Patent Publication Number: US-2020284020-A1

Title: Web or Support Structure and Method for Making the Same

Description:
FIELD AND BACKGROUND OF THE INVENTION 
     The present invention is directed to a web or support structure, and more particularly to a web or support structure that could be utilized to form structural elements. 
     Architects, civil and structural engineers conventionally utilize various web structures for supporting, for example, trusses, floors, columns, etc. Typically, web structures form various lattices or framework that support underlying or overlying supports. In this regard, structural engineers are quite familiar with a “Fink truss” ( FIG. 2 ), the geometry of which encodes an approximation of a “Sierpinski triangle”—the “limit” of the recursive design indicated in  FIGS. 1-3 . 
     It has been observed in the past that the geometry of the hardest substance known to man, namely diamonds, and the modern roof truss encode and represent the approximations to certain fractals. The Fink truss ( FIG. 2 ) is an engineering design that is a level-1 2-web. In nature, carbon-carbon bonding in diamond encodes a level-1 3-web. 
     In my earlier U.S. Pat. No. 6,931,812, which is hereby incorporated herein in its entirety by reference, I disclosed a 4-web structure represented in a 3-dimensional space that at a level-0 contains 10 triangles. In my another U.S. Pat. No. 8,826,602, which is hereby incorporated herein in its entirety by reference, I disclosed a 5-web structure represented in a 3-dimensional space that at level-0 packs or accommodates 15 Fink struts and 20 hyperbolic triangles. 
     The 2-web and 3-web date to circa 1900, while the 4-web from the 4 th  dimension was realized within human vision late in the 19 th  Century, and eventually published in the literature circa early 2003 (Reference No. 3). The 4-web is pictured on the cover of my book (Reference No. 2). Each of the 2-web, 3-web, 4-web, and 5-web are concrete examples of an abstract space that is referenced in the literature as “Lipscomb&#39;s Space” that I invented to solve a half-century old problem in dimension theory. 
     Page 20 of my book (Reference No. 2) contains mathematical details about the lower dimensional webs. In particular, the “w with superscript 5” notation in the book denotes the 5-web, and the “J with subscript  6 ” notation also denotes the 5-web, where the 6=5+1 indicates the number of vertices of the 5-web. In general the “ω with superscript n” denotes an n-web and the “J with subscript n+1” also denotes the n-web, where n+1 indicates the number of vertices of the n-web. 
     Simply put, it has been an open problem to create a picture of an approximation to a 5-web within 3-space (human visual space). In the present disclosure, I use an innovative geometry to show how to visualize within 3-space (human visual space) such approximations to the 5-web. Specifically, one of the new concepts/embodiments extends the earlier 4-web design to contain six (6) vertices and 20 triangles. Perhaps more importantly, however, is an observation in a test that a 4-web structure, embodying my 4-web concept disclosed in my patent, U.S. Pat. No. 6,931,812, yielded a strength of over 6-ton/sq. in., while weighing merely 0.30 oz. It is not difficult, therefore, to note that a 5-web structure with 10 extra triangles would be even stronger, if not, at least twice in strength. Such low weight and super strength would particularly be useful, at least in, for example, the currently used 4-web medical implants. 
     Recalling again the value of “triangles” when it comes to designing high-strength structures, let us also recall that the 3-web level-0 ( FIG. 4 ) has six struts and four triangles. The strength increases as the number of triangles increases. For example, I have shown in my unpublished article (Reference No. 1), that the addition of a single polar strut (compare  FIG. 4  to the top half of  FIG. 7 ), could increase compressive strength by as much as 20%. That is, the polar strut provides more triangles. 
     In order to understand the new 5-web designs (subject of this application), recall that the “4” in “4-web” refers to the “4 th -dimension”—the place where the 4-web originally existed. There are also “2-webs”, which exist in 2-dimensional planes, and “3-webs”, which exist in 3-dimensional space (human visual space). Mathematically, this list of webs and corresponding dimensions continues ad infinitum. Sample illustrations of the webs existing in lower-dimensional space are shown in  FIGS. 1-9 . 
     More specifically,  FIGS. 1-3  depict “levels” of 2-webs. Specifically,  FIGS. 1-3  show a “level-0” (a single triangle), a “level-1” (three level-0 2-webs, illustrated as red, green, and blue), and a “level-2” 2-web (containing three level-1 2-webs), respectively. As the level-numbers increase, these structures approach a “limit”, which is called the “2-web”. 
       FIGS. 4-6  depict “levels” of 3-webs. Specifically,  FIGS. 4-6  show a “level-0” (a single tetrahedron), a “level-1” (four level-0 3-webs, illustrated as red, green, blue, and gold), and a “level-2” 3-web (containing three level-1 3-webs), respectively. As the level-numbers increase, these structures approach a “limit”, which is called the “3-web”. 
       FIGS. 7-9  depict “levels” of 4-webs. Specifically,  FIGS. 7-9  show a “level-0” (a single hexahedron), a “level-1” (five level-0 4-webs, illustrated as red, green, blue, gold, and black), and a “level-2” 4-web (containing five level-1 4-webs), respectively. Again, as the “level numbers” increase these structures approach a “limit” that is called a “4-web”. 
     The key is to observe the inductive process, illustrated in  FIGS. 1-9 . The “inductive process” is a process that allows us to start at a given level, and then move to the next level using the given level. In more detail, the process is a two-step process. First, congruent copies of a given level are made. Second, these congruent copies are positioned so that each is just touching the others. To say that two congruent structures are “just touching” is to say that there exists one and only one point that is contained in both structures. 
     For example, consider the inductive process illustrated in  FIGS. 4-6 . We start with a tetrahedron ( FIG. 4 —four vertices), which is a level-0 3-web. Then, we create four congruent copies (colored red, green, blue, and gold). Next, we position these four copies so that each is just touching the other three. This positioning is shown in  FIG. 5 . To find the just-touching points, simply seek the points where two distinct colors meet. In particular, find the point where the red congruent copy meets the green congruent copy. That point is the “just touching point” for those copies. The construction of congruent copies followed by the “just touching” positioning allows one to move from one level to the next to infinity. Such an algorithm defines the inductive process. 
     In summary, the Fink truss ( FIG. 2 ), which is a level-1 Sierpinski triangle, has been utilized for many years in constructing various support structures. To date, diamond which has the geometry of a level-1 Sierpinski cheese as its basic building structure is known to be the hardest structure. In the present invention, I now disclose geometrical structures that represent the next step over the 4-web structure, i.e., the new 5-web structures, over those disclosed in my earlier patent, U.S. Pat. No. 8,826,602. 
     ASPECTS AND SUMMARY OF THE INVENTION 
     The present disclosure is directed to various aspects of the present invention. 
     One aspect of the present invention is to provide the medical, scientific, engineering, technical, and architectural communities with access to new fundamental designs, i.e., designs that systematically produce homogeneous structures that contain large numbers of triangles constructed with a minimum amount of material. That is, light-weight but exceptionally strong structures. 
     Another aspect of the present invention is to provide a web structure which could be utilized at both macroscopic and microscopic levels to create stronger and more stable structures. On a microscopic scale, for example, a web structure made in accordance with the present invention would produce new compounds and new crystals. Another example is to create structures, such as medical implant devices that enhance bone growth. On a macroscopic scale, for example, a web structure made in accordance with the present invention would create super strong and stable architectural and structural support structures. For example, a web structure of the present invention can be utilized to create super strong and stable trusses, beams, floors, columns, panels, airplane wings, etc. 
     Another aspect of the present invention is to provide the scientific and solid-state physics communities with access to new fundamental web-structure designs that would indicate how to build new compounds and new crystals having utility, for example, in the solid-state electronics industry. 
     Another aspect of the present invention is to provide a web structure that accommodates or packs more triangular shapes into a given volume than conventional web structures. A web structure made in accordance with the present invention could be used in building bridges, large buildings, space-stations, etc. In the space-station case, for example, a basic, modular and relatively small web structure can be made on earth, in accordance with the present invention, and a large space-station could be easily built in space by shipping the relatively small (level-0) web into space, and then joining it with other members according to the “just-touching” feature of web designs. 
     Another aspect of the present invention is to provide a web structure that represents a 5-web in a 3-dimensional space. 
     Another aspect of the present invention is to provide a 5-web structure that packs or accommodates more triangles in a given volume than the previous 4-web structure. 
     Another aspect of the present invention is to provide a web structure that at level-0 packs or accommodates 20 triangles. 
     Another aspect of the present invention is to provide a web structure including six points (or apices or vertices), wherein no two points are equal, no three points lie on a straight line, no four points lie on a plane, each pair of points is connected, by a generally straight segment, which, in pairs, meet in a single common vertex, and, in addition, the structure serves as a level-0 5-web, copies of which may be used to build a level-1 5-web, etc. 
     Another aspect of the present invention is to provide a web structure that includes a generally hexahedron-shaped frame, wherein the frame includes a plurality of points oriented in a manner that no more than three points lie in a common plane. Each pair of the points is connected by a frame segment. A plane includes three of the points, and one frame segment passes through the plane and includes first and second ends, which ends are generally equidistant from the plane. The frame comprises six points or vertices. 
     Another aspect of the present invention is to provide a web structure that includes a generally hexahedron-shaped frame with first and second generally trihedron-shaped portions joined at the bases thereof. The first and second portions include first and second vertices, respectively. The frame includes a plane. A frame segment joins the first and second vertices and passes through the plane. The frame includes third, fourth, fifth and six vertices, wherein the six vertex is situated at approximately 0.429, −0.166, 0.746 coordinates corresponding to x, y, z axes of the frame. 
     Another aspect of the present invention is to provide a web structure that includes a generally hexahedron-shaped frame with first, second, third, fourth, fifth, and sixth vertices generally situated at approximately 0, 0, 1; −0.5, −0.866, 0; 1, 0, 0; −0.5, 0.866, 0; 0, 0, −1.414; and 0.429, −0.166, 0.746 values for x, y, z coordinates, respectively. 
     Another aspect of the present invention is to provide a web structure that includes a frame with first and second triangles, which are spaced from each other by a predetermined distance. The first and second triangles are disposed in first and second generally parallel planes, and the vertices of one of the first and second triangles are offset from the corresponding vertices of the other of the first and second triangles by about 45 degrees. Each of the vertices of one of the first and second triangles is connected to each of the vertices of the other of the first and second triangles by a segment. The frame includes twenty triangles. 
     In summary, the main aspect of the present invention is to provide a 5-web structure in a 3-dimensional space. The invention can be utilized to generate new structural designs that relate to both macroscopic and microscopic structures. These structures would be stronger and more stable than the presently known structures, including diamond and those utilizing the 4-web structure shown in my earlier patent, U.S. Pat. No. 6,931,812. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
       One of the above and other aspects, novel features and advantages of the present invention will become apparent from the following detailed description of the non-limiting preferred embodiment(s) of invention, illustrated in the accompanying drawings, wherein: 
         FIG. 1  illustrates a Sierpinski&#39;s triangle or a level-0 2-web; 
         FIG. 2  illustrates a Fink truss or a level-1 2-web; 
         FIG. 3  illustrates a level-2 2-web; 
         FIG. 4  illustrates a level-0 3-web; 
         FIG. 5  illustrates a level-1 3-web; 
         FIG. 6  illustrates a level-2 3-web; 
         FIG. 7  illustrates a level-0 4-web; 
         FIG. 8  illustrates a level-1 4-web; 
         FIG. 9  illustrates a level-2 4-web; 
         FIG. 10  illustrates a level-0 4-web structure formed in accordance with my earlier invention, shown in U.S. Pat. No. 6,931,812; 
         FIG. 11  is a view of the web structure shown in  FIG. 10 , formed in accordance with a preferred embodiment of the present invention, shown with x, y, z coordinates of six vertices; 
         FIGS. 12-15  illustrate (in purple color) various sets of ten new triangles, in accordance with a preferred embodiment of the present invention; 
         FIG. 16  illustrates another embodiment of a web structure, in accordance with the present invention; and 
         FIGS. 17-20  illustrate (in color) a preferred sequence for the formation of the web structure shown in  FIG. 16 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT(S) OF THE INVENTION 
     As described above, a 3-web may be viewed as a systematic packing of tetrahedra in 3-dimensional space, and a 4-web may be viewed as a systematic packing of hexahedra in 3-dimensional space. As also noted above, the present invention is an extension of my 4-web design described hereafter. 
     As shown in  FIG. 10 , the web structure W includes a generally hexahedron-shaped frame F including an upper generally triangular or trihedron-shaped sub-frame  10  and a lower generally triangular or trihedron-shaped sub-frame  12 . The upper and lower sub-frames  10  and  12  are joined at their bases to form a common equatorial sub-frame  14 . 
     The frame F includes upper and lower points or apices  16  and  18 , respectively, and three equatorial points or apices  20 ,  22 , and  24 . The points  16 ,  18 ,  20 ,  22 , and  24  are oriented in a three-dimensional space in a manner that no more than three points lie in a same plane. The equatorial points  20 ,  22 , and  24  are disposed in a generally common, generally horizontal plane represented by equatorial sub-frame  14 . 
     As illustrated in  FIG. 10 , each pair of the points  16 ,  18 ,  20 ,  22 , and  24 , is connected by a line or frame segment. For instance, equatorial points  20  and  22  are connected by a frame segment  26 , the equatorial points  22  and  24  are connected by a frame segment  28 , and equatorial points  20  and  24  are connected by a frame segment  30 . Likewise, upper and lower points  16  and  18  are connected by a frame segment  32 . In the same manner, the points  16  and  20 ,  16  and  22 ,  16  and  24 ,  18  and  20 ,  18  and  22 , and  18  and  24 , are connected by frame segments  34 ,  36 ,  38 ,  40 ,  42 , and  44 , respectively. 
     The frame segment  32  is disposed preferably generally perpendicular to the plane of sub-frame  14  and passes generally through the geometrical center (GC) thereof. Alternatively, the frame segment  32  may be generally skew or slanted. 
     The frame F forms ten triangles represented by points  16 ,  20 , and  24 ;  16 ,  20 , and  22 ;  16 ,  22 , and  24 ;  20 ,  22 , and  24 ;  18 ,  20 , and  24 ;  18 ,  22 , and  24 ;  18 ,  20 , and  22 ;  16 ,  18 , and  20 ;  16 ,  18 , and  22 ; and  16 ,  18 , and  24 . Each of these triangles functions as a Fink truss when each frame segment thereof is braced in the middle. 
     Preferably, each of the frame segments  26 ,  28 ,  30 ,  32 ,  34 ,  36 ,  38 ,  40 ,  42 , and  44  is a generally straight segment. 
     As illustrated, the 4-web includes five (5) points or apices  16 ,  18 ,  20 ,  22 , and  24 . In an embodiment of the present invention of a web structure WW, shown in  FIG. 11 , a sixth apex or vertex  46  is added at 0.429, −0.166, 0 values of x, y, z coordinates of the frame FF. The sixth apex  46  is connected to each of the other five apices  16 ,  18 ,  20 ,  22 , and  24  by a line or frame segment  48 ,  50 ,  52 ,  54 , and  56 , respectively. The addition of the apex  46  produces a new set of ten (10) triangles in the frame F. Table 1 below lists the ten (10) new triangles with reference to the relevant apices and the associated frame segments, shown progressively in purple in  FIGS. 12-15 , for clarity. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 10 New Triangles 
               
            
           
           
               
               
               
               
               
            
               
                   
                 No. 
                 SETS OF APICES 
                 SEGMENTS 
                 FIGURE 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 1 
                 46, 16,20 
                 48, 34, 52 
                 12 
               
               
                   
                 2 
                 46, 16,24 
                 48, 32, 56 
                 12 
               
               
                   
                 3 
                 46, 16,22 
                 48, 36, 54 
                 12 
               
               
                   
                 4 
                 46, 16, 18 
                 48, 32, 50 
                 12 
               
               
                   
                 5 
                 46, 20, 24 
                 52, 30, 56 
                 13 
               
               
                   
                 6 
                 46, 20, 22 
                 52, 26, 54 
                 13 
               
               
                   
                 7 
                 46, 20, 18 
                 52, 40, 50 
                 13 
               
               
                   
                 8 
                 46, 24, 22 
                 56, 28, 54 
                 14 
               
               
                   
                 9 
                 46, 24, 18 
                 56, 50, 44 
                 14 
               
               
                   
                 10 
                 46, 22, 18 
                 54, 50, 42 
                 15 
               
               
                   
                   
               
            
           
         
       
     
     The formula for calculating the number of triangles is well known. Specifically, the number of combinations of ‘n’ different values taken ‘r’ at a time is calculated by n!/[(r!) (n−r)!] Thus, for the 5-web with six vertices, the number of triangles is calculated to be 6!/[(3!) (6−3)!]=720/[6×6]=720/36=20. 
     Table 2 below lists the preferred coordinates of apices  16 ,  18 ,  20 ,  22 ,  24 , and  46 . 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 X, Y, Z Coordinates of Apices 16, 18, 20, 22 24, 46 
               
            
           
           
               
               
               
            
               
                   
                 APEX 
                 COORDINATES 
               
               
                   
                   
               
               
                   
                 16 
                 0, 0, 1 
               
               
                   
                 18 
                 0, 0, −1 
               
               
                   
                 20 
                 −5, 0.866, 0 
               
               
                   
                 22 
                 −5, −0.866, 0 
               
               
                   
                 24 
                 1, 0, 0 
               
               
                   
                 46 
                 0.429, −0.166, 0 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 16  illustrates an alternative embodiment of the instant invention. As shown, this embodiment of the web structure WWW includes a frame FFF. This embodiment is constructed by positioning two triangles generally along parallel planes, but spaced apart from each other by a distance, about an axis generally perpendicular to the planes. For example, as shown in  FIG. 17 , an upper triangle  58  (shown in red and extending in x-y axes) is spaced from the lower triangle  60  (shown in blue and extending in x-y axes), along the z axis, where GC is the geometric center at 0, 0, 0 coordinates. The triangles  58  and  60  are oriented so as to be congruent. Preferably, the distance D 1  between the triangle  58  and the geometric center GC is equal to or substantially the same as the distance D 2  between the triangle  60  and the geometric center GC. The distances D 1  and D 2  can be measured/selected based on any unit of measurement, such as nanometer, Angstrom, millimeter, centimeter, inch, foot, meter, etc., as an integer or fraction thereof. 
     Preferably, the triangle  60  is then rotated counter-clockwise (arrow AAA) by about 45 degrees to reach the position shown in  FIG. 17  (shown in solid lines). It is noted that the z axis passes through the center of each triangle  58  and  60 . Each of the three vertices  62 ,  64 , and  66  of triangle  58  are then connected to the three vertices  68 ,  70  and  72  of the triangle  60 , as shown progressively in  FIGS. 18-20  (described below). 
     As best shown in  FIG. 18 , vertex  62  of the triangle  58  is connected to the vertices  68 ,  70 , and  72  by a line or frame segment  74 ,  76  and  78 , respectively (shown in green). Likewise, vertex  64  is connected to the vertices  68 ,  70  and  72  by a line or frame segment  80 ,  82  and  84 , respectively (shown in gold in  FIG. 19 ). Finally, vertex  66  of the triangle  58  is connected to the vertices  68 ,  70  and  72  of the triangle  60  by a line or frame segment  86 ,  88 , and  90 , respectively (shown in magenta in  FIG. 20 ). 
     One skilled in the art would appreciate from  FIG. 20 , that the segments  74 ,  76 ,  78 ,  80 ,  82 ,  84 ,  86 ,  88  and  90  do not intersect. 
     Table 3 below lists the twenty (20) triangles formed in the embodiment shown in  FIGS. 16-20 , with reference to the relevant apices and the associated segments. (It is noted herewith that for clarity and ease of understanding the frame segments of triangles  58  and  60  are designated as A, B, C and AA, BB, CC, respectively.) 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 20 Triangles (FIG. 16) 
               
            
           
           
               
               
               
            
               
                 No. 
                 SETS OF APICES 
                 SEGMENTS 
               
               
                   
               
            
           
           
               
               
               
            
               
                 1 
                 66, 72, 70 
                 88, 60, 90 
               
               
                 2 
                 66, 72, 68 
                 90, AA, 86 
               
               
                 3 
                 66, 72, 62 
                 90, 76, A 
               
               
                 4 
                 66, 72, 64 
                 90, 84, B 
               
               
                 5 
                 66, 70, 68 
                 88, CC, 86 
               
               
                 6 
                 66, 70, 62 
                 88, 78, A 
               
               
                 7 
                 66, 70, 64 
                 88, 82, B 
               
               
                 8 
                 66, 68, 62 
                 86, 74, A 
               
               
                 9 
                 66, 68, 64 
                 86, 80, B 
               
               
                 10 
                 66, 62, 64 
                 A, B, C 
               
               
                 11 
                 72, 70, 68 
                 AA, BB, CC 
               
               
                 12 
                 72, 70, 62 
                 BB, 78, 76 
               
               
                 13 
                 72, 70, 64 
                 BB, 82, 84 
               
               
                 14 
                 72, 68, 62 
                 AA, 74, 76 
               
               
                 15 
                 72, 68, 64 
                 AA, 80, 84 
               
               
                 16 
                 72, 62, 64 
                 76, C, 84 
               
               
                 17 
                 70, 68, 62 
                 CC, 78, 74 
               
               
                 18 
                 70, 68, 64 
                 CC, 80, 82 
               
               
                 19 
                 70, 62, 64 
                 78, C, 82 
               
               
                 20 
                 68, 62, 64 
                 74, C, 80 
               
               
                   
               
            
           
         
       
     
     Table 4 below lists three preferred coordinates for apices  62 ,  64 ,  66 ,  68 ,  70 , and  72 . 
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 X, Y, Z Coordinates of Apices 62, 64, 66, 68, 70, and 72 
               
            
           
           
               
               
               
            
               
                   
                 APEX 
                 COORDINATES 
               
               
                   
                   
               
               
                   
                 62 
                 −0.5, −0.866, 1 
               
               
                   
                 64 
                 1, 0, 1 
               
               
                   
                 66 
                 −0.5, 0.866, 1 
               
               
                   
                 68 
                 −0.966, 0.259, −1 
               
               
                   
                 70 
                 0.707, 0.707, −1 
               
               
                   
                 72 
                 0.259, −0.966, −1 
               
               
                   
                   
               
            
           
         
       
     
     As noted above, the present invention is an extension of my 4-web design disclosed in U.S. Pat. No. 6,931,812. In an independent test, the load bearing strength of a rib cage for medical applications, made in accordance with the 4-web design disclosed in U.S. Pat. No. 6,931,812, was determined to be over 6-ton/sq. inch. The present embodiments of 5-web double the number of triangles to 20, from 10 in my earlier 4-web design. One skilled in the art would readily appreciate that a similar article made in accordance with the 5-web design disclosed herein, would therefore be significantly more strong, if not twice in strength. 
     A web structure constructed in accordance with the present invention can be made of any suitable material such as wood, plastic, metal, metal alloy such as steel, fiberglass, glass, polymer, concrete, etc., depending upon the intended use or application, or choice. Further, it can be used alone or part of another structure, or used as a spacer. For example, one or more web structures can be arranged between two or more panels as spacers to add strength to the overall structure. 
     It is noted herewith that while the invention has been described for constructing level-0, level-1 and level-2 5-webs, it may be applied to create webs of higher levels. It is further noted herewith that the invention is not limited in any way to any color choice or scheme, which is used here merely for the purpose of illustration and ease of understanding. 
     While this invention has been described as having preferred/illustrative mathematical levels, sequences, ranges, steps, order of steps, materials, structures, symbols, indicia, graphics, color scheme(s), shapes, configurations, features, components, or designs, it is understood that it is capable of further modifications, uses and/or adaptations of the invention following in general the principle of the invention, and including such departures from the present disclosure as those that come within the known or customary practice in the art to which the invention pertains, and as may be applied to the central features hereinbefore set forth, and fall within the scope of the invention and of the limits of the claims appended hereto or presented later. The invention, therefore, is not limited to the preferred embodiment(s) shown/described herein. 
     REFERENCES 
     The following references, and any cited in the disclosure herein, are hereby incorporated herein in their entirety by reference. (These references are of record in U.S. Pat. No. 8,826,602.)
     1. S. L. Lipscomb,  Compression and Core Geometry of two panels , Unpublished, 2005.   2. S. L. Lipscomb,  Fractals and Universal Spaces in Dimension Theory , Springer Monographs in Mathematics, 2009.   3. J. Perry and S. Lipscomb,  The generalization of Sierpinski&#39;s triangle that lives in  4- space , Huston Journal of Mathematics, vol. 49, No. 3, 2003, pp. 691-710.   4. Greenberg, Marvin J. “Euclidean and Non-Euclidean Geometries” Development and History (second edition). Published by W.H. Freeman and Company. Copyright 1972 by Marvin Jay Greenberg and Copyright 1974, 1980 by W.H. Freeman and Company.