Patent Publication Number: US-6219974-B1

Title: Reversibly expandable structures having polygon links

Description:
RELATED APPLICATIONS 
     This is a continuation-in-part of U.S. patent application Ser. No. 09/154,482, filed Sep. 16, 1998, now U.S Pat. No. 6,082,056. 
    
    
     BACKGROUND OF THE INVENTION 
     U.S. Pat. Nos. 4,942,700 and 5,024,031, hereby incorporated by reference as if fully disclosed herein, disclose a method for constructing reversibly expandable truss-structures in a wide variety of shapes and the teachings therein have been used to build structures for diverse applications including architectural uses, public exhibits and unique folding toys. 
     In accordance with the teaching of the &#39;700 patent, the resulting structures comprise substantially linear, but angulated, strut elements and smaller hub elements that are pivotally connected. The angulated struts always have three pivot points, one central pivot point and two terminal pivot points, and they lie in planes that are essentially orthogonal to the surface of the structure. Utilizing the methods taught in the &#39;700 patent, one may construct foldable structures in a wide variety of shapes. However, certain shapes are more practical to construct in order to maintain a reasonable part count, have good structural integrity and ease of movement. In particular, the method is better suited to structures whose shape has a gentle curvature, rather than sharp corners. Also, the parts that make up a given structural shape will, in general, be unique to that particular shape. Therefore, it is not a simple matter to make a kit of interchangeable parts that may be used in different shaped structures. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention reversibly expandable structures are formed from loop assemblies comprising interconnected pairs of polygonal shaped links which lie essentially on the surface of the structure or parallel to the plane of the surface of the structure. The polygon links in the loop assembly have at least three pivot joints. At least some of the polygon links however, have more than three pivot joints. One of the pivot joints on each link is a center pivot joint for connecting to another link to form a link pair. Each link also has at least one internal pivot joint and one perimeter pivot joint. The internal pivot joints are used for interconnecting adjacent link pairs to form the loop assembly. Finally, loop assemblies can be joined together and/or to other link pairs through the perimeter pivot joints to form structures. 
     In one preferred embodiment of the present invention link pairs may be connected to adjacent link pairs to form a loop assembly through hub elements that are connected at the respective internal pivot joints of the two link pairs. Similarly hubs elements can be used to connect loop assemblies together or loop assemblies to other link pairs through the perimeter pivot joints to form structures. In yet another embodiment of the present invention the pivot joints can be designed as living hinges as described more fully below. 
     Structures built in accordance with the subject invention have specific favorable properties, including: a) The ability to use highly rigid materials rather than bending or distortion of the mechanical links, allowing for a smooth and fluid unfolding process; b) The use of compact, structurally favorable and inexpensive joints in the form of simple pivots; c) Retaining the strength and stability of the structure during folding and unfolding since all movement in the structure is due to the actual deployment process, without floppiness in the structure; d) A wide range of geometries; e) Inexpensive manufacture of structures with flexible hinges that are formed continuously with the links themselves; f) Convenient assembly of structures of many different shapes through kits of the necessary parts; and g) The ability to create a ‘space-filling’ structure by arranging linkages in a three-dimensional matrix. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention will be further described with reference to the accompanying drawings wherein: 
     FIG. 1 is a plan view of the basic polygon link element of the invention. 
     FIGS. 2-3 are plan views of a linked pair of polygon links. 
     FIGS. 4-6 are plan views of one type of two dimensional loop assembly of polygon links in accordance with the present invention, shown in three positions: retracted, partially expanded and fully expanded, respectively. 
     FIGS. 7-9 are plan views of a second type of two dimensional loop assembly of polygon links in accordance with the present invention shown in three positions: retracted, partially expanded and fully expanded, respectively. 
     FIGS. 10-12 are perspective views of a three dimensional loop assembly of polygon links in accordance with the present invention, shown in three positions. 
     FIGS. 13-15 are perspective views of a three dimensional reversibly expandable structure of polygon links in accordance with the present invention, shown in three positions: retracted, partially expanded and fully expanded, respectively. 
     FIG. 16 is a plan view showing an alternate embodiment of a polygon link assembly. 
     FIGS. 17-19 show plan views of a two dimensional embodiment of the invention using a pair of the polygon link assemblies of FIG. 16, shown in three positions: partially expanded, fully expanded and retracted, respectively. 
     FIGS. 20-21 are perspective views of a cylindrical assembly of polygon links in accordance with the present invention shown retracted and expanded, respectively. 
     FIGS. 22-24 are perspective views of a three dimensional reversibly expandable structure of the present invention using polygon links, having an icosahedral shape and shown in three positions: retracted, partially expanded and fully expanded, respectively. 
     FIG. 25A shows a polygon link. 
     FIG. 25B shows a link pair. 
     FIG. 25C shows a loop assembly. 
     FIG. 26 shows the structure  900  in a folded position. 
     FIG. 27 shows the structure  900  in a fully unfolded position. 
     FIG. 28 shows a link pair comprised of a single piece of material. 
     FIG. 29 shows a loop assembly consisting of eight link pairs. 
     FIG. 30 shows a structure  1000  consisting of thirty-two polygon link pairs. 
     FIG. 31 shows structure  1000  in a fully unfolded position. 
     FIGS. 32-34 shows a loop assembly  1200  in a folded position, a partially unfolded position and in a fully unfolded position, respectively. 
     FIGS. 35A and 35B show an alternative embodiment in which separate hub elements are replaced with a ball and socket arrangement. 
     FIGS. 36-37 show front views of an alternate embodiment of the invention, a triangle loop assembly having perimeter corner pivots that are themselves pivotally connected to polygon links. 
     FIGS. 38-39 show perspective views of this embodiment of the invention in its closed and opened states. 
     FIG. 40 shows a detail of the perimeter corner joints. 
     FIGS. 41-42 show front views of a square loop assembly in its closed and unfolded states. 
     FIGS. 43-44 show perspective views of the square loop assembly. 
     FIGS. 45-50 show how loop assemblies having a special perimeter corner joint may function as elements of a “snap-together” kit for making reversibly expandable structures. 
     FIGS. 51-52 show a prism-shaped structure in its closed and opened state. 
     FIGS. 53-54 show another element in a kit for making reversible expandable structures, a scissor pair that may be attached to loop assemblies. 
     FIGS. 55-56 show a prism-shaped structure that incorporates scissor pairs in its closed and opened state. 
     FIGS. 57-59 show means to attach two loop assemblies in a stacked arrangement, and further shows means to attach separate hub elements onto loop assemblies to provide extra attachment points. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention relates to a new reversible expandable loop assembly formed by connecting at least three link pairs, and reversibly expandable structures which are created from multiple interconnected loop assemblies and/or link pairs. Each link pair comprises two links i.e., polygon links, each having a polygonal profile with three or more corners, a central joint and a corner pivot joint proximate to at least two of the three or more corners. The central joint is used to connect the two links together. The corner pivot joints comprise at least one internal corner pivot joint and at least one perimeter corner pivot joint. To form the loop assembly each link pair is connected to at least two adjacent link pairs through at least one of its internal corner pivot joints. 
     When the loop assembly stands alone, the perimeter corner joints of the links are not connected to anything. The perimeter corner joints, however, are used to connect loop assemblies together and/or loop assemblies to link pairs to form expandable structures. 
     The polygon links of the present invention can be made from any suitable material, ascertainable by one skilled in the art. Examples of suitable material include metal, plastic and wood. 
     Loop assemblies formed in accordance with the present invention can expand and retract. In many cases the geometry of the perimeter outline of the loop assembly will remain constant in all positions, with only a change in size. Each loop assembly can be identified by a ring of line-segments formed by intersecting the perimeter corner joints of the link pairs. This property is a result of constructing the loop assembly such that the angle formed between any two line-segments corresponding to a particular two link pairs in a given position of the loop assembly, is the same as the similarly formed angle between the line segments corresponding to the same two link pairs for any other position of the loop assembly. 
     There are two aspects to finding the correct location of pivot points such that this particular property is obtained. 
     First, an arrangement of links must be found such that the loop-assembly does fold freely, that is, it does not lock up. This ability to fold is not guaranteed. For example, by applying the equation to determine the degrees of freedom of a typical planar loop assembly, the result will be negative, indicating a over determined (i.e. locked) condition. 
     Therefore, the ability to fold is dependent on particular geometric conditions. When constructing a planar loop-assembly, an aid to determining possible location of pivot points, is to draw a four sided shape that connects the center joint from one link-pair to two of its interior corner joints, and then in turn connecting those corner joints to the center joint of its neighboring link-pair. According to a typical construction, all such quadrilaterals similarly drawn within a loop-assembly should be parallelograms. 
     If all these parallelograms are similar (have identical angles) the loop-assembly will definitely fold. However, it is possible to construct foldable loop assemblies with dissimilar parallelograms, and indeed to form foldable loop-assemblies where the quadrilaterals, and indeed to form foldable loop-assemblies where the quadrilaterals are not parallelograms at all. These alternative constructions require other symmetric arrangements that may be discovered through deeper study and inquiry. 
     Once a foldable loop-assembly is constructed, the location of the perimeter corner joints must be considered. The goal is to ensure that line segments drawn through paired perimeter corner joints maintain a constant angle relative to one another as the loop assembly is folded. 
     In a similar fashion to finding rules for constructing foldable loop-assemblies, we can find rules for locating perimeter corner joints that will always work. If, for example, each link-pair in a given foldable loop-assembly is comprised of two polygon links having identical relative locations of their perimeter and interior corner-joints, the angles between line segments will remain constant. Generally, paired polygon links that are similar in shape, but different in size will have this property as well. However, there are alternative arrangements that exist as well. 
     As explained above the position of the pivot joints are critical to the function of the loop assemblies and structures of the present invention. The profile of the links, however, are less critical and more design related. It will be apparent to one of ordinary skill in the art that so long as the pivot holes are the same, the links can have most any geometry. The selection of geometries thus is primarily one of creative design choice. However, it will also be obvious to one skilled in the art that certain polygon shapes may restrict the ability of the structure to reach a fully expanded or fully retracted position. 
     The loop assemblies and structures in accordance with the present invention have many applications including: medical devices, toys, architectural design and displays. 
     Referring now more particularly to the drawings, shown in FIG. 1 is a link  10 , which has a triangular shape and four pivot holes. Pivot hole  2  is in the central region of the link hereinafter the “center joint,” and pivot holes  4 ,  6 , and  8  are proximate to the corners of the link. A dashed line  25  is drawn connecting the center of the three corner-pivot holes  4 ,  6 , and  8  hereinafter “corner joints,” forming a triangle. 
     Referring to FIG. 2 the polygon link  10  of FIG. 1 is linked to a second polygon link  20  by center joint  2  to form a link pair. Links  10  and  20  have essentially the same profile and pivot hole locations. A dashed line  24  is shown passing through the center of paired corner joints  4 ,  14 . Similarly, dashed line  26  passes through  6 ,  16 , and dashed line  28  passes through  8 ,  18 . The triangle  30  formed by lines  24 ,  26 , and  28  has essentially the same shape as dashed line triangle  25  shown in FIG.  1 . 
     FIG. 3 shows the link pair  10 ,  20  in a new position having been rotated relative to each other about their center joint. Three dashed lines  34 ,  36  and  38 , are shown passing through paired corner joints  4 ,  14 ;  6 ,  16 ; and  8 ,  18 , respectively. The angle formed between dashed lines  34  and  36  is the same as the angle formed between dashed lines  24  and  26  shown in FIG.  2 . Likewise the angles formed respectively between dashed lines  36 ,  38  and  38 ,  34  are the same as those angles formed respectively between dashed lines  26 ,  28  and  28 ,  24  shown in FIG.  2 . Thus triangle  35  has the same shape as triangle  30  shown in FIG. 2, but larger in size. 
     Referring to FIG. 4 the expanding right triangle is extended to an expanding hexagon by forming a loop assembly  38  consisting of 12 polygon links  40 ,  45 ,  50 ,  55 ,  60 ,  65 ,  70 ,  75 ,  80 ,  85 ,  90  and  95 . These polygon links are respectively joined by center joints  41 ,  51 ,  61 ,  71 ,  81  and  91  into 6 link pairs  49 ,  59 ,  69 ,  79 ,  89  and  99 . The loop assembly  38  is formed by joining the internal corner joint of each top layer to the adjacent internal corner joints of the two adjacent lower polygons on both sides. The internal corner joints are easily seen with reference to FIG.  5 . 
     Thus, referring to FIG. 5 loop assembly  38  is shown unfolded into a different position while maintaining the overall hexagon shape defined by edges drawn between the outer joints of each polygon, as discussed below. In this new position, it is more readily noticeable how adjacent polygon links are connected. For example, link pair  49  is connected to link pair  59  by the two corner joints  42  and  43 . These corner joints are referred to as internal corner joints since they are located on the interior portion of the loop assembly  39 . Likewise link pairs  59  and  69  are connected to each other by internal corner joints  52  and  53 . Similarly link pairs  69 ,  79 ;  79 ,  89 ;  89 ,  99 ; and  99 ,  49  are connected by internal corner joints  62 ,  63 ;  72 ,  73 ;  82 ,  83 ; and  92 ,  93 , respectively. 
     A dashed line  44  is shown passing through corner joints  46  and  48 . These corner joints are located near the outer edge of loop assembly  38  in its unfolded position. These joints are the perimeter corner joints of the loop assembly. Likewise a dashed line  54  is shown passing through perimeter corner joints  56  and  58  and dashed lines  64 ,  74 ,  84  and  94  are shown passing through perimeter corner joints  66 ,  68 ;  76 ,  78 ;  86 ,  88 ; and  96 ,  98 , respectively. These dashed lines through the perimeter corner joints define the edges of the expanding hexagon  100 , mentioned above. 
     Referring to FIG. 6, loop assembly  38  is shown unfolded further, into yet a different position while maintaining the overall hexagonal shape. Dashed lines  47 ,  57 ,  67 ,  77 ,  87  and  97  are shown passing through perimeter corner joints  46 ,  48 ;  56 ,  58 ;  66 ,  68 ;  76 ,  78 ;  86 ,  88 ; and  96 ,  98  respectively, forming hexagon  105 . The commonality between hexagons  100  and  105  is that the angle formed between dashed lines  47  and  57  is the same as the angle formed between  44  and  54  shown in FIG.  5 . Likewise the angles formed between dashed lines that correspond to any two link pairs as shown in FIG. 6 are identical to those angles similarly formed corresponding to the same two link pairs, as shown in FIG.  5 . 
     Referring now to FIG. 7 a different triangle loop assembly  108  is shown consisting of 6 polygon links  110 ,  120 ,  130 ,  140 ,  150  and  160 . These polygon links are respectively joined by center joints  114 ,  134  and  154  into three link pairs  119 ,  139  and  159 . Links  110 ,  120  are joined at  134  to form pair  119 . Links  130 ,  140  are joined at  114  to form pair  139  and links  150  and  160  are joined at  154  to form pair  159 . In FIG. 8 loop assembly  108  is shown unfolded into a different position and as with FIGS. 4-6, the overall triangle shape is maintained. Link pair  119  is connected to link pair  139  by internal corner joints  131  and  141 , link pair  139  is connected to link pair  159  by internal corner joint  161  and link pair  159  is connected to link pair  119  by internal corner joints  121  and  151 . 
     The triangular ring perimeter outline  170  of the loop assembly as shown in FIG. 8 comprises line segments  114 ,  134  and  154 . Dashed line  114  is shown passing through perimeter corner joints  115  and  125  and dashed lines  134  and  154  are shown passing through perimeter corner joints  135 ,  145  and  155 ,  165  respectively. 
     In FIG. 9 the loop assembly  108  is shown in a further unfolded position. Dashed lines  117 ,  137  and  157  are shown passing through paired perimeter corner joints  115 ,  125 ;  135 ,  145 ; and  155 ,  165  respectively, thereby forming triangular ring  180  which is larger in size than ring  170  of FIG.  8 . 
     The angle formed between dashed line-segments  117 ,  137  is the same as the angle formed between  114 ,  134  in FIG.  8 . Similarly, the angles formed between dashed lines  137 ,  157  and lines  157 ,  117  are the same as those angles formed between lines  134 ,  154  and lines  154 ,  114  in FIG.  8 . 
     The loop assemblies shown to this point were all formed by joining adjacent link pairs directly at internal pivot points. The result was a loop assembly with all link pairs lying on parallel planes. It is also possible to add relative dimension to the loop assembly by introducing hub elements between the internal corner pivot joints of adjacent link pairs. As seen in loop assembly  208  of FIG. 10, hub elements such as  299  are used to pivotally connect each link pair to its neighbor. In addition, using hub elements, more than two link pairs can be joined at a single connection point. 
     Other than the hub elements, loop assembly  208  is similar to the other loop assemblies discussed above. Indeed, it will be recognized that the loop assembly of FIGS. 10-12 is similar to that of FIGS. 4-6. In FIGS. 4-6 there are no hub elements and the link pairs lie in parallel planes. In FIGS. 10-12 the hub elements position the same link pairs into non parallel planes. Loop assembly  208  contains polygon links and each has three cover joints and one center joint through which they are paired into link pairs,  249 ,  259 ,  269 ,  279 ,  289  and  299 . 
     A hub element can be any linking material with at least two separate pivot points that are not coaxial with each other. The hub element could have an angle or it could be straight. The axes of the hub pivot points could be parallel, perpendicular or from some other angle therebetween. Each of these variations will impact on the creative design element of the loop assembly including its range of motion. 
     The size of the hub element and the material chosen for its construction will also impact on the durability of the loop assembly. 
     FIG. 11 shows loop assembly  208  unfolded into a different position while the lines crossing the perimeter joints of the polygon links maintain the same polygon shape. Link pair  249  may be seen to be pivotally connected to two hub elements  252  and  253  which connect in turn to link pair  259 . Likewise link pair  259  is connected to link pair  269  via hub elements  262  and  263 . Similarly, link pairs  269 ,  279 ;  279 ,  289 ;  289 ,  299 ; and  299 ,  249  are successively connected by hub elements  272 ,  273 ;  282 ,  283 ;  292 ,  293 ; and  242 ,  243 , respectively. As explained above, these hub elements introduce angles between the planes of adjacent link pairs. 
     The dashed lines  344 ,  354 ,  364 ,  374 ,  384  and  394  lie in the planes of their corresponding link pairs,  349 ,  359 ,  369 ,  379 ,  389  and  399  respectively, and form a three dimensional ring  400 . These lines cross through the perimeter corner joints of their respective links:  240 ,  245  for link pair  249 ;  250 ,  255  for link pair  259 ;  260 ,  265  for link pair  269 ;  270 ,  275  for link pair  279 ;  280 ,  285  for link pair  289 ; and  290 ,  295  for link pair  299 . 
     In FIG. 12 the loop  208  is shown further unfolded into a different position. The dashed lines  444 ,  454 ,  464 ,  474 ,  484  and  494  drawn respectively through the perimeter corner joints of the polygon links  240 ,  245 ;  250 ,  255 ;  260 ,  265 ;  270 ,  275 ;  280 ,  285 ; and  290 ,  295 . As with the other loop assemblies described above, these line segments form a ring  450  that is larger in size than ring  400  shown in FIG.  11 . However, the angle formed between dashed line  444  and  454  is the same as that angle formed between lines  344  and  354  of FIG.  11 . Likewise the angles formed between dashed lines that correspond to any two adjacent link pairs as shown in FIG. 12 are identical to those similarly formed angles corresponding to the same two link pairs as shown in FIG.  11 . Perimeters may be left open or used to connect to another assembly or polygon link pair. 
     As described above, loop assemblies formed in accordance with the present invention can be used in forming three dimensional closed structures. In some instances it will be sufficient to connect two or more loop assemblies together. Other cases may require additional link pairs connected to the loop assemblies to close the structure. 
     Generally, the loop assemblies and/or link pairs are connected together at the perimeter pivot joints described above. It will not always be necessary to use all available perimeter pivot joints. However, the interconnections may only use perimeter corner joints. The interconnections between loop assemblies will generally involve hub elements, although direct pivotal connections are possible, as well as living hinges, as described below. 
     It is important to note that reference to perimeter corner joints has meaning only with respect to a given loop assembly. Once a structure is assembled the perimeter outline of the loop assembly can be drawn with any arbitrary selection of link pairs due to the symmetry inherent in the structure. 
     Referring to FIG. 13 a structure  500  is shown in a folded position. Structure  500  consists of 20 link pairs in interlocking loop assemblies, each link pair comprised of two polygon links. One such loop assembly  510 , within structure  500 , consists of five link pairs  520 ,  530 ,  540 ,  550  and  560 . Link pair  520  is pivotally connected to link pair  530  by two hub elements  522  and  523 . Similarly link pairs  530 ,  540 ,  550 , and  560  are successively joined together by hub elements  532 ,  533 ;  542 ,  543 ; and  552 ,  553 , respectively. Link-pair  560  is connected to link-pair  520  by hub elements  562  and  563 . One may recognize that the loop assembly  510  is similar to that shown in FIGS. 10-12 except that only five link pairs are used and the hub elements have different angles. 
     A structure constructed in accordance with the present invention can include as a creative design element, the formation of a continuous surface. As shown in FIG. 13, in the folded position, structure  500  forms a substantially closed and continuous surface. The degree of continuity will depend on the polygon profile of the links, the number of links in the loop assembly and the angle in the hub elements. 
     FIG. 14 shows structure  500  unfolded into a larger position. Dashed line  524  passes through the perimeter corner joints of link pair  520 . Similarly dashed lines  534 ,  544 ,  554  and  564  respectively pass through the perimeter corner joints of link pairs  530 ,  540 ,  550  and  560 . Dashed line segments  524 ,  534 ,  544 ,  554  and  564  form a five-sided ring  570 . 
     In FIG. 15 the structure  500  is again further unfolded. The dashed lines  526 ,  534 ,  546 ,  556  and  566  pass respectively through the perimeter corner joints of link pairs  520 ,  530 ,  540 ,  550  and  560 , thus forming a five sided ring  580  which is larger in size than ring  570  in FIG.  14 . The angles formed between dashed lines that correspond to any two adjacent link pairs in FIG. 15 are identical to those similarly formed angles corresponding to the same two link pairs in FIG.  14 . In its fully unfolded position, another creative design element resulting from the polygon links that make up structure  500  may be seen. Namely the link pairs separate and create openings that are pentagonally shaped. 
     In addition to the simple pivots shown above for the inter-link connections, either hub or direct, connections can also comprise living hinges. A living hinge is a flexible portion of a material, continuous with, and connecting two or more stiff portions of the material. A change in dimension from the stiff portion gives rise to the flexible portion. FIG. 16 shows a sheet of material  601  that consists of triangular stiff regions of material that act as polygon links, which are connected by thinner flexible regions of material that act as corner joints. While many materials are suitable for living hinges to be used in accordance with the present invention, and those skilled in the art will be readily able to determine the same, polypropylene and nitemol are believed to be especially suitable materials for forming living hinges. 
     FIG. 17 shows a flat structure  600  which consists of two sheets of material  601  as above, and  602  which is the mirror image of  601 . Sheet  601  is joined to sheet  602  by thirty-six pivot joints to create thirty-six link pairs. The folded position of this structure is shown in FIG.  19 . These link pairs are arranged in interlocking loop assemblies. One such loop assembly  605  consists of six link pairs  610 ,  620 ,  630 ,  640 ,  650  and  660 . Dashed line  615  passes through the perimeter corner joints of link pair  610 . Similarly dashed lines  625 ,  635 ,  645 ,  655  and  665  respectively pass through the perimeter corner joints of link pairs  620 ,  630 ,  640 ,  650  and  660 . 
     While FIG. 18 shows living hinges used at internal corner pivot joints, it is also possible to use living hinges at the center pivot joint. An example of a link pair with a living hinge center pivot joint is shown below in connection with FIG.  28 . 
     In FIG. 18 the structure  600  is shown unfolded into a larger position. Dashed line  616  passes through the perimeter corner joints of link pair  610 . Similarly dashed lines  626 ,  636 ,  646 ,  656  and  666  respectively pass through the perimeter corner joints of link pairs  620 ,  630 ,  640 ,  650  and  660 . The angle formed between dashed lines  616  and  626  is identical to the angle formed by dashed lines  615  and  625  shown in FIG. 17, however, unlike the loop assemblies shown in prior FIGS., the shape of the loop assembly changes with folding and unfolding since the size of the edges do not change proportionally. Similarly the angles formed respectively between dashed lines  626 ,  636 ;  636 ,  646 ;  646 ,  656 ; and  656 ,  666  are identical to those angles formed respectively by dashed lines  625 , 635 ;  635 , 645 ;  645 , 655 ; and  655 ,  665  shown in FIG.  17 . 
     Structure  700  shown in FIG. 20 also consists of two sheets of material  701  and  702 . Similar to sheets  601 ,  602  shown in FIG. 16, sheets  701 ,  702  are comprised of triangular stiff regions of material acting as polygon links that are connected by thinner flexible regions of material acting as corner joints. Sheets  701  and  702  have been joined together by a plurality of center pivot connections and are formed into a cylindrical shape. 
     The cylindrical structure can be formed by joining the opposite, parallel edges of a loop assembly much like that of FIGS. 17-19. Alternatively, two cylinders can be formed from a continuous cylindrically shaped material with links cut out much like FIG.  16 . One cylinder can be placed over and around a second cylinder joined by center pivot joints. Yet, a third method would be to cut out link pairs from a single cylindrical material with living hinge center pivot joints. Other embodiments will become apparent to those skilled in the art and fall within the scope and spirit of this invention. 
     In its folded position, the polygon links that make up structure  700  may be seen to form a continuous surface much as described in connection with FIG.  13 . Six dashed lines  710 ,  720 ,  730 ,  740 ,  750  and  760  are shown to pass through the perimeter corner joints of six of the link pairs. 
     FIG. 21 shows the structure  700  in an unfolded position in which it maintains its overall cylindrical shape. Six dashed lines  715 ,  725 ,  735 ,  745 ,  755  and  765  pass through the perimeter corner joints of six link pairs. The angle formed between dashed lines  715  and  725  is identical to the angle formed between dashed lines  710  and  720  shown in FIG.  20 . Similarly, the angles formed between dashed lines that correspond to any two adjacent link pairs as shown in FIG. 21 are the identical to those similarly formed angles corresponding to the same two link pairs as shown in FIG.  20 . 
     FIG. 22 shows yet another structure  800  comprised of interconnected loop assemblies, in a folded position. This structure is comprised of 20 loop assemblies, one of which is loop assembly  810  which is similar to loop assembly  108  of FIG.  8 . 
     FIG. 23 shows the structure  800  in a partially unfolded position. Loop assembly  810  may be seen to be comprised of three link pairs  820 ,  830  and  840 . Dashed line  825  passes through the perimeter corner joints of link pair  820  while dashed lines  835  and  845  respectively pass through the perimeter corner joints of link pairs  830  and  840 . 
     FIG. 24 shows structure  800  in a fully unfolded position, with dashed line  826  passing through the perimeter corner joints of link pair  820  and dashed lines  836  and  846  respectively passing through the perimeter corner joints of link pairs  830  and  840 . The angle formed between dashed lines  826  and  836  is identical to the angle formed by dashed lines  825  and  835  shown in FIG.  23 . Likewise the angles formed between the other adjacent dashed lines shown in FIG. 24 are identical to those similarly formed angles shown in FIG.  23 . 
     FIG. 25A shows a polygon link  901 , which has a center pivot joint  957 , two interior pivot joints  954  and  956 , and a perimeter pivot joint  955 . 
     FIG. 25B shows a link pair  903  consisting of two polygon links  901  and  902  which share the center pivot joint  957 . Also shown are the interior pivot joints for polygon links  901  and  902 , respectively  952 ,  956 ,  958  and  959 . Finally, the perimeter pivot joints for  902  and  903  are shown, being respectively  954  and  955 . 
     FIG. 25C shows a loop assembly  910  in a partially unfolded position. Loop assembly  910  consists of four link-pairs  903 ,  913 ,  923  and  933 , each link-pair comprised of two polygon links. A dashed line  906  passes through perimeter joints  954  and  955  which belong to link-pair  903 . Similarly dashed lines  916 ,  926  and  936  pass through perimeter joints  964 ,  965  and  974 ,  975  and  984 ,  985  respectively, forming a four-sided shape. 
     Loop-assembly  910  (FIG. 26) shows an alternative arrangement for the connection of link-pairs to one another. Rather than all interior corner-joint connections being made between adjacent link-pairs, some interior corner joints are connected to link-pairs that are non-adjacent. 
     Specifically, link-pair  903  (FIG. 27) is connected to adjacent link-pair  913  by its interior corner joint  958 , and likewise to adjacent link-pair  933  by interior corner joint  956 . However, in addition link-pair  903  is connected to non-adjacent link-pair  923  by two interior corner-joints  952  and  959 . 
     FIG. 26 shows the structure  900  in a folded position. This structure is comprised of 6 loop-assemblies, one of which is loop-assembly  910 . 
     FIG. 27 shows the structure  900  in a fully unfolded position. Dashed line  907  passes through the perimeter corner-joints of link-pair  903 . Likewise dashed line  917 ,  927  and  937  respectively pass through the perimeter corner-joints of link-pairs  913 ,  923  and  933 . The angle formed between dashed lines  907  and  917  is identical to the angle formed by dashed lines  906  and  916  shown in FIG.  25 C. Similarly, the angles formed between dashed lines that correspond to any two adjacent link-pairs as shown in FIG. 27 are identical to those similarly formed angles corresponding to the same two link-pairs as shown in FIG.  25 C. 
     FIG. 28 shows a link-pair  1001  that is comprised of a single piece of material, cut to form two polygon links  1002  and  1003 . Center-joint  1004  is comprised of a region of flexible material which is formed in a continuous manner with links  1003  and  1004 . Thus link  1003  can rotate relative to link  1004  by flexing the center-joint  1004 . 
     In FIG. 29 is shown the loop assembly  1005  consisting of eight link pairs  1011 ,  1021 ,  1031 ,  1041 ,  1051 ,  1061 ,  1071  and  1081 . Similar to link-pair  1001  shown in FIG. 28, each link pair is formed of two polygon links that are connected by a center-joint comprised of a flexible region of material formed continuously with the polygon links. 
     Link-pair  1071  is connected to adjacent link-pair  1081  by two interior corner-joints  1022  and  1023 . Joint  1022  is comprised of a region of flexible material that is formed continuously with links  1072  and  1082 . Likewise joint  1023  is formed continuously with  1073  and  1083 . 
     Thus loop-assembly  1005  is formed from a unitary piece of material comprised of essentially rigid regions acting as polygon links and flexible regions acting as pivot connections. 
     Also shown in FIG. 29 is a dashed line  1015  which passes through the perimeter corner joints of link-pair  1011 . Similarly dashed lines  1025 ,  1035 ,  1045 ,  1055 ,  1065 ,  1075  and  1085  respectively pass through the perimeter corner joint of link pairs  1011 ,  1021 ,  1031 ,  1041 ,  1051 ,  1061 ,  1071  and  1081 , forming an eight-sided ring of dashed lines. 
     In FIG. 30 shown structure  1000  consisting of thirty-two polygon link-pairs, each link-pair being similar to link pair  1001  of FIG.  28 . These link pairs are grouped as four assemblies of eight link pairs each. One of these loop assemblies of structure  1000  is loop assembly  1005  in its fully folded position. 
     Structure  1000  is formed of a unitary piece of material consisting of stiff regions acting as polygon-links and relatively flexible regions acting as corner-joints or center-joints. 
     In FIG. 31 the structure  1000  is shown in a fully unfolded position with dashed line  1016  passing through the two perimeter corner-joints of link-pair  1011 . Similarly dashed lines  1026 ,  1036 ,  1046 ,  1056 ,  1066 ,  1076  and  1086  may be seen to pass through the perimeter corner-joints of link-pairs lines  1021 ,  1031 ,  1041 ,  1051 ,  1061 ,  1071  and  1081  respectively. The perimeter corner joints of loop assembly  1005  are living hinge joints which joint it to adjacent loop-assembly  1006  (shown as a shaded region). 
     The angle formed between dashed lines  1016  and  1026  is identical to the angle formed between dashed lines  1015  and  1025  shown in FIG.  29 . Likewise the angles respectively formed between each dashed line and its neighbor in FIG. 31 is identical to the angles respectively formed between dashed lines shown in FIG.  30 . 
     The interior corner-joints of link-pair  1051  may be seen to be pivotally joined to the interior corner-joints of adjacent link pair  1061 . Additionally, the interior corner joints are joined to link pairs  1091  and  1101  which belong to adjacent loop assembly  1006 , which is shown in the shaded portion of FIG.  31 . Thus each polygon link belonging to link pairs  1051 ,  1061 ,  1091  and  1101  is joined to three other polygon links. Similar multiple connections between polygon-links in structure  1000  thus form a three-dimensional matrix of link-pairs. 
     FIG. 32 shows a loop assembly  1200  in a folded position. Loop assembly  1200  is made up in part of four link pairs  1210 ,  1230 ,  1250  and  1270 . In addition to these four link pairs,  1200  contains eight connecting links  1221 ,  1222 ,  1241 ,  1242 ,  1261 ,  1262 ,  1281  and  1282 . 
     FIG. 33 shows loop assembly  1200  in a partially unfolded position. Link pair  1210  is comprised of two polygon links  1211  and  1212  which are connected by center joint  1215 . Link  1211  has an interior corner joint  1213  connecting it to polygon link  1272 , and link  1212  is connected to link  1231  by corner joint  1214 . Similarly, link-pairs  1230 ,  1250  and  1270  are connected one to the other via their respective interior corner joints. 
     In addition to said interior corner joint connections, the four link pair that comprise  1200  are connected one to the other via eight additional connecting links. In particular, link pair  1210  is connected to link pair  1270  by connecting links  1221  and  1282 , which are pivotally attached to one another. Similarly, link pair  1270  is connected to  1250  by links  1281  and  1262 . Link pairs  1250 ,  1230  and  1230 ,  1210  are connected respectively by connecting links  1261 ,  1242  and  1241 ,  1222 . In addition to these four connections, connecting link  1221  may be seen to be pivotally attached to  1222 . Likewise,  1241 ,  1242  and  1261 ,  1262  and  1281 ,  1282  are pivotally attached to one another. Thus the eight connecting links form a closed loop that is pivotally connected in eight places to the four link pairs. Each connecting link has one pivotal connection to one polygon link, each connection link has one center joint and two terminal joints, and is pivotally connected to its neighboring connecting link via one of its terminal joints to form a closed loop. 
     These eight connecting links serve to synchronize the motion of the loop assembly. This is sometimes necessary in the case where polygon links are connected to one another by only a single interior corner joint, as is true for loop assembly  1200 . 
     Dashed line  1218  passes through perimeter joints  1216  and  1217  which belong to link pair  1210 . Similarly dashed lines  1238 ,  1258  and  1278  pass through the perimeter joints of link pairs  1230 ,  1250  and  1270  respectively. 
     FIG. 34 shows  1200  in a fully unfolded position. Dashed lines  1219 ,  1239 ,  1259  and  1279  pass through the perimeter joints of link pairs  1230 ,  1250  and  1270  respectively. The angle formed between dashed lines  1219  and  1239  is the same as that formed between  1218  and  1238  shown in FIG.  33 . This similarity of angles holds for the other dashed lines as well. 
     FIGS. 35A and 35B show four triangular loop assemblies with the assembly shown in  35 A being folded and the assembly shown in  35 B being unfolded. In this embodiment, the connection between loop assemblies includes separate hub elements, shown at  1301  and  1302 , which connect adjacent assemblies with the perimeter joint of each polygon link being either a ball or a socket. This embodiment allows loop assemblies to be connected directly without the need of a separate hub element. This particular embodiment provides a reduction of part count (i.e., no hub elements) and favorable structural characteristics as forces between assemblies are transferred directly, rather than indirectly. 
     FIG. 36 shows a triangular loop assembly  1400 , comprised of polygon links  1401 ,  1402 ,  1403 ,  1404 ,  1405  and  1406 . Perimeter corner joint  1411  is a separate element that is pivotally connected to link  1401 . The axis of this pivot connection lies essentially within the plane of link  1401 . Similarly, perimeter corner joints  1412 ,  1413 ,  1414 ,  1415  and  1416  are pivotally connected to polygon links  1402 ,  1403 ,  1405  and  1406  respectively, each of their axes lying within the plane of their corresponding link. Dashed line  1410  passes through the six perimeter corner joints. 
     FIG. 37 shows loop assembly  1400  in its opened state. Dashed line  1420 , which passes through all six perimeter corner joints, contains similar angles to those in dashed line  1410 . 
     FIGS. 38 and 39 show perspective views of  1400  in its folded and unfolded state. 
     FIG. 40 shows a perspective view of links  1403 ,  1404  with their corresponding perimeter corner joint  1413 ,  1414 . Corner joint  1413  has a pivotal connection to  1403  whose axis lies essentially within the plane of link  1403 . Thus, corner joint  1413  may rotate as indicated by arrow  1432 . Corner joint  1414  is pivotally connected to link  1404  in a similar fashion, and rotate as indicated by arrow  1431 . 
     FIGS. 41 and 42 show another embodiment of the invention, a square loop assembly  1500 , in its folded and unfolded state respectively. Similar to loop assembly  1200  shown in FIGS. 32-34, assembly  1500  is comprised of eight polygon links, as well as eight additional connecting links. 
     In FIG. 42 are shown perimeter corner joints  1511  and  1512 , which are pivotally connected to polygon links  1501  and  1502  respectively. The axes of said pivots lie essentially within the plane of these two polygon links. 
     Also shown in FIGS. 41 and 42 are dashed lines  1510  and  1520  respectively, which pass through the perimeter corner joints of loop assembly  1500 . Lines  1510  and  1520  contain similar angles to each other. 
     FIGS. 43 and 44 show perspective views of  1500  in its folded and unfolded state. 
     FIGS. 45 through 50 show how loop assemblies  1400  and  1500  may function as elements of a kit for building reversibly expandable structures. In FIG. 45, loop assembly  1400  is placed adjacent to loop assembly  1500 , such that corner joint  1413  is proximate to corner joint  1511 . Similarly, joint  1414  is proximate to  1512 . 
     FIG. 46 shows corner joint  1414  being pressed over corner joint  1512 . As joint  1414  is pressed, two flexible regions  1415 ,  1416  spread apart, while two flexible regions  1515 ,  1516  of joint  1512  are pressed together. 
     FIG. 47 shows corner joints  1414  and  1512  having been brought into alignment and joined together, the flexible of both joints having snapped back into their unrestrained condition. Joints  1414  and  1512  may now rotate relative to one another as indicated by arrow  1622 . This combination of perimeter corner joints  1414  and  1512  having been connected together may now be considered to form a combined hub element  1650 . Said hub element has three intersecting axes of rotation indicated by arrows  1621 ,  1622 ,  1623  and acts essentially as a ball joint with regards to its range of motion. 
     FIG. 48 shows loop assemblies  1400  and  1500  having been connected, where corner joints  1511  and  1512  are respectively attached to joints  1413  and  1414 , thereby forming combined hub elements  1650 ,  1651  respectively. Loop assembly  1400  is free to pivot relative to loop assembly  1500  as indicated by arrow  1620 . 
     FIG. 49 shows loop assembly  1400  having been rotated relative to loop assembly  1500  thereby introducing an angle between the planes of each loop assembly. Combined hub elements  1650 ,  1651  may be squeezed towards each other as indicated by arrows  1630  and  1640 . 
     FIG. 50 shows loop assemblies  1400  and  1500  in an expanded condition after combined hub elements  1650 ,  1651  have been squeezed together. 
     In FIG. 51 are shown two triangle-shaped loop assemblies  1401 ,  1402  which are connected by their perimeter corner joints to three square-shaped loop assemblies  1501 ,  1502 ,  1503  to form a prism-shaped structure  1600 . Structure  1600  is shown in its folded state in FIG.  44 . 
     FIG. 52 shows structure  1600  in its unfolded state. 
     FIG. 53 shows scissor-paid  1700  comprised of two links  1731  and  1732  which are pivotally connected together, said pivot lying in a central region of each link. Terminal pivots  1710  and  1711  are themselves pivotally connected to link  1732 ; terminal pivots  1720  and  1721  are pivotally connected to link  1731 . All end-joints are free to pivot around axes that lie within the plane of scissor-pair  1700 . 
     FIG. 54 shows a perspective view of scissor-pair  1700 , where arrows  1725  and  1726  indicate the axis of rotation of end-joints  1710  and  1720  respectively. 
     FIG. 55 shows prism-shaped structure  1750  in its folded position. Structure  1750  is comprised of three square-shaped loop assemblies  1501 ,  1502 ,  1503 ; two triangle-shaped assemblies  1401 ,  1402 ; and six scissor-pairs  1700 ,  1701 ,  1702 ,  1703 ,  1704  and  1705 . Loop assembly  1402  is attached at two points each to three scissor-pairs  1700 ,  1701 ,  1702 , which are in turn attached to two points each of three loop assemblies  1501 ,  1502  and  1503 . Likewise, loop assembly  1401  is attached at two points each to scissor-pairs  1703 ,  1704 ,  1705 , which are in turn attached to two pints each of loop assemblies  1501 ,  1502  and  1503 . 
     In FIG. 56, structure  1750  is shown in its unfolded state. It may be seen that addition of scissor-pairs augment the increase in size of structure  1750  relative to structure  1600  as shown in its unfolded state in FIG.  52 . 
     FIG. 57 shows two adjacent square loop assemblies  1501 ,  1502 , such that the planes that each assembly lies in are parallel to one another. Shown between assemblies  1501 ,  1502  are eight tube elements  1800 . Said tube elements may be attached to assemblies  1501 ,  1502  by pressing them over features in the assemblies such as the post  1505 . 
     Also shown in FIG. 57 are two separate hub elements  1530 ,  1531 , which may be attached to loop assembly  1501  by pressing them over features such as post  1506 . 
     In FIG. 58, loop assemblies  1501 ,  1502  are shown attached to one another via the eight tube elements  1800  so that each of the eight polygon link pairs in assembly  1501  are connected to a corresponding polygon link pair in assembly  1502 . The two loop assemblies are parallel to one another, thereby forming a stacked arrangement. 
     Also shown in FIG. 58 are separate hub elements  1530 ,  1531  attached to assembly  1501 , thereby providing additional connecting points to the loop assembly. 
     FIG. 59 shows loop assemblies  1501 ,  1502  in an unfolded state, tube elements  1800  serving to synchronize the motion of the two assemblies. 
     It will be appreciated that the instant specification and claims set forth by way of illustration and not limitation, and that various modifications and changes may be made without departing from the spirit and scope of the present inventions.