Patent Publication Number: US-2013236707-A1

Title: Patterning Technique

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 13/364,619, filed Feb. 2, 2012, which is a continuation of U.S. patent application Ser. No. 13/086,680, filed Apr. 14, 2011, now U.S. Pat. No. 8,141,214, which is a divisional of U.S. patent application Ser. No. 12/583,449, filed Aug. 20, 2009, now U.S. Pat. No. 7,968,165, which is a continuation of International Application No. PCT/US2008/002361, filed Feb. 22, 2008, which claims the benefit of U.S. Provisional Application No. 60/903,113, filed Feb. 23, 2007, all of which are incorporated by reference herein in their entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure is directed generally to a patterning technique. More particularly, the disclosure is directed to a patterning technique for textiles or other materials that facilitates alignment of the pattern along seams and, in doing so, creates new, visually pleasing patterns. 
     BACKGROUND 
     Patterned textiles and fabrics are used frequently in a variety of applications, including, for example, carpet, garments, wallpaper, and upholstery. In many instances, the alignment of patterns between two or more textile pieces results in a substantial amount of waste. Thus, there is a need for a patterning technique that facilitates alignment of patterns between textiles and reduces the amount of waste generated. 
     SUMMARY 
     This disclosure relates generally to a patterning technique for textiles and patterns formed according to the patterning technique. The patterning technique facilitates alignment of two or more textile pieces, thereby simplifying use of the textiles and reducing the amount of waste associated with creating a visually pleasing pattern along and across seams. 
     The textile pattern generally comprises a design or motif repeated across the length and width of a textile. The motif comprises an arrangement of substantially square, substantially identical design modules variously oriented with respect to one another in fixed positions within the motif. Each design module includes at least two visually distinct hues. 
     A module used in accordance with the invention generally includes one or more features or characteristics that allow the module to form a visual connection or link with an adjacent module in the motif. The visual connection may be a “perfect” edge alignment or an “imperfect” edge alignment that is nonetheless visually pleasing. 
     To form a pattern according to the invention, a design module having an initial orientation is rotated and/or inverted to prepare a plurality of new module orientations. The variously oriented modules are arranged in a tiled configuration such that adjacent modules are contiguous with one another (e.g., abutting or touching) within the tiled array of modules, with each module in the motif having the initial orientation or one of the new orientations. The collective design of the oriented modules in the array defines a motif, which may be repeated across the length and width of a textile web. 
     A patterned textile according to the invention may be used or installed readily. With a slight adjustment of one or more adjacent pieces of the textile, alignment of the patterns along and cross a seam can be achieved without having to remove and/or discard a significant portion of the textile piece. Each of the various possible alignments may result in a different overall pattern for the abutted and/or adjoined textile pieces. Nonetheless, the vague or seemingly random nature of the pattern within the motif renders the resulting overall pattern both unique and visually pleasing. 
     The motif then may be used to form a textile (not shown) according to any suitable technique, method, or process. Typically, the textile is formed as a roll good. However, textile sheets and other structures are contemplated. In one example, the textile is a carpet including a plurality of tufted yarns. In another example, the textile is a fabric for a garment, upholstery, linens, or other application. In still other examples, the textile is a rug, carpet tiles, or other woven structure. Numerous other textile applications are contemplated. 
     To convert the roll good into a carpet installation, garment, or other product, pieces of the textile are cut, aligned, and/or joined as needed. In a typical carpet installation, pieces of the carpet are abutted along respective edges to fill the desired space, for example, a hallway or room. The carpet pattern is aligned along seams to create a visually pleasing, seemingly continuous piece of carpet. However, alignment of the pattern along the seams often results in a significant amount of waste and/or unsatisfactory installation. The present invention addresses this problem by designing the module and, therefore, the motif, such that when a first piece of carpet is installed, an adjacent piece need only be adjusted slightly to align the pattern across the seam. In doing so, a variety of overall carpet patterns may be created, each of which is visually pleasing. 
     Likewise, to form a garment, upholstery, or other fabric-based product, the various pieces are cut as needed, abutted and/or overlapped as needed, and optionally joined to form seams. If desired the visual appearance of the seams may be accentuated or minimized by aligning the pattern of the textile across the seams. In doing so, the patterning technique of the invention facilitates alignment of the textile pattern and, therefore, minimizes waste. 
     Other aspects, features, and advantages of the present invention will become apparent from the following description and accompanying figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The description refers to the accompanying schematic drawings, in which like reference characters refer to like parts throughout the several views, and in which: 
         FIG. 1A  depicts a generic “R” module used to illustrate various aspects of the invention; 
         FIG. 1B  depicts various orientations of the module of  FIG. 1A ; 
         FIG. 1C  depicts a tiled arrangement of module orientations; 
         FIG. 1D  depicts an exemplary arrangement of “R” modules having various orientations corresponding to the arrangement of module orientations of  FIG. 1C ; 
         FIG. 2A  depicts an exemplary module that may be used to form a textile pattern; 
         FIG. 2B  depicts various orientations of the module of  FIG. 2A ; 
         FIGS. 2C and 2D  schematically depict the orientations of  FIG. 2B  being brought together to form a design; 
         FIG. 2E  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a textile repeat unit or motif; 
         FIG. 2F  depicts an exemplary textile repeat unit or motif formed by replacing the “R” modules of  FIG. 2E  with the module of  FIG. 2A  in same orientation as the “R” modules; 
         FIGS. 2G-2N  illustrate a method of aligning textile pieces formed using the motif of  FIG. 2F ; 
         FIGS. 2O and 2P  illustrate the alignment of two textile pieces; 
         FIGS. 2Q-2Z  illustrate the replacement of a textile piece; 
         FIG. 3A  depicts another exemplary module that may be used to form a textile pattern; 
         FIG. 3B  depicts various orientations of the module of  FIG. 3A ; 
         FIG. 3C  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 3D  depicts an exemplary motif formed by replacing each “R” module of  FIG. 3C  with the module of  FIG. 3A  oriented in the same manner as the respective “R” module; 
         FIGS. 3E-3H  illustrate the alignment of textile pieces formed using the motif of  FIG. 3D ; 
         FIG. 4A  depicts yet another exemplary module that may be used to form a textile pattern; 
         FIG. 4B  depicts various orientations of the module of  FIG. 4A ; 
         FIG. 4C  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 4D  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 4C  with the modules of  FIG. 4A  oriented in the same manner as the “R” modules; 
         FIGS. 4E and 4F  illustrate the alignment of textile pieces formed using the motif of  FIG. 4D ; 
         FIG. 5A  depicts still another exemplary module that may be used to form a textile pattern; 
         FIG. 5B  depicts various orientations of the module of  FIG. 5A ; 
         FIG. 5C  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 5D  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 5C  with the modules of  FIG. 5A  oriented in the same manner as the “R” modules; 
         FIGS. 5E and 5F  illustrate the alignment of textile pieces formed using the motif of  FIG. 5D ; 
         FIG. 6A  depicts another exemplary module that may be used to form a textile pattern; 
         FIG. 6B  depicts various orientations of the module of  FIG. 6A ; 
         FIG. 6C  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 6D  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 6C  with the modules of  FIG. 6A  oriented in the same manner as the “R” modules; 
         FIGS. 6E and 6F  illustrate the alignment of textile pieces formed using the motif of  FIG. 6D ; 
         FIG. 7A  depicts still another exemplary module that may be used to form a textile pattern; 
         FIG. 7B  depicts various orientations of the module of  FIG. 7A ; 
         FIG. 7C  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 7D  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 7C  with the modules of  FIG. 7A  oriented in the same manner as the “R” modules; 
         FIGS. 7E and 7F  illustrate the alignment of textile pieces formed using the motif of  FIG. 7D ; 
         FIG. 8A  depicts still another exemplary module that may be used to form a textile pattern; 
         FIG. 8B  depicts various orientations of the module of  FIG. 8A ; 
         FIG. 8C  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; and 
         FIG. 8D  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 5C  with the module of  FIG. 8A  oriented in the same manner as the “R” modules; 
         FIG. 5E  depicts an exemplary alignment of textile pieces formed from the motif of  FIG. 8D ; 
         FIG. 9A  depicts another exemplary module that may be used to form a textile pattern; 
         FIG. 9B  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 9C  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 9B  with the module of  FIG. 9A  oriented in the same manner as the “R” modules; 
         FIG. 10A  depicts another exemplary module that may be used to form a textile pattern; 
         FIG. 10B  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 10C  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 10B  with the module of  FIG. 10A  oriented in the same manner as the “R” modules; 
         FIG. 11A  depicts another exemplary module that may be used to form a textile pattern; 
         FIG. 11B  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 11C  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 11B  with the module of  FIG. 11A  oriented in the same mariner as the “R” modules; 
         FIG. 12A  depicts another exemplary module that may be used to form a textile pattern; 
         FIG. 12B  depicts an exemplary arrangement of variously oriented “R” modules that may be used to form a motif; 
         FIG. 12C  depicts an exemplary motif formed by replacing the “R” modules of  FIG. 12B  with the module of  FIG. 12A  oriented in the same manner as the “R” modules; and 
         FIG. 12D  depicts various orientations of the module of  FIG. 12A . 
     
    
    
     DESCRIPTION 
     This disclosure is directed generally to various textile patterns and a patterning technique for forming such patterns. The patterning technique and resulting textile patterns facilitate alignment of textile pieces along and across seams and reduce the amount of waste typically associated with such alignment. 
     The various patterns comprise a design or motif repeated, for example, along the length and width of a textile. Each occurrence of the motif in the overall textile pattern includes a plurality of design elements that serve as “connection points” that are capable of being abutted with other design elements along a seam to define one or more new elements or designs. Although the new elements may disrupt the pattern of repeating motifs, the connection points in each piece allow the overall pattern across adjacent pieces to be perceived as continuous. 
     Each motif can be divided into a plurality of design “modules” arranged in a tiled (i.e. block repeat) configuration or array. Each module is substantially square in shape and includes at least two visually distinct hues. Typically, the more prominent feature within the module defines a foreground pattern of the textile, with the remainder of the textile pattern comprising a field. However, the converse is contemplated. The particular arrangement of hues in the module forms a portion or segment of the motif. 
     Each module within a motif is substantially identical to each other module within the motif, but the modules are variously, and sometimes randomly, oriented with respect to one another in fixed positions within the motif. The orientation of each module may be upright, inverted, and/or rotated 90°, 180°, or 270° with respect to one another. Each arrangement of variously oriented modules forms a unique motif. Thus, for a given module design, countless motifs may be formed. The motif may be symmetrical or asymmetrical, depending on the particular design of the module and the arrangement of modules. 
     A module suitable for use with the patterning technique may have one or more of various features that facilitate the formation of an optical or visual connection with the edge of an adjacent, variously oriented module, such that the design elements of one module are contiguous with or complementary to the design elements on an adjacent module. The connection may comprise a “match” of adjacent design elements or a visually pleasing “mismatch.” As a result, when two or more variously oriented modules are arranged edge-to-edge, each module forms a portion of an overall, harmonious design. 
     More particularly, each module includes at least one edge having an arrangement of design elements and/or hues that corresponds to or coordinates with the design elements and/or hues along at least one other edge. The lateral alignment of such edges may result in a “perfect” match of design elements and/or hues, such that the design elements and/or hues appear to extend continuously or “flow” from one module to the other, or an “imperfect” match, in which the design elements and/or hues do not flow continuously from one module to the other. A perfect edge alignment generally results in the linkage of design elements across the respective modules, while an imperfect edge alignment may create the appearance of a break or discontinuity in the flow of a particular design element. Nonetheless, the lateral alignment of any module edge with any other module edge is visually pleasing. Accordingly, any module having any orientation may be placed adjacent to any other module having any orientation and still make a visually pleasing connection. Depending on the particular characteristics of the module, the resulting motif and overall textile pattern may be relatively “open” (i.e., may have a greater % area that comprises the field) or may be relatively “closed” or interconnected (i.e., may have a greater % area that comprises foreground elements). 
     In some examples, one or more edges may be characterized as having bidirectional symmetry, such that the arrangement of design elements and/or hues along the respective edge is symmetric about a midpoint of the respective edge. Despite such edge symmetry, a lateral alignment of modules may result in an imperfect match or a perfect match, depending on the hues associated with each edge. In one particular example, each of the edges has a substantially identical bidirectional alignment of hues, such that any edge readily forms a perfect visual connection with any other edge. 
     Although some examples may feature such edge symmetry, it will be understood that the overall symmetry of the module may vary. Generally, each module may be asymmetric across at least one centerline that bisects the module. The module may have an overall degree of symmetry (“symmetry degree”) of 0, such that the module is asymmetric across any bisecting centerline,  1 , such that the module is symmetric across one bisecting centerline, or  2 , such that the module is symmetric across two bisecting centerlines. It will be understood that where the module includes at least one line of symmetry, the module also will have at least two edges with a substantially identical arrangement of hues. However, the module may have one or more edges with bidirectional symmetry without being symmetrical across any bisecting centerline. The degree of symmetry determines the number of distinct orientations of the module and contributes to the appearance of the overall design created by the variously oriented modules in the motif, as will be illustrated with reference to the examples. 
     Since each motif comprises an array of optically connecting modules, adjacent motifs also are capable of optically connecting to one another to form a visually continuous design. As a result, regardless of the orientation of each particular module, any module can be placed next to any other module without disrupting the overall pattern of the textile. The number available connection points between adjacent motifs depends on the number of modules in the motif. For example, a motif that comprises 16 modules across the array (i.e., in a row) and 16 modules down the array (i.e., in a column) has 16 connection points in each row and 16 connection points in each column. Accordingly, the pattern on adjacent textile pieces may be aligned across a seam by making only a minor adjustment of one piece relative to another to bring a module of the first piece into alignment with a module on the second piece. Thus, adjacent textile pieces may be aligned readily with little waste. 
     In some instances, the design elements on one module of a motif may align with adjacent design elements to form all or a portion of a new design element. The new design element may have a closed shape (i.e. no open ends) or open shape (i.e. one or more open ends capable of further connections), and generally differs in appearance from a mere side-by-side tiling of similarly oriented modules. Although the new element may not be present in the motif and/or may not conform to the arrangement of elements within the motif, the presence of the new element is difficult to discern. Thus, two or more textile pieces can be arranged in numerous ways and still form a seemingly continuous overall design. 
     Various aspects of the patterning technique may be understood with reference to the figures. For purposes of simplicity, like numerals may be used in the figures to describe like features. It will be understood that where a plurality of similar features are depicted, not all of such features are necessarily labeled on each figure. 
     It is noted that each module is described as being substantially “square” in shape with a plurality of peripheral edges or boundaries including a first or “top” edge, a second or “bottom” edge, a third or “left” edge, and a fourth or “right” edge. However, it will be understood that, the arrangement of hues may not include a square shaped border or defined edges. Rather, the modules are defined in this manner to provide a convenient means of describing the arrangement of the hues within the confines of the module and to assist with understanding the patterning technique and patterns of the invention. 
     Likewise, each module is characterized as having a plurality of centerlines, including a longitudinal centerline, a transverse centerline, a first diagonal centerline, and a second diagonal centerline, each of which bisects the module, only some of which may be labeled on the figures. The longitudinal centerline and transverse centerline extend between respective pairs of opposed edges of the module, while the first and second diagonal centerlines extend between respective pairs of opposed corners of the module. It will be understood that these positional and directional characterizations are made for discussion purposes only, and are not intended to be limiting in any manner. 
       FIG. 1A  depicts a generic “R” module  100  that can be used to illustrate various aspects of the patterning technique. The module  100  can be characterized as having a substantially square shape defined by a plurality of peripheral boundaries or edges  102 ,  104 ,  106 ,  108 , each of which is shown as a black dashed line. With the module in this initial, upright orientation, opposed edges  106 ,  108  extend substantially in a first direction D 1  (also referred to throughout as a longitudinal direction) and are substantially parallel to one another. Opposed edges  102 ,  104  extend substantially in a second direction D 2  (also referred to throughout as a transverse direction) and are substantially parallel to one another. The first direction D 1  and the second direction D 2  are substantially perpendicular to one another. 
     The module  100  includes a foreground design element  110  and a field  112 . In this example, the design element  110  is shown as the letter “R” in black and the field  112  is shown as solid white. However, it will be understood that various other hues and combinations of hues may be used. Thus, the design element may be lighter or darker than the field, and the field may be lighter or darker than the design element. It also is contemplated that with some modules, it may be difficult to discern which elements comprise the foreground and the field. The precise characterization of each design element is not critical to the invention, as will be evident from the examples. 
     The module  100  can be reoriented in numerous ways, as illustrated in  FIG. 1B . Orientation  1  is the module in its original orientation with the “R” in an upright configuration. Orientation  2  is the module rotated 90 degrees to the right relative to orientation  1 . Orientation  3  is the module rotated an additional 90 degrees relative to orientation  2 . Orientation  4  is the module rotated another 90 degrees relative to orientation  3 . Orientation  5  is the module in its original orientation inverted or flipped downward. Orientation  6  is the module of Orientation  5  rotated 90 degrees to the right relative to orientation  5 . Orientation  7  is the module rotated an additional 90 degrees relative to orientation  6 . Orientation  8  is the module further rotated an additional 90 degrees relative to orientation  7 . 
     Since the “R” module is wholly asymmetric (symmetry degree 0), each orientation has a different appearance than each other orientation. However, as will be seen with reference to the remaining examples, a module having a degree of symmetry other than 0 typically results in some orientations that have the same appearance as some other orientations. Stated differently, a module having a symmetry degree of 0 typically has in eight distinct orientations, while a module with at symmetry degree of at least 1, 2, or 3 typically has in fewer than eight distinct orientations. In particular, a module having a symmetry degree of 1 (i.e., symmetric across one centerline) typically has four distinct orientations, and a module having a symmetry degree of 2 (i.e., symmetric across two centerlines) typically has two distinct orientations. A module having a symmetry degree of 3 (i.e., symmetric across three centerlines) is wholly symmetric and typically has only one distinct orientation. 
     Returning to the figures, a tiled arrangement or array  114  of variously oriented modules  100  may be prepared. The selection of each orientation may be made manually or by using a computer or other device, and may be purposeful or random. Each position in the array corresponds to a position identifier, as shown in  FIG. 1C , which can be replaced with the corresponding module, as shown in  FIG. 1D . In the example shown in  FIGS. 1C and 1D , the array includes 4 rows and 4 columns and therefore may be referred to as a “4 by 4 array” (or “4×4 array”) of modules. In this example, the arrangement includes a total of 16 modules. However, other arrangements are contemplated by the invention. 
     The generic “R” module can be replaced with various modules to form numerous textile motifs and textile patterns, some of which are presented in the following examples. It will be understood that countless other motifs and patterns may be formed according to the patterning technique, and that such patterns are contemplated by the invention. 
     Example 1 
       FIG. 2A  depicts an exemplary module  200  that may be used in accordance with the patterning technique. For purposes of simplicity and not limitation, the module  200  is depicted as having a substantially square shape defined by a plurality of theoretical peripheral boundaries or edges  202 ,  204 ,  206 ,  208 , each of which is shown in dashed form. However, the module  200  does not include a defined border, as will be evident from the remaining figures. A first arc  210  extends between edge  202  and edge  208 , with the endpoints of the arc  210  being substantially centered along the length of edges  202 ,  208 . Likewise, a second arc  212  extends between edges  204 ,  206 , with the endpoints of the arc  212  being substantially centered along the length of edges  204 ,  206 , respectively. The module also includes a field  214 , shown in black. The module  200  is symmetrical along a first diagonal centerline CD 1  and asymmetrical across the remaining centerlines CT, CL, and CD 2 , such that the module  200  has an overall degree of symmetry of 1. 
     It will be appreciated that the various components that comprise a particular module may be described and/or represented in numerous ways. For example, in this illustration, the first arc  210  is shown in white. The second arc  212  could be described as being black with a white border or, alternatively, could be described as a pair of white arcs, each having endpoints along the respective edges, with the black interior space being part of the field. For ease of discussion, and not limitation, the second arc  212  is characterized herein as a single arc having white edges and a black interior space. It also will be appreciated that arcs  210 ,  212  and the field  214  may vary in color. Any combination of colors may be used as desired, with the field color being lighter or darker than the foreground pattern. 
     As shown in  FIG. 2A , each edge  202 ,  204 ,  206 ,  208  can be divided into various segments a, b, c, c′, with respective segments a having a substantially equal length, respective segments b having a substantially equal length, and respective segments c, c′ having a substantially equal length, such that the arrangement of segment lengths is symmetrical along each edge  202 ,  204 ,  206 ,  208 . 
     Each segment may be associated with a particular portion of a design element and/or a particular hue. In this example, the arrangement of design elements and/or hues is substantially identical and symmetrical along edges  202 ,  208 , such that edges  202 ,  208  readily form a perfect alignment with one another. Likewise, the arrangement of design element and/or hues is substantially identical and symmetrical along edges  204 ,  206 , such that edges  204 ,  206  readily form a perfect alignment with one another. In contrast, other edge combinations (i.e.,  202  or  208  with  204  or  206 ) result in the imperfect alignment of segments c, c′. Nonetheless, there is a seemingly continuous flow of elements from one module to the next, as will be apparent from the remaining figures. 
     Turning to  FIG. 2B , the module  200  can be reoriented in a manner similar to that described in connection with  FIG. 1B . For clarity and ease of illustration, the corresponding “R” orientation is provided above each oriented module. By examining the various oriented modules, it will be evident that for some module designs, some rotations and/or inversions will result in modules having the same appearance. In this example, orientation  1  appears the same as orientation  8 , orientation  2  appears the same as orientation  5 , orientation  3  appears the same as orientation  6 , and orientation  4  appears the same as orientation  7 . Thus, there are four distinct orientations. 
     As stated above, each module may be positioned next to the same module having any orientation. By way of example, as shown in  FIGS. 2C and 2D , the various module orientations  1 - 8  may be brought together such that adjacent modules are contiguous with one another (e.g., abutting or touching) to define a pattern of sinuous shapes against a black field. As shown in  FIG. 2D , each module forms a perfect or imperfect optical connection with the edge of the adjacent module(s). Further, the design elements on each of the variously oriented adjacent modules respectively and collectively define a plurality of new elements, for example, circles. 
     Each arrangement of modules having various orientations will define a unique arrangement of design elements or shapes. By way of example,  FIG. 2E  illustrates a 16×16 array  216  of variously oriented modules  200 , again illustrated using the letter “R” and the numeric position indicator for simplicity. The array includes a total of 256 modules having various orientations. 
     The generic “R” modules then may be replaced with the corresponding orientations of module  200  to form a textile motif or repeat unit  218 , as shown in  FIG. 2F . As shown in  FIG. 2F , the motif comprises the design module repeated to form a unitary array of design modules, with the design modules of the unitary array of design modules having various orientations with respect to one another in fixed positions within the unitary array to define the motif  218 . The unitary array of design modules (i.e., the motif  218 ) includes at least a first design module having a first orientation, and a second design module having a second orientation, where the design module having the first orientation is visually distinct from the design module having the second orientation, as illustrated schematically in  FIG. 2F . 
     In this example, the motif  218  includes a plurality of arcs with aligned endpoints that form various new design elements, including circles  220 , double circles  222 , triple circles  224 , and numerous other sinuous shapes, for example, shape  226 , each of which is set against a field  228 . Each design element is depicted as being completely white, completely black (with a white border), or some combination of both white and black (with a white border). However, other color configurations are contemplated. Notably, there are no “incomplete” or “open” shapes (i.e. ones with available endpoints), except along the periphery of the motif  218 . 
     The motif  218  may then be repeated along a length and a width of a textile web (e.g., a continuous textile web), such that the textile web comprises at least a first occurrence of the motif and a second occurrence of the motif. Each occurrence or repeat of the motif along the length and/or width of the textile web comprises the unitary array of design modules of the motif  218 , as shown schematically in  FIG. 2F . 
       FIGS. 2G-2Z  illustrate how the patterning technique of the invention facilitates alignment of two or more textile pieces. In this and other examples, it is noted that some of the textile pieces depicted in the figures also may correspond to a single repeat unit or motif. However, it will be understood that the repeat unit is continuously repeated over the length and width of the textile web, and that the alignment of textile pieces illustrated herein may be achieved using any textile piece patterned according to the invention, regardless of where the particular pieces are taken from the textile web. 
     Viewing  FIG. 2G , a first textile piece  230 , for example, a first piece of fabric or carpet, includes a plurality of peripheral edges including edge  232 , which extends generally in a first direction D 1 . The endpoints of one or more arcs, for example, endpoints  234   a ,  234   b ,  234   c , that form the various elements of the motif, abut at least one of the edges, for example, edge  232 . Likewise, a second textile piece  236 , for example, a second piece of fabric or carpet, includes a plurality of peripheral edges including edge  238 , which extends generally in the first direction D 1 . The endpoints of one or more arcs, for example, arcs  240   a ,  240   b ,  240   c , abut at least one of the edges, for example, edge  238 . 
     The basic lateral alignment of the textile pieces  230 ,  236  is illustrated in  FIGS. 2G and 2H . As the pieces  230 ,  236  are brought towards one another in a direction D 2 , it is evident that arc ends  234   a ,  234   b ,  234   c  on the first piece  230  will align readily with arc ends  240   a ,  240   b ,  240   c  on the second piece  236 . In doing so, a seemingly random pattern of shapes may be formed, some of which are adjoined across a seam S (shown sometimes herein as a dashed line extending beyond the dimensions of the textile pieces when needed for clarity), as shown in  FIG. 2H . Although the alignment of textile pieces formed a plurality of new shapes or design elements across the seam, the patterning technique of the invention creates an overall impression of continuity. As a result, it is difficult to discern the boundary both between adjacent motifs and textile pieces. 
     In many cases, however, this simple side-by-side alignment of similar textile pieces is not practicable. For example, where the shape of an item dictates different size pieces (for example, the shape of a garment or room), the pieces must be cut and therefore aligned differently. Additionally, where pieces are cut from a roll of the textile, the pieces rarely are cut to include exactly one repeat unit of the pattern. Further, where a portion of the textile piece is replaced due to damage or wear, which often occurs with carpet, it is unlikely that the replacement piece will consist precisely of a textile repeat unit. 
       FIGS. 2I-2N  illustrate how the patterning technique of the invention facilitates alternate alignments between pieces. Starting with the alignment illustrated in  FIG. 2H , the second piece  236  may be moved in the first direction D 1  relative to the first piece  230 , as shown in  FIG. 2I . Initially, the arcs are misaligned along the seam S. However, because of the edge symmetry of each module, further movement of the second piece  236  quickly results in alignment of the arcs to form a seemingly random pattern of interconnected arcs, as shown in  FIG. 2J . It is noted that this alignment results in a different pattern being formed along the seam S than that illustrated in  FIG. 2H . Still, the visually ambiguous nature of the overall design formed by the abutted pieces  230 ,  236  is aesthetically pleasing. 
       FIGS. 2K-2N  illustrate further movement of the second piece  236  in the second direction D 2  with alignment occurring, for example, as shown in  FIGS. 2J and 2L . Each alignment results in a different overall design, each being visually pleasing. It will be understood that numerous other alignments are contemplated. In this example, the pieces will align at least once per module, or in this case, at least 16 times per motif. 
     Likewise, as shown in  FIGS. 2M-2N , the patterning technique of the invention facilitates the alignment of the textile pieces  230 ,  236  when the second piece  236  is moved in the second direction D 2 . Beginning with the alignment illustrated in  FIG. 2L  and viewing  FIG. 2M , the available arcs on the second piece  236  initially are not in overlapping alignment with the pattern of arcs on the first piece  230 . However, alignment is achieved readily by moving the second piece  236  further in the second direction D 2 , as depicted in  FIG. 2N . 
     As another example,  FIGS. 2O and 2P  illustrate the alignment of the design elements on two irregularly shaped pieces  242 ,  244 , each including a portion of at least one textile repeat unit. 
     Thus, numerous possible alignments between two or more textile pieces may be made by merely adjusting one or more of the pieces until the theoretical boundaries of adjacent modules are brought into alignment. As a result, the amount of waste generated in aligning the design on adjacent pieces is minimized. For example, where the module is about 4 inches by 4 inches, the textile pieces need only be adjusted up to about 4 inches to bring the respective designs into alignment. In sharp contrast, typical patterns often require a significantly larger portion to be removed, and often wasted, to align the textile design along a seam. 
     The patterning technique also facilitates replacement of a worn or damaged portion of a textile. For example, carpets and upholstery often are soiled and are not capable of being cleaned. Typically, replacement of a section or piece of the carpet or fabric requires the use of excess carpet or fabric to achieve proper alignment of the pattern with the existing installation. However, the patterning technique affords greater flexibility in alignment and better potential for use of smaller pieces or scrap materials, as will be discussed in connection with  FIGS. 2Q-2Z . 
     Turning to  FIGS. 2Q and 2R , a portion (not shown) of a textile has been removed from an installation  246 , thereby forming an opening or void  248  for receiving a replacement piece. A scrap  250  ( FIG. 2R ) of the textile is available for use in replacing the damaged section of the installed textile. The scrap may be larger than the opening, as shown in  FIG. 2R , or may be smaller if desired. In the latter case, multiple scraps likely will be needed to complete the installation. 
     With the various patterns formed according to the patterning technique, numerous replacement pieces may be cut from the scrap piece. As mentioned above, alignment of the design occurs at each module. Thus, it would be prudent to measure the module and size the replacement piece to be slightly larger than the actual size of the void to be filled, for example, at least one module length and width larger than the size of the void. 
       FIGS. 2S and 2T  illustrate exemplary replacement pieces  252 ,  254  (defined by dashed lines) that may be removed from the textile scrap  250 . The replacement piece  252 ,  254  may be installed in numerous ways, examples of which are shown respectively in  FIGS. 2U-W  and  FIGS. 2X-2Z , in which the edges of the respective replacement piece  252 ,  254  are shown in dashed lines. In either example, the replacement piece may be adjusted as necessary to achieve the best fit and alignment with the existing design. Any excess textile may be removed from the replacement piece before permanently installing it in the void. 
     Example 3 
       FIG. 3A  illustrates another exemplary module  300  that may be used to form a textile pattern. The module  300  is depicted as having a substantially square shape defined by theoretical edges  302 ,  304 ,  306 ,  308 , each of which is shown as a dashed line. However, the module  300  does not include a defined border, as will be evident from the remaining figures. 
     The module  300  includes a somewhat L-shaped element  310  extending between edges  302 ,  308 , with the endpoints of the element  310  being substantially centered along the length of respective edges  302 ,  308 . An innermost edge  312  of the element  310  lies substantially along a first diagonal centerline CD 1  of the module  300 . The module  300  also includes a somewhat trapezoidal element  314  that extends between edges  304 ,  306 . The endpoints of the trapezoidal element  314  are substantially centered along the length of respective edges  304 ,  306 . An innermost edge  316  of element  314  is substantially parallel to the innermost edge  312  of element  310 . The remainder of the module  300  comprises a field  318 , shown in white. The module  300  is substantially symmetrical across a second diagonal centerline CD 2  and asymmetrical across the various other centerlines, such that the module  300  has a degree of symmetry of 1. 
     Each of edges  302 ,  304 ,  306 ,  308  can be divided into segments a, b having a substantially equal length, as illustrated with respect to edge  308 . It will be evident from  FIG. 3A  that the center segment b of each edge  302 ,  304 ,  306 ,  308  has a first hue, in this example, black, defined by elements  310 ,  314 , while the end segments a have a second hue, in this example, white, defined by the field  318 . Each edge  302 ,  304 ,  306 ,  308  can be characterized as having bidirectional symmetry, such that any edge will form a perfect alignment with any other edge. 
     Various orientations of the module  300  are illustrated in  FIG. 3B , in which the module again is illustrated with a theoretical peripheral boundary (shown with dashed lines). The oriented modules may be prepared in the manner described in connection with  FIG. 1B . For clarity and ease of illustration, the corresponding “R” module for each orientation is provided above each orientation. In this example, orientation  1  appears the same as orientation  8 , orientation  2  appears the same as orientation  5 , orientation  3  appears the same as orientation  6 , and orientation  4  appears the same as orientation  7 . Thus, there are four distinct orientations. 
       FIG. 3C  illustrates an exemplary 16×16 array  320  of “R” modules. With the “R” module replaced by module  300 , the resulting motif  322  resembles a maze, as shown in  FIG. 3D . The design comprises a plurality of closed and open shapes, for example, substantially hexagonal closed shape  324 , substantially octagonal closed shape  326 , and irregular closed shape  328 . Each of the open shapes, for example, shape  330 , includes at least one available endpoint, for example, endpoint  332 , along a peripheral edge  334  of the motif  322 . 
       FIGS. 3E-3H  illustrate how the patterning technique facilitates alignment of two or more textile pieces  336 ,  338 . As shown in  FIGS. 3E and 3F , endpoints  340   a ,  340   b  of shape  342  on textile piece  336  readily align with endpoints  348 ,  350  of respective shapes  352 ,  354  on textile piece  338  to form a seemingly random pattern of shapes, some of which are formed across a seam S. 
     Numerous other alignments may be made in both directions D 1  and D 2 . The pieces will align at least once per module in each direction, or in this case, at least 16 times per motif in each direction. For example,  FIG. 3G  depicts an example of lateral and vertical misalignment. As textile piece  338  is, moved in the direction of the arrows, the various shapes in both textile pieces  336 ,  338  readily align, as shown in  FIG. 3H , to form a seemingly random pattern of interconnected shapes. Although this alignment results in a different design being formed along and across seams than that illustrated in  FIG. 3F , the overall pattern formed by the abutted pieces  336 ,  338  appears to be consistent with the remainder of the textile pattern. 
     Example 4 
       FIG. 4A  illustrates yet another exemplary module  400  that may be used to form a textile pattern. As with the various other examples, the module  400  is depicted as having a substantially square shape defined by theoretical edges  402 ,  404 ,  406 ,  408 , each of which is shown as a dashed line. 
     The module  400  includes a plurality of elements  410 ,  412 ,  414  (shown in black) arranged between a plurality of substantially square corner elements  416 ,  418 ,  420 ,  422  (shown in white). Elements  410 ,  412  are spaced apart by a bar  424  (shown in white) extending in a first direction D 1  between corner elements  416 ,  420 , and elements  410 ,  414  are spaced apart by a bar  426  extending in a second direction D 2  between corner elements  420 ,  422 . 
     Element  410  abuts and/or at least partially defines edges  402 ,  408  and generally resembles a square having a notched corner defined by corner element  418 . Element  412  is substantially rectangular in shape and abuts and/or at least partially defines edge  406 . Element  414  also is substantially rectangular in shape and abuts and/or at least partially defines edge  404 . The module is substantially symmetrical along a diagonal centerline CD and asymmetrical across the remaining centerlines (not labeled), such that the module  400  has an overall degree of symmetry of 1. 
     Each edge  402 ,  404 ,  406 ,  408  can be divided into segments a, b, as illustrated with respect to edge  408 , with respective segments a having a substantially equal length and respective segments b having a substantially equal length, such that the arrangement of segment lengths is symmetrical along each edge  402 ,  404 ,  406 ,  408 . The center segment b of each edge  402 ,  404 ,  406 ,  408  has a first hue (black) defined by elements  410 ,  412 ,  414 , while the end segments a of each edge  402 ,  404 ,  406 ,  408  have a second hue (white) defined by the corner elements  416 ,  418 ,  420 ,  422 . Each edge  402 ,  404 ,  406 ,  408  can be characterized as having bidirectional symmetry, such that any edge will form a perfect alignment with any other edge. 
     Various orientations of the module  400  are illustrated in  FIG. 4B  with the corresponding “R” module for each orientation. Orientation  1  appears the same as orientation  8 , orientation  2  appears the same as orientation  5 , orientation  3  appears the same as orientation  6 , and orientation  4  appears the same as orientation  7 . Thus, there are four distinct orientations. 
       FIG. 4C  illustrates an exemplary 16×16 array  424  of “R” modules. The “R” module may be replaced by module  400  to form a motif  426  that resembles a plurality of black overlapping zigzags, for example, zigzags  428 ,  430 , separated by a plurality of white interconnected bars and squares, for example, bars  432 ,  434  and squares  436 ,  438 , as shown in  FIG. 4D . 
       FIGS. 4E and 4F  illustrate how the patterning technique facilitates alignment of textile pieces  440 ,  442 , with  FIG. 4E  illustrating a misalignment of the designs on the respective pieces  440 ,  442  and  FIG. 4F  depicting an alignment of the designs on the respective pieces  440 ,  442 . The overall design appears to be continuous, despite the presence of new elements created across the seams S. 
     Example 5 
       FIG. 5A  illustrates yet another example of a module  500  that may be used to form a textile pattern. The module  500  is depicted as having a substantially square shape defined by theoretical boundaries or edges  502 ,  504 ,  506 ,  508 , each of which is shown as a dashed line. In this configuration, opposed edges  506 ,  508  extend substantially in the first direction D 1  and are substantially parallel to one another, while opposed edges  502 ,  504  extend substantially in the second direction D 2  and are substantially parallel to one another. 
     The module  500  includes a plurality of spaced, substantially rectangular bars  510 ,  512 ,  514 , each of which is substantially equal in length and width. Bar  510  extends in the first direction D 1  substantially between theoretical edges  502 ,  504  and is substantially perpendicular to bars  512 ,  514 . Bars  512 ,  514  extend in the second direction D 2  substantially between theoretical edges  506 ,  508  and are substantially parallel to one another. Bar  514  intersects bars  510 ,  512  at a point P offset a distance D from a longitudinal centerline CL drawn through the module  500 . A pair of substantially square shaped voids  516 ,  518  respectively interrupt a portion of overlapping bars  510 ,  512  and  510 ,  514 . The module  500  is substantially symmetrical along a transverse centerline CT and asymmetrical across the remaining centerlines (not labeled), such that the module  500  has an overall degree of symmetry of 1. The remaining spaces and the voids  516 ,  518  define a field  520  of the module  500 , shown in white. 
     Each edge  502 ,  504 ,  506 ,  508  can be divided into segments a, b, b′ c, with respective segments a having a substantially equal length, respective segments b, b′ having a substantially equal length, and respective segments e having a substantially equal length, such that the arrangement of segment lengths is symmetric along each edge  502 ,  504 ,  506 ,  508 . Segments a, c, and b′ of edges  502 ,  504 ,  506 ,  508  each have a first hue (white) defined by the field  520 , and segments b have a second hue (black) defined by the endpoints of bars  510 ,  512 ,  514 . The respective arrangement of hues is symmetric and identical along edges  506 ,  508 , such that edges  506 ,  508  readily form a perfect alignment with one another. In contrast, the arrangement of hues is identical but asymmetric along edges  502 ,  504 . As a result, some alignments of edges  502 ,  504  will result in a perfect alignment, while others will instead define a plurality of shapes that terminate within the motif (best seen in  FIG. 5D ). 
     Various orientations of the module  500  are illustrated in  FIG. 5C  with the corresponding “R” module. In this example, orientation  1  appears the same as orientation  5 , orientation  2  appears the same as orientation  6 , orientation  3  appears the same as orientation  7 , and orientation  4  appears the same as orientation  9 . Thus, there are four distinct orientations. 
       FIG. 5D  illustrates an exemplary 8×8 array  522  of “R” modules. The “R” module may be replaced by module  500  to form a motif  524  including a plurality of interconnected bars that resemble a somewhat open lattice structure or trellis, as shown in  FIG. 5D . 
       FIGS. 5E and 5F  illustrate how the patterning technique facilitates alignment of textile pieces  526 ,  528 , with  FIG. 5E  illustrating a misalignment of the designs on the respective pieces  526 ,  528  and  FIG. 5F  depicting an alignment of the designs on the respective pieces  526 ,  528 . The overall design appears to be somewhat random, but continuous, despite the presence of new elements created across the seams S between the pieces  526 ,  528 . Numerous other alignments may be made with the textile pieces  526 ,  528 . In this example, the pieces will align at least once per module when moved in the first or second direction, in this case, at least 8 times per motif in each direction. 
     Example 6 
       FIG. 6A  illustrates yet another example of a module  600  that may be used to form a textile pattern. The module  500  is depicted as having a substantially square shape defined by theoretical peripheral edges  602 ,  604 ,  606 ,  608 , each of which is shown as a dashed line. The module  600  includes a first, somewhat L-shaped element  610  and a second element  612  that resembles a zigzag. The first and second elements  610 ,  612  are arranged in a somewhat nested configuration. The end points of element  610  are substantially centered along and/or at least partially define theoretical edges  602 ,  608 . Likewise, the endpoints of element  612  are substantially centered along and/or at least partially define edges  604 ,  606 . 
     In this example, each of the first element and the second element is shown as having more than one color, with the darker color (shown as black) being proximate the nestled edges of each and the lighter color (shown as gray) being distal the nestled edges of each. However, it is contemplated that the elements may have only one hue, may each have a different hue, or may each have multiple hues and combinations thereof. The remainder of the module  600  comprises a field  614 , shown in white. However, other hues and hue combinations may be used. The module  600  is substantially symmetrical along a diagonal centerline CD and asymmetrical along the various other centerlines. Thus, the module  600  has an overall degree of symmetry of 1. 
     As shown in  FIG. 6A , each edge  602 ,  604 ,  606 ,  608  can be divided into segments a, b, c, with respective segments a having a substantially equal length and respective segments IP having a substantially equal length, and respective segments c having a substantially equal length. Although each edge is divided into the same segments, segments b and c are in opposite positions on opposed edges of the module  600 . The endpoints of elements  610 ,  612  define combined respective center segments b+c, while the field  614  defines the respective end segments a. 
     Despite the use of multiple hues in elements  610 ,  612 , this pseudo-symmetrical arrangement of elements along each edge  602 ,  604 ,  606 ,  608  ensures that elements  610 ,  612  with align with each other to create a seemingly continuous design, while the field  614  will align with itself. However, the bi-tonal nature of elements  610 ,  612  results in some perfect alignments and some imperfect alignments of segments. For example, viewing the various orientations of the module  600  in  FIG. 6B  with the corresponding “R” module provided above each orientation, the lateral alignment of orientations  1  and  6 , for example, will result in an imperfect alignment of segments b and c, while the lateral alignment of orientations  1  and  7  will result in a perfect alignment of segments b and c. It is noted that, in this example, orientation  1  as appears the same as orientation  8 , orientation  2  as appears the same as orientation  5 , orientation  3  as appears the same as orientation  6 , and orientation  4  as appears the same as orientation  7 . Thus, there are four distinct orientations. 
       FIG. 6C  illustrates an exemplary 16×16 array  616  of “R” modules. The resulting motif  618  includes a plurality of interconnected elements that resemble a lattice structure or overlapping staircases, as shown in  FIG. 6D  (in which the dual tones are difficult to discern). 
       FIGS. 6E and 6F  illustrate how the patterning technique facilitates alignment of textile pieces  620 ,  622 , with  FIG. 6E  illustrating a misalignment of the designs on the respective pieces  620 ,  622  and  FIG. 6F  depicting an alignment of the designs on the respective pieces  620 ,  622 . As with the various other examples, the overall design appears to be somewhat random, yet continuous, despite the presence of new elements formed across the seams. 
     Example 7 
       FIG. 7A  illustrates still another module  700  having a substantially square shape defined by theoretical edges  702 ,  704 ,  706 ,  708 , each of which is shown as a dashed line. The module  700  includes a plurality of substantially V-shaped elements  710 ,  712 ,  714 ,  716 , each of which is shown in white, and a field  718 , shown in black. Elements  710 ,  712  each abut edge  702  and elements  714 ,  716  each abut edge  704 . Each of the elements  710 ,  712 ,  714 ,  716  is positioned within the module  700  with the narrowest part of the “V” proximate to a longitudinal centerline CL. The module  700  is substantially symmetrical along the longitudinal centerline CL and a transverse centerline CT, and is asymmetrical across each of the diagonal centerlines (not labeled), such that the module  700  has an overall degree of symmetry of 2. 
     Each edge  702 ,  704 ,  706 ,  708  can be divided into segments a, b, c, with respective segments a having a substantially equal length, respective segments b, having a substantially equal length, and respective segments c having a substantially equal length, such that the arrangement of segment lengths is symmetrical along each edge  702 ,  704 ,  706 ,  708 . Segments a, b, and c have a first hue (black) defined by the field  718 . In contrast, respective segments b′ have a white hue defined by elements  710 ,  712 ,  714 ,  716 . Each edge features bidirectional symmetry, with edges  702 ,  704  being identical to one another and edges  706 ,  708  being identical to one another. As such, edges  702 ,  704  form perfect alignments with one another and edges  706 ,  708  form perfect alignments with one another. Other alignments result in imperfect matching of segments b and b′ ( FIG. 7D ). 
     Various orientations of the module  700  are illustrated in  FIG. 7B  with the corresponding “R” module provided above each orientation. In this example, orientations  1 ,  3 ,  5 , and  7  appear the same, and orientations  2 ,  4 ,  6 , and  8  appear the same. Thus, there are two distinct orientations. 
       FIG. 7C  illustrates an exemplary 16×16 array  720  of “R” modules. With the “R” module replaced with module  700 , the resulting textile repeat unit  722  includes a plurality of square shaped arrangements of v&#39;s and other new elements that resemble w&#39;s, m&#39;s, and zigzags, as shown in  FIG. 7D . 
       FIGS. 7E and 7F  illustrate how the patterning technique facilitates alignment of textile pieces  724 ,  726 , with  FIG. 7E  illustrating a misalignment of the respective designs and  FIG. 7F  depicting an alignment of the respective designs. The overall design appears to be somewhat random, but consistent with the repeating motif, despite the presence of new elements created across the seams S. 
     Example 8 
       FIG. 8A  illustrates yet another example of a module  800  that may be used to form a textile pattern. The module  800  has a generally square shape defined by a plurality of theoretical peripheral edges  802 ,  804 ,  806 ,  808  shown in dashed form. 
     The module  800  includes a plurality of curvilinear elements (i.e., arcuate or curved lines) extending in a first or longitudinal direction D 1  substantially between edges  802 ,  804 , including outermost lines  810 ,  812  respectively closest to edges  806 ,  808 . Each of the longitudinal elements, including elements  810 ,  812 , converges slightly towards a center C of the module  800 , which also corresponds to a midpoint of the longitudinal centerline CL and a midpoint of the transverse centerline CT. 
     The module  800  also includes a plurality of curvilinear elements  814  (i.e. curves and lines) extending in a second or transverse direction D 2  substantially between edge  806  and longitudinal curved line  810 , and a plurality of curvilinear elements  816  (i.e. curves and lines) extending in the second or transverse direction D 2  substantially between edge  808  and longitudinal curved line  812 . The outermost elements  814 ,  816  respectively proximate to edges  802 ,  804  are substantially linear, while the innermost elements  814 ,  816  proximate to the transverse centerline CT are substantially curved. However, other arrangements are contemplated. 
     Each of the plurality of elements  814  is respectively aligned in the longitudinal direction  131  with a corresponding element of the plurality of elements  816 . The module is symmetrical along a transverse centerline CT and a longitudinal centerline CL and asymmetrical across each of the diagonal centerlines (not labeled), such that the module  800  has an overall degree of symmetry of 2. 
     Each edge  802 ,  804 ,  806 ,  808  can be divided into 13 segments with respective segments along each edge  802 ,  804 ,  806 ,  808  having a substantially equal length. The segments alternate a between a first hue (black) and second hue (white), such that the respective arrangement of segment lengths and hues is symmetrical and identical along each edge  802 ,  804 ,  806 ,  808 . Thus, each edge  802 ,  804 ,  806 ,  808  forms a perfect alignment with each other edge  802 ,  804 ,  806 ,  808 . 
     Various orientations of the module  800  are illustrated in  FIG. 8B  with the corresponding “R” module. In this example, orientations  1 ,  3 ,  5 , and  7  appear the same, and orientations  2 ,  4 ,  6 , and  8  appear the same. Thus, there are two distinct orientations. 
       FIG. 8C  illustrates an exemplary 8×8 array  820  of “R” modules that may be replaced by module  800  to form the motif  822  depicted in  FIG. 8D . The pattern generally resembles a basket weave. 
     It will be appreciated that the motif  822  may be used to form various textiles, and therefore textile pieces, that may be aligned readily by making minor adjustments to the positioning of one or multiple pieces relative to one another, as described in connection with the various other examples set forth herein. One example of an alignment of textile pieces  824 ,  826  is shown in  FIG. 8E . 
       FIGS. 9A and 9B  respectively depict yet another exemplary module  900  and exemplary array  902  of “R” modules that may be used to form a motif  904  ( FIG. 9C ). In this example, each edge of the module  900  exhibits bidirectional symmetry. The module has no bisecting lines of symmetry, and therefore has an overall degree of symmetry of 0. Thus, the module has eight distinct orientations (not shown). 
       FIGS. 10A and 10B  respectively depict another exemplary module  1000  and exemplary array  1002  of “R” modules that may be used to form a motif  1004  ( FIG. 10C ). In this example, each edge of the module  1000  exhibits bidirectional symmetry. The module has no bisecting lines of symmetry, and therefore has an overall degree of symmetry of 0. The module has eight distinct orientations (not shown). 
       FIGS. 11A and 11B  respectively depict still another exemplary module  1100  and exemplary array  1102  of “R” modules that may be used to form a motif  1104  ( FIG. 11C ). In this example, each edge of the module  1100  exhibits bidirectional symmetry. The module has no bisecting lines of symmetry, and therefore has an overall degree of symmetry of 0. Thus, the module has eight distinct orientations (not shown). 
       FIGS. 12A and 12B  respectively depict yet another exemplary module  1200  and exemplary array  1202  of “R” modules that may be used to form a motif  1204  ( FIG. 12C ). In this example, each edge of the module  1200  exhibits bidirectional symmetry. The module has an apparent line of symmetry CD, but the over-and-under pattern of the design elements removes the actual appearance of symmetry. However, there is a “hidden” symmetry, because there are only four distinct orientations, as shown in  FIG. 12D . 
     Although certain embodiments of this invention have been described with a certain degree of particularity, those skilled in the art could make numerous alterations to the disclosed embodiments without departing from the spirit or scope of this invention. All directional references (e.g., upper, lower, upward, downward, left, right, leftward, rightward, top, bottom, above, below, vertical, horizontal, clockwise, and counterclockwise) are used only for identification purposes to aid the reader&#39;s understanding of the various embodiments of the present invention, and do not create limitations, particularly as to the position, orientation, or use of the invention unless specifically set forth in the claims. Joinder references (e.g., joined, attached, coupled, connected, and the like) are to be construed broadly and may include intermediate members between a connection of elements and relative movement between elements. As such, joinder references do not necessarily imply that two elements are connected directly and in fixed relation to each other. 
     Accordingly, it will be readily understood by those persons skilled in the art that, in view of the above detailed description of the invention, the present invention is susceptible of broad utility and application. Many adaptations of the present invention other than those herein described, as well as many variations, modifications, and equivalent arrangements will be apparent from or reasonably suggested by the present invention and the above detailed description thereof, without departing from the substance or scope of the invention as set forth in the following claims. 
     While the present invention is described herein in detail in relation to specific aspects, it is to be understood that this detailed description is only illustrative and exemplary of the present invention and is made merely for purposes of providing a full and enabling disclosure of the present invention and to provide the best mode contemplated by the inventor or inventors of carrying out the invention. The detailed description set forth herein is not intended nor is to be construed to limit the present invention or otherwise to exclude any such other embodiments, adaptations, variations, modifications, and equivalent arrangements of the present invention.