Abstract:
This invention relates to interlocking rectangular and triangular tiles for use in tessellating a plane, teaching certain aspects of geometry and trigonometry, and creating decorative objects. The tiles comprising the invention, some with a design on them, tessellate the plane with periodic and non-periodic patterns based on the design as the basic unit of repetition. In particular; the tiles can be assembled to create any of the 17 possible plane periodic patterns. The tiles of the present invention can be used as a teaching tool for a partially structured exploration of plane periodic patterns, and for learning the mathematical notation for these patterns. They can also be used for teaching basic geometry to younger children through exploratory play. This invention can also be used as a stencil assembly, so that the patterns created can be transferred to a plane surface and used as decorations.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This applications claims the benefit of provisional patent application Ser. No. 61/422,037, filed 2010 Dec. 10 by the present inventor. 
    
    
     BACKGROUND 
     Prior Art 
     The following is a tabulation of some prior art that presently appears relevant: 
     
       
         
               
             
               
               
               
               
             
           
               
                   
               
               
                 U.S. Patents 
               
             
          
           
               
                 Patent  
                 Kind  
                   
                   
               
               
                 Number 
                 Code 
                 Issue Date 
                 Patentee 
               
               
                   
               
               
                 909603  
                   
                 Jan. 12, 1909 
                 Albert S. Janin 
               
               
                 3633286 
                   
                 Jan. 11, 1972 
                 Donald J. Maurer 
               
               
                 4133152 
                   
                 Jan. 9, 1979 
                 Roger Penrose 
               
               
                 4620998 
                   
                 Nov. 4, 1986 
                 Haresh Lalvani 
               
               
                 5203706 
                   
                 Apr. 20, 1993 
                 Amos Zamir 
               
               
                 5368301 
                   
                 Nov. 29, 1994 
                 Dennis E. Mitchell 
               
               
                 5619830 
                   
                 Apr. 15, 1997 
                 John A. L. Osborn 
               
               
                 6309716 
                 B1 
                 Oct. 30, 2001 
                 Adrian Fisher et al. 
               
               
                 7833077 
                 B1 
                 Nov. 16, 2010 
                 Felix J. Simmons, Jr. 
               
               
                   
               
             
          
         
       
     
     NONPATENT LITERATURE DOCUMENTS 
     
         
         Amenta, N., &amp; Phillips, M.  Java Kali . Retrieved from website: &lt;http://www.geom.uiuc.edu/java/Kali/welcome.html&gt;, 1996. 
         Conway, J. H., Burgiel, H., &amp; Goodman-Strauss, C. (2008).  The Symmetries of Things . Wellesley, Mass.: A K Peters, Ltd. 
         Gallian, J. A. (2009).  Contemporary Abstract Algebra . Belmont, Calif.: Brooks Cole. 
         National Council of Teachers of Mathematics.  Geometry Standards . Retrieved from website: &lt;http://standards.nctm.org/document/appendix/geom.htm&gt;, 2011. 
         Nicolas Schoeni, N., Hardake, W., &amp; Chapuis, G.  Escher Web Sketch . Retrieved from website: &lt;http://escher.epfl.ch/escher/&gt;, 1987. 
       
    
     BACKGROUND OF THE INVENTION 
     Plane periodic patterns have been used as decorations all over the world for millennia. Apart from being used in the design of floor tiles, wallpapers, textiles, and other articles, plane periodic patterns also occur in nature as cross sections of crystal lattices. As such, they have been studied by crystallographers since the 19 th  century. It was at then that scientists discovered that all the plane periodic patterns fall into just 17 categories. Mathematicians consider two patterns in the same category to be “the same”, even if the patterns look somewhat different to a lay person. This begs the question: what criteria are used to determine whether any two given patterns are in the same category, or, in other words, “the same”? The mathematically correct answer is that two patterns are “the same” if their groups of symmetry are the same. A pattern&#39;s group of symmetry is the set of all the moves (such as rotations, reflections, translations, or glide reflections) one can perform on a pattern without changing it. This definition of “the same” with regard to plane periodic patterns will be used throughout this document. 
     Some definitions are in order. The following notation, henceforth referred to as the Conway notation, will be used for the 17 plane periodic patterns: *632, 632, 3*3, *333, 333, *442, 442, 4*2, 2*22, 22*, 2222, *2222, **, *X, XX, 22X, O. The patterns, together with their notation, are shown in  FIG. 7 . Patterns *632 and *442 will be referred to as the “parent” patterns, since all the other 15 patterns can be derived from them. 
     The “fundamental domain” of a pattern is the smallest part of the pattern based on which the entire pattern can be constructed. Some patterns have a uniquely defined fundamental domain, while others have fundamental domains of more than one possible shape. 
     Some of the tiles comprising the present invention differ from one another only in terms of the presence of a design. The tiles that do not have a design will be referred to as “plain”. 
     A tile and its “complement” are the same size and shape. The only difference between them is that the genders of their corresponding male and female coupling elements are opposite. Two complementary tiles fit together so that the designs on each tile are mirror images of one another. 
     DISCUSSION OF KNOWN ART 
     The “17 wallpaper patterns,” as they are commonly known, have been the subject of rigorous mathematical research, of which the most recent and complete exposition can be found in (Conway, Burgiel, &amp; Goodman-Strauss, The Symmetries of Things, 2008, pp. 15-49). This work describes one of the standard notations for the patterns, known as the “Conway notation” and shows how this notation holds the key to the proof of why there are only 17 possible patterns. Conway et al. go further to show the relationships between the various patterns and the different ways they can be colored. They put these findings in a mathematical framework using group theory, which is an extremely important and powerful tool in the study of mathematics. 
     Given both the visual and mathematical appeal of planar patterns, they have been a popular subject of study for students of a variety of levels. At the college level, students are taught to identify each of the 17 patterns in textbooks such as (Gallian, 2009, pp. 467-478). For students wishing to create an electronic version of any of the 17 patterns based on a single motif, there is a number of free software packages available on the intemet. These include Escher Web Sketch (Nicolas Schoeni, Hardake, &amp; Chapuis) and Java Kali (Amenta &amp; Phillips, 1996). Both of these programs can create any of the 17 patterns in seconds, making them effective tools for demonstration. However, the speed and the method with which the patterns are drawn on the screen make it difficult for most users to learn about the structure and identification of the patterns from these programs. All the elements of the tiling array are drawn on the screen simultaneously, so that the user has a hard time grasping the symmetries (i.e rotations, translations, glide reflections, or reflections) that generate the entire pattern. 
     There are infinitely many ways to create tessellations, and much work has been done in the area of designing tessellating articles. Some examples can be seen in U.S. Pat. No. 909,603 (1909) to Janin, U.S. Pat. No. 4,133,152 (1979) to Penrose, U.S. Pat. No. 4,620,998 (1986) to Lalvani, U.S. Pat. No. 5,619,830 (1997) to Osborn, and U.S. Pat. No. 6,309,716 (2001) to Fisher. The above patents disclose sets of tiles that can be assembled into either periodic or non-periodic planar patterns. However, they neither attempt to construct the complete set of plane periodic patterns, nor claim that this is possible. 
     U.S. Pat. No. 5,368,301 (1994) issued to Mitchell discloses a puzzle composed of isosceles right triangle pieces that are similar in shape to some of the pieces in the present invention, as well as equilateral triangles. It is not possible to construct the complete set of all 17 repeating plane patterns with this set, even if one were to work with a multitude of identical tiles. Moreover, the isosceles right triangle pieces in the present invention have male and female coupling elements that are spaced differently than those in Mitchell&#39;s art. This allows for more ways to assemble the tiles. 
     U.S. Pat. No. 7,833,077 (2010) due to Simmons, Jr. discloses interlocking blocks that can be assembled in a variety of ways. Their cross sections are somewhat similar to the tiles in the present invention, but they cannot be assembled into all of the 17 repeating planar patterns. 
     U.S. Pat. No. 3,633,286 (1972) to Maurer and U.S. Pat. No. 5,203,706 (1993) to Zamir disclose interlocking tiles with a stencil portion. Both of these inventions can be used to draw a very limited number of repeating planar patterns due to the shape of the tiles (rectangular) and the placement of the interlocking elements. 
     The present invention differs from all prior art in the fields of tessellation and stencils in that it (1) combines the tessellating and stenciling capabilities to create any of the 17 possible plane periodic patterns (2) is a teaching tool for learning about the basic symmetries as well as the plane periodic patterns. Math education experts recommend that students starting in 3 rd  grade “predict and describe the results of sliding, flipping, and turning two-dimensional shapes” (National Council of Teachers of Mathematics). They can learn to do this by exploring the various juxtapositions of the tiles comprising the invention. The invention is also a tool for helping students learn the properties of 30°-60°-90° triangles 45°-45°-90° triangles. 
     The present invention has an advantage over polygonal tiles with straight edges in that the possibilities of combinations are reduced by the arrangement of the male and female coupling elements. Thus, the students are somewhat guided toward discovering mathematical properties by the way the tiles mate with each other. Because of this guiding structure built into the tiles, less teacher supervision is required with using the present invention compared to using traditional straight-edged polygons. 
     OBJECTS OF THE INVENTION 
     It is the primary object of the present invention to provide a set of tiles as a manipulative educational aid for studying plane periodic patterns through the process of constructing them, and for learning the mathematical notation for these patterns. 
     It is another object of the invention to provide a set of tiles as a manipulative educational aid for learning basic geometrical and trigonometric concepts. 
     It is an object of the invention to provide a set of tiles as a manipulative educational aid for learning the four types of symmetry: translation, rotation, reflection, and glide reflection. 
     It is a further object of the invention to provide a stenciling assembly tracing planar patterns for the purpose of decoration and amusement. 
     SUMMARY 
     A set of tiles for covering a surface consists of nine types of tile. The basic shape of each tile is a 30°-60°-90° triangle, a 45°-45°-90° triangle, or a rectangle. Some tiles have an asymmetrical design cut out in the fashion of a stencil, while others are plain. The tiles have jigsaw puzzle-like edges, with the male and female coupling elements placed so that the tiles can be juxtaposed only in certain prescribed configurations. Either periodic or non-periodic patterns can be assembled using the tiles. With periodic patterns, a fundamental domain can always be constructed using one or more tiles. This has the advantage of reinforcing the concept of a fundamental domain in the mind of the user. After creating a repeating pattern, the user can replace some of the tiles with designs on them with identically shaped plain tiles, thereby creating a new repeating pattern that is related to the original one he created. By working in this fashion and restricting himself to working with one shape at a time (e.g. isosceles triangle), the user can gradually discover the various families of patterns, along with their Conway notation, all of which comprise complete set of plane periodic patterns. The user then has the option of tracing the patterns in the stenciled tiles. This feature serves two functions. First, the user can keep a record of his design and utilize it for decorative or other purposes. Second, the pattern can be colored, creating yet more repeating patterns. The tiles can also be used by school age children for exploring basic geometry. Children can be challenged to create a multitude of different polygons or patterns by juxtaposing a set of tiles. In addition, the tiles are a tool in teaching the properties of 30°-60°-90° triangle and 45°-45°-90° triangles, which are key elements in the study of trigonometry. 
    
    
     
       DRAWINGS 
         FIG. 1  is a perspective view of a typical tile. 
         FIG. 2A  is a top view of the tile in  FIG. 1 . 
         FIG. 2B  is a top view tile complementary to the tile in  FIG. 2A . 
         FIG. 3A  is a top view of the plain version of the tile in  FIG. 2A . 
         FIG. 3B  is a top view of the plain version of the tile in  FIG. 2B . 
         FIGS. 4A and 4B  show a top view of the right isosceles triangle tile and its complement. 
         FIG. 5  is a top view of the plain version of the tiles in  FIGS. 4A and 4B . 
         FIGS. 6A and 6B  show a top view of the rectangular tile and its complement. 
         FIG. 7  lists the 17 plane periodic patterns along with their labels in the Conway notation. 
         FIG. 8  is a top view of the *632 pattern. The fundamental domain is outlined in bold. 
         FIG. 9  is a top view of the *333 pattern. The fundamental domain is outlined in bold. 
         FIG. 10  is a top view of the 3*3 pattern. A fundamental domain is outlined in bold. 
         FIG. 11  is a top view of the 632 pattern. A fundamental domain is outlined in bold. 
         FIG. 12  is a top view of the 333 pattern. A fundamental domain is outlined in bold. 
         FIG. 13  is a top view of the *442 pattern. The fundamental domain is outlined in bold. 
         FIG. 14  is a top view of the 4*2 pattern. A fundamental domain is outlined in bold. 
         FIG. 15  is a top view of the *2222 pattern. The fundamental domain is outlined in bold. 
         FIG. 16  is a top view of the 2*22 pattern. A fundamental domain is outlined in bold. 
         FIG. 17  is a top view of the 442 pattern. A fundamental domain is outlined in bold. 
         FIG. 18  is a top view of the 2222 pattern. A fundamental domain is outlined in bold. 
         FIG. 19  is a top view of the 22* pattern. A fundamental domain is outlined in bold. 
         FIG. 20  is a top view of the ** pattern. A fundamental domain is outlined in bold. 
         FIG. 21  is a top view of the *X pattern. A fundamental domain is outlined in bold. 
         FIG. 22  is a top view of the XX pattern. A fundamental domain is outlined in bold. 
         FIG. 23  is a top view of the O pattern. A fundamental domain is outlined in bold. 
         FIG. 24  is a top view of the 22X pattern. A fundamental domain is outlined in bold. 
         FIG. 25  is a top view of the O pattern made with the rectangular tiles. 
         FIG. 26  is a top view of the *2222 pattern made with the rectangular tiles. 
         FIG. 27  is a top view of the XX pattern made with the rectangular tiles. 
         FIG. 28  is a top view of the *X pattern made with the rectangular tiles. 
         FIG. 29  is a top view of the ** pattern made with the rectangular tiles. 
         FIG. 30  is a top view of the 2*22 pattern made with the rectangular tiles. 
         FIG. 31  is a top view of the 22* pattern made with the rectangular tiles. 
         FIG. 32  is a top view of the 22X pattern made with the rectangular tiles. 
         FIG. 33  is a top view of the 2222 pattern made with the rectangular tiles. 
         FIG. 34  is a top view of the XX pattern made with the rectangular tiles. 
         FIG. 35  is a top view of the XX pattern made with the rectangular tiles configured in a different way than in  FIG. 34 . 
         FIG. 36  is a top view of the XX pattern made with the tiles shown in  FIGS. 4A ,  4 B, and  5 . This is a different layout of the XX pattern than the one shown in  FIGS. 22 ,  34 , and  35 . 
         FIGS. 37A ,  37 B,  37 C, and  37 D demonstrate the four distinct ways in which the rectangular tiles of  FIGS. 6A and 6B  can be juxtaposed along their long sides. 
         FIGS. 38A ,  38 B,  38 C, and  38 D illustrate some of some of the ways in which the tiles fit together. 
         FIGS. 39A and 39B  illustrate how tiles in  FIG. 5  can be assembled into squares so as to demonstrate the derivation of √{square root over (2)}. 
         FIGS. 40A ,  40 B,  40 C, and  40 D illustrate how the tiles in  FIGS. 3A and 3B  can be juxtaposed so as to demonstrate the properties of a 30°-60°-90° triangle. 
         FIGS. 41A and 41B  show both sides of an alternative embodiment of a typical puzzle piece. In contrast with the piece depicted in  FIG. 1 , here the design is printed on both side of each piece, rather than cut out. 
         FIGS. 42A ,  42 B,  42 C, and  42 D show a top view of an alternative embodiment of the 30°-60°-90° triangle tiles that were depicted in  FIGS. 2A ,  2 B,  3 A and  3 B. 
         FIGS. 43A and 43B  show alternative embodiments of the rectangular tiles in terms of the placement of their male and female coupling elements. 
         FIG. 43C  shows the juxtaposition of two identical tiles pictured in  FIGS. 43A . 
         FIGS. 44A and 44B  show another alternative embodiment of the rectangular tiles in terms of the placement of their male and female coupling elements. 
     
    
    
     REFERENCE NUMERALS 
     
         
         
           
               50  Stencil aperture 
               51  Male coupling element 
               52  Female coupling element 
           
         
       
    
     DETAILED DESCRIPTION 
       FIG. 1  is an enlarged left perspective view of a single tile. This particular embodiment has an aperture  50  in the shape of a paisley, but any other asymmetrical shape may be used in place of the paisley. The dimensions obey the relationships y=√{square root over (3)}x and z=2x. The absolute length of the sides x, y, z and its thickness, t, are of sufficient size for the tile to be comfortably used as a stencil by children and adults. The male and female coupling elements can be realized as tongues and recesses, markers, dowel pins and recesses, or other connectors with clearly identifiable male and female elements. In the preferred embodiment, pictured in  FIG. 1 , tongues and recesses form the male and female coupling elements, respectively. The male  51  and female  52  coupling elements are each symmetric about their respective midlines. 
     The tiles comprising the invention can be made of acrylic, polycarbonate, other plastics, chipboard, cardboard, glass, foam, wood, or other sufficiently rigid materials. Depending on the materials used and the size and thickness of the tiles, a variety of methods can be used to manufacture the tiles, such as cutting by water jet or laser, stamping with a die or injection molding. 
       FIG. 2A  is a top view of the tile in  FIG. 1 , and  FIG. 2B  is its complement. The dimensions in this embodiment of the invention are as follows. a=b=d=x/2, e=x where x was defined in  FIG. 1 . c is determined by the fact that 2c+b=y. So 
     
       
         
           
             c 
             = 
             
               
                 
                   y 
                   - 
                   b 
                 
                 2 
               
               . 
             
           
         
       
     
       FIGS. 3A and 3B  depict the plain version of the tiles in  FIGS. 2A and 2B , respectively. 
       FIGS. 4A and 4B  depict the right isosceles triangle tile and its complement. 2c′+b′=2c+b=y, where y was defined in  FIG. 1 . In the preferred embodiment, b′=b, c′=c and g=2f. b and c are the same dimensions as in  FIGS. 2A and 2B . Given that the tiles in question are shaped like right isosceles triangles, the length of each hypotenuse must be √{square root over (2)} times the length of either side. This translates into the equation 
                 2   ⁢           ⁢   f     +   g     =         2     ⁢     (       2   ⁢           ⁢     c   ′       +     b   ′       )       =       2     ⁢     y   .               
Together with g=2f, this gives f=√{square root over (2)}y/4 and g=√{square root over (2)}y/2.
 
       FIG. 5  depicts the plain version of both of the tiles in  FIGS. 4A and 4B . 
       FIGS. 6A and 6B  is a top of the rectangular tile and its complement. In the preferred embodiment, b″=b, c″=c, j=b, and h=j/2, with b and c defined in  FIGS. 2A and 2B . 
       FIG. 7  depicts of all the 17 planar periodic patterns based on a motif of a half-apple as the basic unit of repetition. The Conway notation is used to label each pattern. 
       FIG. 8  is a top view of the pattern denoted as *632 in the Conway notation. The inside of the paisley shape will be shown hatched for clarity. The pattern is assembled using the tiles in  FIGS. 2A and 2B . The fundamental domain of the pattern is shown in bold. If we disregard the male and female coupling elements of the tiles, the fundamental domain of this pattern corresponds exactly to a single tile. 
       FIG. 9  is a top view of the pattern denoted as *333 in the Conway notation. It is assembled using the tiles in  FIGS. 2A ,  2 B,  3 A, and  3 B. The fundamental domain consists of the two tiles as shown in bold. 
       FIG. 10  is a top view of the pattern denoted as 3*3 in the Conway notation. It is assembled using the tiles in  FIGS. 2A ,  2 B,  3 A, and  3 B. This pattern has more than one fundamental domain. The bold outline shows one of these fundamental domains. 
       FIG. 11  is a top view of the pattern denoted as 632 in the Conway notation. It is assembled using the tiles in  FIGS. 2A ,  2 B,  3 A, and  3 B. A fundamental domain is shown in bold. 
       FIG. 12  is a top view of the pattern denoted as 333 in the Conway notation. It is assembled using the tiles in  FIGS. 2A ,  2 B,  3 A, and  3 B. A fundamental domain is shown in bold. 
       FIG. 13  is a top view of the pattern denoted as *442 in the Conway notation. It is assembled using the tiles in  FIGS. 4A and 4B . The only fundamental domain is shown in bold. *442 the second of the two “parent” patterns. It can be used to construct the patterns *2222, 2*2, 4*2, and 442. This is accomplished by systematically replacing groups of tiles with identically shaped plain tiles. 
       FIG. 14  is a top view of the pattern denoted as 4*2 in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 15  is a top view of the pattern denoted as *2222 in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . The fundamental domain is shown in bold. The rest of the patterns, 2222, 22*, 22X, **, *X, XX, O can be derived from *2222 by replacing some of the patterned tiles with identically shaped plain tiles. 
       FIG. 16  is a top view of the pattern denoted as 2*22 in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 17  is a top view of the pattern denoted as 442 in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 18  is a top view of the pattern denoted as 2222 in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 19  is a top view of the pattern denoted as 22* in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 20  is a top view of the pattern denoted as ** in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 21  is a top view of the pattern denoted as *X in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 22  is a top view of the pattern denoted as XX in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 23  is a top view of the pattern denoted as O in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 24  is a top view of the pattern denoted as 22X in the Conway notation. It is assembled using the tiles in  FIGS. 4A ,  4 B, and  5 . A fundamental domain is shown in bold. 
       FIG. 25  is a top view of the O pattern made with the rectangular tiles in  FIG. 6B . This is a different way of making the pattern than the one shown in  FIG. 23 . 
       FIG. 26  is a top view of the *2222 pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 27  is a top view of the XX pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 28  is a top view of the *X pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 29  is a top view of the ** pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 30  is a top view of the 2*22 pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 31  is a top view of the 22* pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 32  is a top view of the 22X pattern made with the rectangular tiles in  FIGS. 6A and 6B . 
       FIG. 33  is a top view of the 2222 pattern made with the rectangular tiles in  FIG. 6B . 
       FIG. 34  is a top view of the XX pattern made with the rectangular tiles in  FIG. 6B . 
       FIG. 35  is a top view of the pattern denoted as XX in the Conway notation. In contrast with the XX pattern shown in  FIG. 27 , here the pattern is assembled entirely of the tiles in  FIG. 6A . Although the patterns in  FIGS. 27 and 35  do not appear to be identical, they are considered to be the “same” from a mathematical point of view. That is because they have the same group of symmetry. 
       FIG. 36  shows how the tiles in  FIGS. 4A ,  4 B, and  5  can be juxtaposed to make the XX pattern. A different arrangement of these tiles to make the XX pattern was already shown in  FIGS. 27 ,  34 , and  35 . In this configuration, the tiles are staggered with respect to each other, utilizing the placement of the male and female coupling elements along the hypotenuse of the triangular tiles in the preferred embodiment. 
       FIGS. 37A ,  37 B,  37 C, and  37 D illustrate how the invention can be used to teach students translation, gyration, reflection, and glide reflection. In  FIG. 37A  one gets from one tile to the other by horizontally translating. In  FIG. 37B , one rotates, or gyrates, about the midpoint of the line joining the tiles to get from one tile to the other. Both translation and rotation are orientation preserving transformations, in the sense that they transform an image without mirror-reflecting it. This property is embedded in the invention in the following way. The user can perform all the orientation preserving transformations with just one type of tile; that is, without using the complement of the tile. However, if the user wants to perform an orientation reversing transformation, such as a reflection about an axis ( FIG. 37C ) or a glide reflection along an axis ( FIG. 37D ), he will need to use both the tile and its complement. 
       FIGS. 38A and 38B  show two different ways in which a rhombus may be made with two pairs of complementary triangles pictured in  FIGS. 3A and 3B .  FIG. 38C  shows how the same two pairs of complementary triangles can be rearranged to form a larger 30°-60°-90° triangle.  FIG. 38D  illustrates how a variety of tile shapes can be combined to form a trapezoid. 
       FIG. 39A  shows how two tiles pictured in  FIG. 5  fit together to form a square. The square shown in  FIG. 39B  obviously has area equal to twice that of the square in  FIG. 39A . Therefore, if we assume, for simplicity, that the area of the square in  FIG. 39A  is 1 unit, then the area of the square in  FIG. 39B  is 2 units. It follows that the side length of the square in  FIG. 39B  is a number whose square is equal to 2, in other words, √{square root over (2)}. By going through this reasoning, the student can “discover” the concept of the square root. 
       FIGS. 40A ,  40 B,  40 C, and  40 D form a sequence of figures that show how a student can be guided to “discover” the 30°-60°-90° triangle and its properties. This triangle is extremely important in the study of trigonometry. The only prior knowledge required is that the full revolution around any vertex spans 360°. By juxtaposing the tiles in  FIGS. 3A and 3B  as shown in  FIG. 40A , the student discovers that 12 of the smaller angles in the triangle make 360°. Therefore, the smaller angle must equal 360°/12=30°. Similarly, the construction in  FIG. 9B  demonstrates that the larger angle of the triangle must equal 360°/6=60°. Consequently, the student finds that two 30°-60°-90° triangles can be assembled to make an equilateral triangle.  FIGS. 40C and 40D  illustrate the fact that the shortest side of the 30°-60°-90° triangle has length that is half of the length of its hypotenuse. First, the user can verify that segments B′E and EA′ have equal length by stacking the triangles EFB′ and EFA′. Then the user discovers that the segments AB and A′B′ have the same lengths since the tiles fit together as shown in  FIG. 40D . Therefore, the length of EA′ is half of the length of AB. Since the length of AB is the same as the length of A′F, EA′ is also half of the length of A′F. 
     FIGS.  41 A- 45 B—Alternative Embodiments. 
       FIGS. 41A and 41B  show the front and back side of a version of the invention without the stencil capacity. Instead of being a cutout, as in  FIG. 1 , the design is printed or drawn on both sides of the tile. 
       FIGS. 42A ,  42 B,  42 C, and  42 D show an alternative embodiment of the tiles shown in  FIGS. 2A ,  2 B,  3 A, and  3 B, using the same dimensions a and b as earlier. 
       FIGS. 43A and 43B  show an alternative embodiment of the rectangular tiles shown in  FIGS. 6A and 6B . When the user starts out with a stack of two tiles from  FIG. 43A  and then juxtaposes them as pictured in  FIG. 43C , he physically performs a glide reflection. Thus, the tiles in  FIGS. 43A and 43B  are an aid for kinetic learning of the concept of glide reflection. Unlike the case in  FIG. 40D , here a glide reflection can be demonstrated using the same type of tile (as opposed to a tile and its complement). 
       FIGS. 44A and 44B  show a second embodiment of the rectangular tiles shown in  FIGS. 6A and 6B . In this embodiment, each tile mates with a glide reflected version of itself, where the gliding can be parallel to either side of the rectangle. 
     Operation 
     There is no single correct way to use this invention. In fact, one of the main purposes of the invention is for the user to explore the different ways of combining the tiles on his or her own. The male and female coupling elements of the tiles have been designed so as to provide clues for the user to engage in tile placements in a structured way. While there are no hard and fast rules of operation, some suggestions on how to use the invention effectively are outlined below. 
     Suggested Use, Version 1. 
     In this method of operation the user is guided to discover the various types of patterns he or she can make with the tiles while noting the relationships these patterns have with one another. While there is no mathematical prerequisite for seeing these relationships, the user schooled in abstract algebra will recognize the fact that the patterns derived from the “parent” pattern have symmetry groups that are subgroups of the symmetry group of the “parent” pattern. Following the steps below, the user can construct the complete set of 17 plane periodic patterns. 
     First, one starts with *632 as the “parent” pattern. 
     Step 1: The user is only permitted to work only with the tiles in  FIGS. 2A and 2B . He is given an equal number of tiles of each type. He is asked to create a pattern such that each tile is adjacent to its complementary tile. The user will eventually come up with the pattern *632, shown in  FIG. 8 . 
     Step 2: The user is given an option of creating a pattern with a design and its mirror image or one without the mirror image. In the latter case, the simplest thing to do is to replace all the tiles that are identical to each other with the corresponding plain tiles. This results in the 632 pattern, as shown in  FIG. 11 . In the former case, the user can do one of two things. If he replaces every other equilateral triangle made of 2 tiles with a plain one (so that each 2-tile equilateral triangle with a design is surrounded by 3 plain equilateral triangles), he will obtain 3*3 ( FIG. 10 ). On the other hand, if he replaces every other equilateral triangle made of 6 tiles with a plain one, he will obtain *333 ( FIG. 9 ). 
     Step 3: Now the user is asked to take either 3*3 or *333 and replace all the tiles that are identical to each other with plain tiles. This will result in the pattern 333 ( FIG. 12 ). 
     Now one starts with *442 as the “parent” pattern. 
     Step 4: Restrict the user to working only with the types of tiles in  FIGS. 4A and 4B . He is to make a pattern such that each tile is adjacent to its complementary tile. The user will eventually come up with the pattern *442, shown in  FIG. 13 . 
     Step 5: The user is given an option of creating a pattern with a design and its mirror image or one without the mirror image. In the latter case, the simplest thing to do is to replace all the tiles that are identical to each other with the plain tiles in  FIG. 5 . This results in the 442 pattern ( FIG. 17 ). In the former case, the user can do one of three things. The first option is to replace every other 4-tile square with one made of plain tiles. This will result in *2222 ( FIG. 15 ). The second option is to replace every other 2-tile isosceles triangle with one made of plain tiles. This will give 4*2 ( FIG. 14 ). The last option is for the user can replace every other 2-tile square with a blank one. This will result in 2*22, shown in  FIG. 16 . 
     Step 6: Starting with *2222, the user is asked to continue replacing patterned tiles with plain ones in the same vein as in Step 5. He will eventually come up with the remainder of the patterns: 2222, 22*, **, *X, XX, O, and 22X ( FIGS. 18-24 , respectively). 
     Stencil feature: At any point in the series of steps described above, the user is free to trace the pattern using the stencil capability of each tile. This can serve several purposes:
         1. Each user can keep a record of his work, and subsequently compare the patterns she obtained with those of other users.   2. After tracing the pattern, the user Can color it in 2, 3, or any prime number of colors to obtain new patterns. This topic is covered in (Conway, Burgiel, &amp; Goodman-Strauss, The Symmetries of Things, 2008, pp. 135-169).
 
Suggested Use, Version 2.
       

     In contrast with version 1, here the user starts out with fewer types of tiles and gradually increases the number of tiles available to her. This process guides her to discover the various features of the patterns and seeing how these features form an integral part of the Conway notation in (Conway, Burgiel, &amp; Goodman-Strauss, 2008, pp. 15-49). This version provides another way to guide the user to construct the set of 17 plane periodic patterns. 
     Step 1: To begin with, the user is permitted to work only with the rectangular tiles shown in  FIG. 6B .
         (a) She is asked to make all possible repeating patterns using these tiles without flipping them over. This will result in patterns O and 2222, as shown in  FIGS. 25 and 33 . These patterns have symmetries involving only translations and rotations. They are essentially extensions of  FIGS. 37A and 37B . Using just one type of rectangular tile without flipping it over corresponds to those patterns which, in the Conway notation, do not have any “X” or “*”.   (b) The user is still restricted only to the tiles pictured in  FIG. 6B , but now she is permitted to flip tiles over. This will result in the addition of the patterns XX and 22X, shown in  FIGS. 27 and 32 , respectively. X stands for “glide reflection”. The user will experience glide reflections as she moves the tiles. Flipping over the tiles and sliding them (otherwise they will not mate with the other tiles) corresponds to patterns with an “X” (but without the “*”) in the Conway notation.       

     Step 2: Now the user is permitted to work with tiles in  FIGS. 6A and 6B , and she is permitted to flip the tiles over. She will obtain the patterns *2222, *X, **, 2*22, and 22*, shown in FIGS.  26  and  28 - 31 , respectively. Working with both a tile and its complement corresponds to mirror reflecting, which is denoted with a “*” in the Conway notation. 
     Step 3: Now the user is permitted to work with tiles in  FIGS. 2A and 2B . She follows Steps 1-3 in version 1 to construct the patterns *632, 632, 3*33, *333, and 333. 
     Step 4: The user is permitted to work only with the tiles in  FIG. 4A  and  FIG. 4B . Following the Steps 4-6 in version 1 will enable her to construct the remaining patterns: *442, 4*2, and 442. 
     Suggested Use, Version 3. 
     In this version, the user needs to learn the notation for the 17 wallpaper pattern, as shown in  FIG. 7 . Then the invention can be used in a classroom or other group setting as a game for practicing identifying the notation. Here are some specific examples: 
     1) Each user is asked to make a pattern and also identify the patterns made by the other users.
         2) Given a restricted set of tiles to work with, the user can be asked to make as many distinct patterns as he can with those tiles. He is then asked to determine which of his patterns are mathematically the same.   3) Given one of the 17 possible patterns and the complete set of tiles, the user is asked to make the pattern in as many different ways as possible.
 
Suggested Use, Version 4.
       

     The plain tiles can be used by students learning basic geometry and trigonometry. For instance, given a set of tiles, a student can be challenged to find all the different shapes into which these tiles can be arranged. This is illustrated in  FIGS. 38A and 38C . This exercise can be made more challenging by asking the student to find all the different ways in which the tiles can be arranged to make a given shape, as seen in  FIG. 38B . In addition, a student can be asked to make a given shape with the tiles, such as, for example, a trapezoid shown in  FIG. 38D .  FIGS. 39A ,  39 B,  40 A,  40 B,  40 C, and  40 D and their detailed description illustrate how the tiles can be used to teach students some basic properties of important right triangles.  FIGS. 37A ,  37 B,  37 C, and  37 D, along with their detailed description, show how the tiles can be used to teach the basic 4 symmetry types of translation, gyration, reflection, and glide reflection.