiles. (c) Simplest polyomino by expansion.
squares are equivalent 4-fold rotation centers and open squares are the other equivalent 4-fold rotation centers. Small rectangles represent half-turn centers. Two independent translations that generate the translation symmetry group of the pattern are shown by the vectors; these vectors are orthogonal and have the same length. The full p4 symmetry group is generated by two 4-fold rotations about adjacent non-equivalent 4-fold centers. We assume our tile has no 4-fold rotational symmetry with respect to a given pattern of symmetry elements of p4. Thus at least four tiles have to meet at a filled square or at an open square, and two different 4-fold centers are connected by the boundary of the
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tile. Consequently, the tiles are arranged on a regular square lattice as shown in Fig. 2 (b), where tiles are shown symbolically by dotted lines that connect two different 4-fold centers. The simplest tile in this arrangement is the square which is obtained by expanding the dotted line segment to the shaded square tile as shown in Fig. 2 (c). In order to produce a more complex polyomino in this context, we consider the pattern of Fig. 2 (b) turned and superimposed on a reduced copy of the pattern as shown in Fig. 3 (a), where the reduced pattern is shown by solid line segments which represent the reduced tiles. It is always possible to make a superimposed pattern by joining two 4-fold centers of the reduced pattern by a dotted line.
l
(a)
(b)
(c)
(d)
Fig. 3. (a) The reduced pattern with an original pattern superimposed. (b) Generator. (c) Generated tile. (d) Tiling by generated tile.
Then the p4 symmetry elements of the superimposed pattern classify the line segments of the reduced pattern into L2 (1) n= 2 l equivalence classes, where l is the length of a line segment in the reduced pattern and L is the length of a dotted segment of the superimposed pattern. For example the line segments of the reduced pattern in Fig. 3 (a) are classified into n = 12 + 22 = 5 classes. Thus, instead of the dotted line connecting two 4-fold centers, we can use a set of n line segments of the reduced pattern, which we will call a ‘generator’. Such a generator must satisfy the following conditions: 1. A generator consists of n = L2 /l2 line segments, where L and l are the lengths of a dotted line segment and a line segment in the generator, respectively. We call a line segment in the generator simply an ‘element’ of the generator. Note that n, which is the ratio of the area of a square in the overlaid lattice to the area of a square in the reduced lattice, is limited to n=
L2 = x2 + y 2 , x, y = 0, 1, 2, . . . , (x, y) = (0, 0), l2
(2)
i.e., n = 1, 2, 4, 5, 8, . . ., where x and y are the horizontal and vertical components of a dotted segment on the reduced lattice pattern. 2. No two elements of the generator belong to the same equivalence class defined by the p4 symmetry elements of the superimposed pattern.
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3. A generator connects both ends of the dotted line segment. 4. A generator is connected. 5. If the degree of a node of a generator is two, the two elements from this node should not be parallel. An example of a generator for the overlaid pattern of Fig. 3 (a) is shown in Fig. 3 (b). It is clear that if we expand each element in the generator as we did for the dotted line segment in Fig. 2 (c), we obtain the corresponding tile with shaded area shown in Fig. 3 (c). Fig. 3 (d) shows the tiling produced by applying 90◦ rotations about the two different 4-fold centers joined by the dotted line segment in Fig. 3 (c).
1-1
2-1
4-1
5-8
8-1
8-12
8-13
8-25
8-37
8-26
8-38
4-2
4-3
5-1-2
8-3
8-4
8-2
8-14
8-27
8-15
8-28
8-39
5-1
8-16
5-2-2
8-5
8-17
8-29
8-40
5-2
8-18
8-42
5-3
8-6
8-30
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5-2-3
8-7
8-19
8-31
8-44
5-5
8-8
8-20
8-32
8-43
5-4
8-9
8-21
8-33
5-6
8-10
8-22
8-34
5-7
8-35
5-7-2
8-11
8-23
8-24
8-36
8-45
Fig. 4. List of polyominoes produced by the algorithm for n ≤ 8. The labels indicate n and the tile number for that n.
In Fig. 4, all polyominoes obtained by the above procedure for n ≤ 8 are listed. The first number in the 2-number label identifies n and the second number identifies the number of the tile for that n. For example, 8-17 identifies the 17th 8-omino listed by the computer. The tile in Fig. 3 (c) is listed as 5-1 in Fig. 4. For each tile, its corresponding p4 tiling is obtained by using the filled and open squares (shown as circles in Fig. 4) as 4-fold rotation centers. For example, tile 8-22 produces the tiling shown in Fig. 1. If the same polyomino can have different 4-fold rotation centers, a third number is added after the 2-number label. For example, 5-1 and 5-1-2 are the same polyomino but have differently-placed rotation centers. We note that in the list some tiles may produce tilings with higher symmetry than p4. For example, 1-1 and 4-3, which are simple squares, are tiles for p4m tilings, while 8-7 and 8-26 are tiles for p4g tilings.
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3. Polyiamonds for tilings with 3-fold rotation symmetry We can obtain polyiamonds for p3 tilings in a similar manner. Fig. 5 (a) shows the symmetry elements of a p3 pattern where black triangles, open triangles and gray triangles represent equivalent 3-fold rotation centers, respectively. The vectors shown are two independent translations that generate the translation subgroup of the p3 symmetry group.
L
(a)
(b)
(c)
Fig. 5. (a) Symmetry elements of a p3 pattern. (b) Arrangement of tiles. (c) Simplest polyiamond by expansion.
We assume that tiles have no 3-fold rotational symmetry with respect to a given pattern of symmetry elements, and so at least two different 3-fold centers are connected by the boundary of a tile. Thus we may represent tiles by dotted line segments that join adjacent black and open triangles; this gives an arrangement of segments that form a regular hexagonal tiling as shown in Fig. 5 (b). Expansion of a dotted segment to a diamond as shown in Fig. 5 (c) produces the simplest polyiamond tile; this corresponds to Fig. 2 (c) for the polyomino case. Following a procedure similar to that for polyominoes, we superimpose the pattern in Fig. 5 (b) on a reduced copy of the pattern as shown in Fig. 6 (a), by connecting two 3-fold centers by a dotted line (each dotted line representing a larger tile). Using the reduced hexagonal pattern we can define a generator for a tile in a manner similar to the p4 case. Fig. 6 (b) shows an example of a generator consisting of n = L2 /l2 = 7 elements using the overlaid pattern shown in Fig. 6 (a). Note that n for this case is limited to n = x2 + y 2 + xy, x, y = 0, 1, 2, . . . , (x, y) = (0, 0),
(3)
i.e., n = 1, 3, 4, 7, 9, . . . instead of Eq. (2) for p4. Fig. 6 (c) is the corresponding tile gotten by expanding each element of the generator in the manner shown in Fig. 5 (c). This tile is a 14-iamond, composed of 14 congruent equilateral triangles. The tiling produced by applying 120◦ rotations about the two different 3-fold centers joined by the dotted segment is shown in Fig. 6 (d) In Fig. 7, all 2n-iamonds obtained in this manner for n ≤ 9 are listed. The tiling produced by tile 9-1-3 is shown in Fig. 1. We note that the list includes some tiles that generate tilings with reflection symmetry as well; for example, tile 9-6 produces a p31m tiling. In addition, some tiles produce tilings that also have 6-fold rotation symmetry. For example, tiles 1-1 and 3-1 produce p6m tilings, and tile 7-5 produces a p6 tiling.
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(a)
(b)
(c)
(d)
Fig. 6. (a) The reduced pattern with an original pattern superimposed. (b) Generator. The dotted line has length L. (c) Generated tile. (d) Tiling by generated tile.
1-1
3-1
7-8
9-2-3
9-5-3
4-1
7-8-2
9-3
9-6
7-1
7-2
7-9
9-3-2
9-7
7-10
9-3-3
9-7-2
7-2-2
7-3
9-1
9-1-2
9-4
9-7-3
7-4
7-5
9-1-3
7-6
9-2
9-4-2
9-4-3
9-5
9-8
9-8-2
9-8-3
7-7
7-7-2
9-2-2
9-5-2
Fig. 7. List of polyiamonds produced by the p3 algorithm for n ≤ 9. The labels indicate n and the tile number for that n.
4. Polyiamonds for tilings with 6-fold rotation symmetry Finally, we consider polyiamond tiles for p6 tilings. We can proceed as in the previous sections. Fig. 8 (a) shows the symmetry elements of a p6 pattern where filled circles represent equivalent 6-fold rotation centers. Note that in p6 patterns all 6-fold centers are identical. Triangles and rectangles represent 3-fold rotation centers and half-turn centers, respectively. The vectors shown are two independent translations that generate the translation subgroup of the p6 symmetry group. We assume that tiles have no 6-fold rotational symmetry, hence tiles (represented by dotted lines) are arranged on the regular triangular lattice as shown in Fig. 8 (b). Expanding a line as shown in Fig. 8 (c) produces the simplest polyiamond for this case; this expansion corresponds to Fig. 2 (c) for p4 or to Fig. 5 (c) for p3.
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L
(a)
(b)
(c)
Fig. 8. (a) Symmetry elements of a p6 pattern. (b) Arrangement of tiles. (c) Simplest tile by expansion.
(a)
(b)
(c)
(d)
Fig. 9. (a) The reduced pattern with an original pattern superimposed. (b) Generator. The dotted line has length L. (c) Generated tile. (d) Tiling by generated tile.
As before, we reduce our pattern in Fig. 8 (b) so that we can superimpose the original pattern, matching 6-fold centers, as illustrated in Fig. 9 (a). Any such overlaid pattern can be produced by connecting two 6-fold centers of the reduced pattern by a dotted line (the dotted line representing a larger tile). Using the overlaid pattern we can define a generator as before. Fig. 9 (b) shows an example of a generator which consists of n = L2 /l2 = 7 elements of the reduced pattern in Fig. 9 (a). Since all 6-fold centers are identical and the tile must have half-turn symmetry (as indicated by the half-turn center at the midpoint of the dotted line in Fig. 9 (b)), the generator for this tile also must have half-turn symmetry. Fig. 9 (c) is the corresponding tile gotten by expanding each element of the generator in the manner shown in Fig. 8 (c). For this p6 case, n also satisfies Eq. (3). Fig. 9 (d) shows the tiling by the tile in Fig. 9 (c), generated by 60◦ rotations about the 6-fold centers at the endpoints of the generator. In Fig. 10, all 2n-iamonds obtained in this manner for n ≤ 13 are listed. As indicated in the previous section, tile 1-1 in this list produces a p6m tiling. Also, some tiles in this list also appear in the list in Fig. 7. This is because all p6 tilings have 3-fold rotations, and the algorithm used to produce polyiamonds with generators that connect 3-fold centers sometimes produces tiles with 6-fold centers on the boundary. For example, tile 7-2 in Fig. 10 is the same as tile 7-5 in Fig. 7; Fig. 10 shows the 6-fold centers on the tile, while Fig. 7 shows the 3-fold centers.
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1-1
3-1
12-1
13-6
7-1
12-2
13-7
7-2
9-1
13-1
13-2
13-8
13-9
9-2
9-3
13-3
13-10
9-4
13-4
13-11
13-5
13-12
Fig. 10. Polyiamonds produced by the p6 algorithm for n ≤ 13. The labels indicate n and the tile number for that n.
5. Additional comments The algorithm presented in Section 2 produces polyominoes that generate isohedral tilings by 4-fold rotations. Such tilings will have p4 rotational symmetry and perhaps additional symmetry (i.e., they might have p4g or p4m symmetry group). In addition, the algorithm produces all polyominoes that are fundamental domains for p4 isohedral tilings, that is, minimal regions which, when acted on by the symmetry group of the tiling, generate the whole tiling. This is because we know from Heesch’s work [5] that a fundamental domain for a p4 isohedral tiling must be one of three types, and only two of these types are possible for polyominoes. A polyomino that is a p4 fundamental domain will have either four or five vertices at which three or four tiles meet in the tiling and have non-equivalent 4-fold centers at two of these vertices. When the tile has four of these vertices, the two other vertices are half-turn centers of the tiling; when the tile has five of these vertices, the edge that does not meet a 4-fold center has a half-turn center at its midpoint. Our polyomino construction using a generator that connects two 4-fold centers produces tiles that satisfy these conditions. Conversely, any polyomino that is a fundamental domain for a p4 tiling can easily be converted into its generator as defined in Section 2. Thus our list in Fig. 4 contains, for n ≤ 8, all n-ominoes that are fundamental domains for p4 isohedral tilings (as well as many other n-ominoes that are not fundamental domains for the p4 isohedral tiling they generate). Also, given a polyomino without 4-fold symmetry, we can determine whether or not it can produce a p4 isohedral tiling by checking to see if it can be converted to a generator by taking as 4-fold centers a suitable pair of vertices of the polyomino. For polyiamonds, our algorithms in Sections 3 and 4 produce only a partial list of such tiles that can produce isohedral tilings with p3 or p6 symmetry. A main reason is that the unit shapes we have chosen, shown in Fig. 5 (c) and Fig. 8 (c), are not equilateral triangles but diamonds, and hence can never produce an m-iamond for m odd. In addition, there are m-iamonds for even m that do not arise from our algorithm. For example, our algorithms produce no 6-iamonds, but the 6-iamond in Fig. 11 produces a p31m tiling by rotating about the labeled 3-fold centers. For the p6 case, our algorithm always produces a tile with half-turn symmetry, and the tile can be dissected into two congruent polyiamonds that generate a p6 isohedral
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tiling by rotating about the 6-fold center and the half-turn center on the boundary of the half-tile. While these dissections produce some m-iamonds for odd m, it is not likely that all can be produced in this manner. And there are still other polyiamonds that do not have half-turn symmetry and do not arise as half of a centro-symmetric polyiamond, yet tile the plane in a p6 isohedral manner. A simple example is the 4-iamond in the shape of a chevron, shown in Fig. 11, along with its p6 tiling, produced by rotating about its labeled 6-fold and 3-fold centers.
Fig. 11. (Left) A 6-iamond and its p31m isohedral tiling. (Right) A 4-iamond and its p6 isohedral tiling.
However, it might be possible to enumerate polyiamonds that produce p3 or p6 isohedral tilings by taking an equilateral triangle as the unit shape. Also, if we take a regular hexagon as a unit shape then it might be possible to enumerate polyhexes that produce p3 isohedral tilings. We will address these problems in further investigations. We note that the method discussed here may be generalized to tilings having other symmetry groups. Reference [2] considers some of these generalizations. We also note that the method used in our algorithm, of overlaying a pattern of symmetry elements on a reduced copy of the pattern, was employed by Rinus Roelofs [7] for quite a different purpose: to transform an Escher tessellation to a different Escher tessellation having the same symmetry group. References 1. H. S. M. Coxeter, and W. O. J. Moser, Generators and Relations for Discrete Groups, (Springer-Verlag, New York 1965). 2. H. Fukuda, T. Betumiya, S. Nishiyama and G. Nakamura, KATACHI ∪ SYMMETRY (Springer, Tokyo 1996) 231–238. 3. S. Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings, revised ed., (Princeton University Press, 1994). 4. B. Gr¨ unbaum, and G. C. Shephard, Tilings and Patterns (W.H. Freeman, 1987). 5. H. Heesch and O. Kienzle, Fl¨ achenschluss. System der Formen l¨ uckenlos aneinanderschliessender Flachteile (Springer-Verlag, 1963). 6. G. E. Martin, Polyominoes: A Guide to Puzzles and Problems in Tiling (Mathematical Association of America, 1991). 7. R. Roelofs, “Not the Tiles, but the Joints: A little Bridge Between M.C. Escher and Leonardo da Vinci”, in M. C. Escher’s Legacy, A Centennial Celebration, Eds. D. Schattschneider and M. Emmer (Springer-Verlag, 2003) 252–264.
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8. D. Schattschneider, American Mathematical Monthly 85 (1978) 439–450.
Received: December 22, 2002 Final version received: December 22, 2002
School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka, 4228526 JAPAN Research Institute of Education, Tokai University, 2-28-4 Tomigaya, Shibuya-ku Tokio, JAPAN Mathematics Dept. PPHAC Moravian College, 1200 Main St. Bethlehem, PA 18018-6650
Abstract. We show a simple method to generate polyominoes and polyiamonds that produce isohedral tilings with p3, p4 or p6 rotational symmetry by using n line segments between lattice points on a regular hexagonal, square and triangular lattice, respectively. We exhibit all possible tiles generated by this algorithm up to n = 9 for p3, n = 8 for p4, and n = 13 for p6. In particular, we determine for n ≤ 8 all n-ominoes that are fundamental domains for p4 isohedral tilings. Key words. Polyominoes, Polyiamonds, Isohedral Tilings, Rotational Symmetry
1. Polyominoes, polyiamonds, and isohedral tilings Polyominoes and polyiamonds are among the simplest shapes for tiles and are easily produced by computer or by hand. A polyomino (or n-omino) is a tile made up of n congruent squares joined at their edges; the name is a natural extension of “domino”, which is two squares joined at an edge. A polyiamond (or n-iamond) is a tile made up of n congruent triangles joined at an edge; this name is an extension of “diamond”, which is two congruent triangles joined at an edge. There is a rich store of problems (both solved and unsolved) concerning these tiles. For example, how many n-ominoes and how many n-iamonds are there for a given n? The answers are known for at least n ≤ 28, but not known for large n, nor is there any formula for the general case. [6] Most of the problems about these tiles are about their tiling properties – for example, the different ways in which a complete set of n-ominoes for a given n can fill a rectangle or other shape, or which of the polyominoes or polyiamonds can tile the plane. [3, 6] In this article, we focus on isohedral tilings of the plane by these tiles in which the tilings have rotation symmetry – for polyominoes, the tilings will have 4-fold rotation symmetry, and for polyiamonds, the tilings will have 3-fold or 6-fold rotation symmetry. An isohedral tiling of the plane is one in which congruent copies of a single tile fill the plane without gaps or overlaps, and the symmetry group of the tiling acts transitively on the tiles. Send offprint requests to:
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We describe an algorithm (which we have implemented by computer) that can systematically produce polyominoes and polyiamonds that will tile the plane isohedrally, and the tilings have the desired rotation symmetry. In our discussion, we assume knowledge of the lattice of rotation centers of the symmetry groups p3, p4, and p6. References [1, 8] give details on these and the other 14 symmetry groups of plane tilings. Figure 1 shows examples of such tilings with symmetry groups p4 and p3.
Fig. 1. An isohedral p4 tiling by an 8-omino and an isohedral p3 tiling by an 18-iamond.
2. Polyominoes for tilings with 4-fold rotation symmetry We first consider the case of producing polyominoes that can tile the plane by repeated 4-fold rotations. The symmetry group of such a tiling will be of type p4 or it will contain a p4 subgroup. In Fig. 2 (a), symmetry elements of a p4 pattern are shown. Filled
L
(a)
(b)
(c)
Fig. 2. (a) Symmetry elements of a p4 pattern. (b) Arrangement of tiles. (c) Simplest polyomino by expansion.
squares are equivalent 4-fold rotation centers and open squares are the other equivalent 4-fold rotation centers. Small rectangles represent half-turn centers. Two independent translations that generate the translation symmetry group of the pattern are shown by the vectors; these vectors are orthogonal and have the same length. The full p4 symmetry group is generated by two 4-fold rotations about adjacent non-equivalent 4-fold centers. We assume our tile has no 4-fold rotational symmetry with respect to a given pattern of symmetry elements of p4. Thus at least four tiles have to meet at a filled square or at an open square, and two different 4-fold centers are connected by the boundary of the
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tile. Consequently, the tiles are arranged on a regular square lattice as shown in Fig. 2 (b), where tiles are shown symbolically by dotted lines that connect two different 4-fold centers. The simplest tile in this arrangement is the square which is obtained by expanding the dotted line segment to the shaded square tile as shown in Fig. 2 (c). In order to produce a more complex polyomino in this context, we consider the pattern of Fig. 2 (b) turned and superimposed on a reduced copy of the pattern as shown in Fig. 3 (a), where the reduced pattern is shown by solid line segments which represent the reduced tiles. It is always possible to make a superimposed pattern by joining two 4-fold centers of the reduced pattern by a dotted line.
l
(a)
(b)
(c)
(d)
Fig. 3. (a) The reduced pattern with an original pattern superimposed. (b) Generator. (c) Generated tile. (d) Tiling by generated tile.
Then the p4 symmetry elements of the superimposed pattern classify the line segments of the reduced pattern into L2 (1) n= 2 l equivalence classes, where l is the length of a line segment in the reduced pattern and L is the length of a dotted segment of the superimposed pattern. For example the line segments of the reduced pattern in Fig. 3 (a) are classified into n = 12 + 22 = 5 classes. Thus, instead of the dotted line connecting two 4-fold centers, we can use a set of n line segments of the reduced pattern, which we will call a ‘generator’. Such a generator must satisfy the following conditions: 1. A generator consists of n = L2 /l2 line segments, where L and l are the lengths of a dotted line segment and a line segment in the generator, respectively. We call a line segment in the generator simply an ‘element’ of the generator. Note that n, which is the ratio of the area of a square in the overlaid lattice to the area of a square in the reduced lattice, is limited to n=
L2 = x2 + y 2 , x, y = 0, 1, 2, . . . , (x, y) = (0, 0), l2
(2)
i.e., n = 1, 2, 4, 5, 8, . . ., where x and y are the horizontal and vertical components of a dotted segment on the reduced lattice pattern. 2. No two elements of the generator belong to the same equivalence class defined by the p4 symmetry elements of the superimposed pattern.
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3. A generator connects both ends of the dotted line segment. 4. A generator is connected. 5. If the degree of a node of a generator is two, the two elements from this node should not be parallel. An example of a generator for the overlaid pattern of Fig. 3 (a) is shown in Fig. 3 (b). It is clear that if we expand each element in the generator as we did for the dotted line segment in Fig. 2 (c), we obtain the corresponding tile with shaded area shown in Fig. 3 (c). Fig. 3 (d) shows the tiling produced by applying 90◦ rotations about the two different 4-fold centers joined by the dotted line segment in Fig. 3 (c).
1-1
2-1
4-1
5-8
8-1
8-12
8-13
8-25
8-37
8-26
8-38
4-2
4-3
5-1-2
8-3
8-4
8-2
8-14
8-27
8-15
8-28
8-39
5-1
8-16
5-2-2
8-5
8-17
8-29
8-40
5-2
8-18
8-42
5-3
8-6
8-30
8-41
5-2-3
8-7
8-19
8-31
8-44
5-5
8-8
8-20
8-32
8-43
5-4
8-9
8-21
8-33
5-6
8-10
8-22
8-34
5-7
8-35
5-7-2
8-11
8-23
8-24
8-36
8-45
Fig. 4. List of polyominoes produced by the algorithm for n ≤ 8. The labels indicate n and the tile number for that n.
In Fig. 4, all polyominoes obtained by the above procedure for n ≤ 8 are listed. The first number in the 2-number label identifies n and the second number identifies the number of the tile for that n. For example, 8-17 identifies the 17th 8-omino listed by the computer. The tile in Fig. 3 (c) is listed as 5-1 in Fig. 4. For each tile, its corresponding p4 tiling is obtained by using the filled and open squares (shown as circles in Fig. 4) as 4-fold rotation centers. For example, tile 8-22 produces the tiling shown in Fig. 1. If the same polyomino can have different 4-fold rotation centers, a third number is added after the 2-number label. For example, 5-1 and 5-1-2 are the same polyomino but have differently-placed rotation centers. We note that in the list some tiles may produce tilings with higher symmetry than p4. For example, 1-1 and 4-3, which are simple squares, are tiles for p4m tilings, while 8-7 and 8-26 are tiles for p4g tilings.
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3. Polyiamonds for tilings with 3-fold rotation symmetry We can obtain polyiamonds for p3 tilings in a similar manner. Fig. 5 (a) shows the symmetry elements of a p3 pattern where black triangles, open triangles and gray triangles represent equivalent 3-fold rotation centers, respectively. The vectors shown are two independent translations that generate the translation subgroup of the p3 symmetry group.
L
(a)
(b)
(c)
Fig. 5. (a) Symmetry elements of a p3 pattern. (b) Arrangement of tiles. (c) Simplest polyiamond by expansion.
We assume that tiles have no 3-fold rotational symmetry with respect to a given pattern of symmetry elements, and so at least two different 3-fold centers are connected by the boundary of a tile. Thus we may represent tiles by dotted line segments that join adjacent black and open triangles; this gives an arrangement of segments that form a regular hexagonal tiling as shown in Fig. 5 (b). Expansion of a dotted segment to a diamond as shown in Fig. 5 (c) produces the simplest polyiamond tile; this corresponds to Fig. 2 (c) for the polyomino case. Following a procedure similar to that for polyominoes, we superimpose the pattern in Fig. 5 (b) on a reduced copy of the pattern as shown in Fig. 6 (a), by connecting two 3-fold centers by a dotted line (each dotted line representing a larger tile). Using the reduced hexagonal pattern we can define a generator for a tile in a manner similar to the p4 case. Fig. 6 (b) shows an example of a generator consisting of n = L2 /l2 = 7 elements using the overlaid pattern shown in Fig. 6 (a). Note that n for this case is limited to n = x2 + y 2 + xy, x, y = 0, 1, 2, . . . , (x, y) = (0, 0),
(3)
i.e., n = 1, 3, 4, 7, 9, . . . instead of Eq. (2) for p4. Fig. 6 (c) is the corresponding tile gotten by expanding each element of the generator in the manner shown in Fig. 5 (c). This tile is a 14-iamond, composed of 14 congruent equilateral triangles. The tiling produced by applying 120◦ rotations about the two different 3-fold centers joined by the dotted segment is shown in Fig. 6 (d) In Fig. 7, all 2n-iamonds obtained in this manner for n ≤ 9 are listed. The tiling produced by tile 9-1-3 is shown in Fig. 1. We note that the list includes some tiles that generate tilings with reflection symmetry as well; for example, tile 9-6 produces a p31m tiling. In addition, some tiles produce tilings that also have 6-fold rotation symmetry. For example, tiles 1-1 and 3-1 produce p6m tilings, and tile 7-5 produces a p6 tiling.
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(a)
(b)
(c)
(d)
Fig. 6. (a) The reduced pattern with an original pattern superimposed. (b) Generator. The dotted line has length L. (c) Generated tile. (d) Tiling by generated tile.
1-1
3-1
7-8
9-2-3
9-5-3
4-1
7-8-2
9-3
9-6
7-1
7-2
7-9
9-3-2
9-7
7-10
9-3-3
9-7-2
7-2-2
7-3
9-1
9-1-2
9-4
9-7-3
7-4
7-5
9-1-3
7-6
9-2
9-4-2
9-4-3
9-5
9-8
9-8-2
9-8-3
7-7
7-7-2
9-2-2
9-5-2
Fig. 7. List of polyiamonds produced by the p3 algorithm for n ≤ 9. The labels indicate n and the tile number for that n.
4. Polyiamonds for tilings with 6-fold rotation symmetry Finally, we consider polyiamond tiles for p6 tilings. We can proceed as in the previous sections. Fig. 8 (a) shows the symmetry elements of a p6 pattern where filled circles represent equivalent 6-fold rotation centers. Note that in p6 patterns all 6-fold centers are identical. Triangles and rectangles represent 3-fold rotation centers and half-turn centers, respectively. The vectors shown are two independent translations that generate the translation subgroup of the p6 symmetry group. We assume that tiles have no 6-fold rotational symmetry, hence tiles (represented by dotted lines) are arranged on the regular triangular lattice as shown in Fig. 8 (b). Expanding a line as shown in Fig. 8 (c) produces the simplest polyiamond for this case; this expansion corresponds to Fig. 2 (c) for p4 or to Fig. 5 (c) for p3.
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L
(a)
(b)
(c)
Fig. 8. (a) Symmetry elements of a p6 pattern. (b) Arrangement of tiles. (c) Simplest tile by expansion.
(a)
(b)
(c)
(d)
Fig. 9. (a) The reduced pattern with an original pattern superimposed. (b) Generator. The dotted line has length L. (c) Generated tile. (d) Tiling by generated tile.
As before, we reduce our pattern in Fig. 8 (b) so that we can superimpose the original pattern, matching 6-fold centers, as illustrated in Fig. 9 (a). Any such overlaid pattern can be produced by connecting two 6-fold centers of the reduced pattern by a dotted line (the dotted line representing a larger tile). Using the overlaid pattern we can define a generator as before. Fig. 9 (b) shows an example of a generator which consists of n = L2 /l2 = 7 elements of the reduced pattern in Fig. 9 (a). Since all 6-fold centers are identical and the tile must have half-turn symmetry (as indicated by the half-turn center at the midpoint of the dotted line in Fig. 9 (b)), the generator for this tile also must have half-turn symmetry. Fig. 9 (c) is the corresponding tile gotten by expanding each element of the generator in the manner shown in Fig. 8 (c). For this p6 case, n also satisfies Eq. (3). Fig. 9 (d) shows the tiling by the tile in Fig. 9 (c), generated by 60◦ rotations about the 6-fold centers at the endpoints of the generator. In Fig. 10, all 2n-iamonds obtained in this manner for n ≤ 13 are listed. As indicated in the previous section, tile 1-1 in this list produces a p6m tiling. Also, some tiles in this list also appear in the list in Fig. 7. This is because all p6 tilings have 3-fold rotations, and the algorithm used to produce polyiamonds with generators that connect 3-fold centers sometimes produces tiles with 6-fold centers on the boundary. For example, tile 7-2 in Fig. 10 is the same as tile 7-5 in Fig. 7; Fig. 10 shows the 6-fold centers on the tile, while Fig. 7 shows the 3-fold centers.
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1-1
3-1
12-1
13-6
7-1
12-2
13-7
7-2
9-1
13-1
13-2
13-8
13-9
9-2
9-3
13-3
13-10
9-4
13-4
13-11
13-5
13-12
Fig. 10. Polyiamonds produced by the p6 algorithm for n ≤ 13. The labels indicate n and the tile number for that n.
5. Additional comments The algorithm presented in Section 2 produces polyominoes that generate isohedral tilings by 4-fold rotations. Such tilings will have p4 rotational symmetry and perhaps additional symmetry (i.e., they might have p4g or p4m symmetry group). In addition, the algorithm produces all polyominoes that are fundamental domains for p4 isohedral tilings, that is, minimal regions which, when acted on by the symmetry group of the tiling, generate the whole tiling. This is because we know from Heesch’s work [5] that a fundamental domain for a p4 isohedral tiling must be one of three types, and only two of these types are possible for polyominoes. A polyomino that is a p4 fundamental domain will have either four or five vertices at which three or four tiles meet in the tiling and have non-equivalent 4-fold centers at two of these vertices. When the tile has four of these vertices, the two other vertices are half-turn centers of the tiling; when the tile has five of these vertices, the edge that does not meet a 4-fold center has a half-turn center at its midpoint. Our polyomino construction using a generator that connects two 4-fold centers produces tiles that satisfy these conditions. Conversely, any polyomino that is a fundamental domain for a p4 tiling can easily be converted into its generator as defined in Section 2. Thus our list in Fig. 4 contains, for n ≤ 8, all n-ominoes that are fundamental domains for p4 isohedral tilings (as well as many other n-ominoes that are not fundamental domains for the p4 isohedral tiling they generate). Also, given a polyomino without 4-fold symmetry, we can determine whether or not it can produce a p4 isohedral tiling by checking to see if it can be converted to a generator by taking as 4-fold centers a suitable pair of vertices of the polyomino. For polyiamonds, our algorithms in Sections 3 and 4 produce only a partial list of such tiles that can produce isohedral tilings with p3 or p6 symmetry. A main reason is that the unit shapes we have chosen, shown in Fig. 5 (c) and Fig. 8 (c), are not equilateral triangles but diamonds, and hence can never produce an m-iamond for m odd. In addition, there are m-iamonds for even m that do not arise from our algorithm. For example, our algorithms produce no 6-iamonds, but the 6-iamond in Fig. 11 produces a p31m tiling by rotating about the labeled 3-fold centers. For the p6 case, our algorithm always produces a tile with half-turn symmetry, and the tile can be dissected into two congruent polyiamonds that generate a p6 isohedral
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tiling by rotating about the 6-fold center and the half-turn center on the boundary of the half-tile. While these dissections produce some m-iamonds for odd m, it is not likely that all can be produced in this manner. And there are still other polyiamonds that do not have half-turn symmetry and do not arise as half of a centro-symmetric polyiamond, yet tile the plane in a p6 isohedral manner. A simple example is the 4-iamond in the shape of a chevron, shown in Fig. 11, along with its p6 tiling, produced by rotating about its labeled 6-fold and 3-fold centers.
Fig. 11. (Left) A 6-iamond and its p31m isohedral tiling. (Right) A 4-iamond and its p6 isohedral tiling.
However, it might be possible to enumerate polyiamonds that produce p3 or p6 isohedral tilings by taking an equilateral triangle as the unit shape. Also, if we take a regular hexagon as a unit shape then it might be possible to enumerate polyhexes that produce p3 isohedral tilings. We will address these problems in further investigations. We note that the method discussed here may be generalized to tilings having other symmetry groups. Reference [2] considers some of these generalizations. We also note that the method used in our algorithm, of overlaying a pattern of symmetry elements on a reduced copy of the pattern, was employed by Rinus Roelofs [7] for quite a different purpose: to transform an Escher tessellation to a different Escher tessellation having the same symmetry group. References 1. H. S. M. Coxeter, and W. O. J. Moser, Generators and Relations for Discrete Groups, (Springer-Verlag, New York 1965). 2. H. Fukuda, T. Betumiya, S. Nishiyama and G. Nakamura, KATACHI ∪ SYMMETRY (Springer, Tokyo 1996) 231–238. 3. S. Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings, revised ed., (Princeton University Press, 1994). 4. B. Gr¨ unbaum, and G. C. Shephard, Tilings and Patterns (W.H. Freeman, 1987). 5. H. Heesch and O. Kienzle, Fl¨ achenschluss. System der Formen l¨ uckenlos aneinanderschliessender Flachteile (Springer-Verlag, 1963). 6. G. E. Martin, Polyominoes: A Guide to Puzzles and Problems in Tiling (Mathematical Association of America, 1991). 7. R. Roelofs, “Not the Tiles, but the Joints: A little Bridge Between M.C. Escher and Leonardo da Vinci”, in M. C. Escher’s Legacy, A Centennial Celebration, Eds. D. Schattschneider and M. Emmer (Springer-Verlag, 2003) 252–264.
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8. D. Schattschneider, American Mathematical Monthly 85 (1978) 439–450.
Received: December 22, 2002 Final version received: December 22, 2002
