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This application is based on and claims priority of provisional application Ser. No. 60/512,633, filed Oct. 20, 2003 
   U.S. Pat. No. 5,024,031 hereby incorporated by reference as if fully disclosed herein teaches methods for constructing transformable truss-structures in a variety of shapes. The teachings therein have been used to build structures for diverse applications including architectural uses, public exhibits and unique folding toys. 
   One basic embodiment disclosed in U.S. Pat. No. 5,024,031 are loop-assemblies comprised of scissor-pairs which are in turn comprised of angulated strut elements. Such loop-assembles are foldable in the sense that they expand and contract in a synchronized fashion when a relative motion is imposed between any two links. 

   BACKGROUND OF THE INVENTION 
   SUMMARY OF THE INVENTION 
   In accordance with the present invention, a new way to create loop assemblies comprised of links pivotally joined end to end such that the motion of the assembly is provided. The synchronization may be accomplished in a variety of methods such as gears, belts or pulleys. All methods have in common the linking of every second link in the loop assembly such that the relative rotation of every second link is synchronized. 
   One key benefit of the invention is a reduction in the number of individual elements as compared with those structures disclosed in U.S. Pat. No. 5,024,031. Rather than all links being “doubled” in the form of scissor-pairs, a single loop of links suffices. The addition of gears or pulleys represents only minor additional material. 
   The invention has a second useful feature as well. For structures disclosed in &#39;031, they move between a contracted state and expanded state. As the structure expands, its members rotate approximately ninety degrees. When the structure is fully expanded, the members are prevented from rotating further because the hub elements contact each other. 
   According to the current invention, structures are disclosed such that its members rotate approximately one hundred and eighty degrees. Thus, the structure starts in a contracted state, where its members are in a radial configuration, to an expanded state where its members form an extended loop, and then it can be continuously folded again so that it reaches a second, unique contracted state. 
   This unusual ability to “flip” between two unique folded states allow for structures to be built that display a pleasing visual transformation. 
   A third useful feature of the current invention is that it provides a mechanism whereby a circular ring that has flight on characteristics can transform into a boomerang. 
   It is thus an object of the invention to provide an improved ring linkage system. 
   Another object of the invention is to provide a linkage system whose motion is synchronized. 
   Yet a further object of the invention is to provide an improved linkage having a plurality of links in geared contact with one another. 
   Other objects and advantages of the invention will, in part, be obvious and will, in part, be apparent from the following description. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a fuller understanding of the invention, reference is made to the following description, taken in connection with the accompanying drawings in which: 
       FIG. 1  is a perspective view of three links pivotally connected to each other and made in accordance with the invention; 
       FIG. 2  is a plan view of the three links of  FIG. 1  in a partially folded condition; 
       FIG. 3  is a plan view of the three links of  FIG. 1  in a partially extended condition; 
       FIG. 4  is a plan view of the three links of  FIG. 1  in a third angled condition; 
       FIG. 5  is a perspective view of a first embodiment of the linkage assembly of the invention; 
       FIG. 6  is a plan view of the linkage of  FIG. 5 ; 
       FIG. 7  is also a plan view of the linkage assembly of  FIG. 5  and similar to  FIG. 6 ; 
       FIG. 8  is a plan view of the linkage assembly of  FIG. 5  shown in a different position; 
       FIG. 9  is a plan view of the linkage assembly of  FIG. 5  in a folded condition; 
       FIG. 10  is a plan view of the linkage assembly of  FIG. 5  and similar to  FIG. 9 ; 
       FIG. 11  is an exploded perspective view of a second embodiment of the linkage assembly of the invention; 
       FIG. 12  is a perspective view of the linkage assembly of  FIG. 1  in a closed position; 
       FIG. 13  is a perspective view of the assembly of  FIG. 11  in a partially open condition; 
       FIG. 14  is a perspective view of the linkage assembly of  FIG. 11  in a fully open condition; 
       FIG. 15  is a plan view of the linkage assembly of  FIG. 11 ; 
       FIG. 16  is a plan view of the linkage assembly of  FIG. 11  in a partially open condition; 
       FIG. 17  is a plan view of the linkage assembly of  FIG. 11  in a fully open condition; 
       FIG. 18  is a plan view of the linkage assembly of  FIG. 11  in a second partially open condition; 
       FIG. 19  is a plan view of the linkage assembly of  FIG. 11  in a second closed condition; 
       FIG. 20  is a plan view of a third embodiment of a linkage assembly in accordance with the invention; 
       FIG. 21  is a plan view of the linkage assembly of  FIG. 20  in a partially open condition; 
       FIG. 22  is a plan view of the linkage assembly of  FIG. 20  in a fully open condition; 
       FIG. 23  is an exploded perspective view of covering panels suitable for attaching to the linkage assembly of  FIG. 20 ; 
       FIG. 24  is a perspective view showing the covering panels of  FIG. 23  forming a ring; 
       FIG. 25  is a perspective view showing the covering panels of  FIG. 23  in a partially open condition; 
       FIG. 26  shows the covering panels of  FIG. 23  in a fully closed condition; 
       FIG. 27  is a perspective view of another linkage assembly made in accordance with the invention: 
       FIG. 28  is a plan view of the linkage assembly of  FIG. 27  in an extended condition; 
       FIG. 29  is a plan view of the linkage assembly of  FIG. 27  in a partially angled condition; 
       FIG. 30  is a plan view of the linkage assembly of  FIG. 27  in a folded condition; 
       FIG. 31  is a plan view of still another embodiment of the linkage assembly made in accordance with the invention; 
       FIG. 32  is a plan view of the linkage assembly of  FIG. 31  in a partially open condition; 
       FIG. 33  is a plan view of the linkage assembly of  FIG. 31  in a fully open condition; 
       FIG. 34  is a perspective view of yet a further embodiment of the linkage assembly of the invention; 
       FIG. 35  is a plan view of the linkage assembly of  FIG. 34 ; 
       FIG. 36  is a plan view of the linkage assembly of  FIG. 34  in a partially folded condition; 
       FIG. 37  is a plan view of the linkage assembly of  FIG. 34  in an alternative folded condition; 
       FIG. 38  is a plan view of yet another embodiment of a linkage assembly of the invention; 
       FIG. 39  is a plan view of a linkage assembly of  FIG. 38  in a partially closed condition; 
       FIG. 40  is a top plan view of a further embodiment of a linkage assembly of the invention; 
       FIG. 41  is a top plan view showing the linkage assembly of  FIG. 40  in a different position; and 
       FIG. 42  is a top plan view showing the linkage assembly of  FIG. 40  in yet another position. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     FIG. 1  shows a perspective view of mechanical assembly  10  consisting of three links  12 ,  14  and  16  which are pivotally connected to each other end-to-end. Link  12  has two ends  11  and  13  each of which has a geared profile. Similarly the ends of link  16  has two ends  15  and  17  ends each of which have a geared profile. 
   Gear  18  is pivotally connected to link  14 . Gear end  13  engages with gear  18 ; likewise gear end  15  engages with gear  18 . 
     FIG. 2  shows a plan view of assembly  10 . The two dashed lines shown passing through the end pivots of links  12  and  16  form an angle  15  between them. 
     FIG. 3  shows assembly  10  in a different position, links  12  and  16  having been rotated relative to link  14 . Links  12  and  16  form an angle  15  between them, said angle being unchanged from the angle shown in  FIG. 2 . The reason that this angle remains unchanged is because gear  18  synchronizes the relative motion of links  12  and  16 . 
     FIG. 4  shows assembly  10  in a third position. The relative angle between links  12  and  16  remains unchanged. 
     FIG. 5  shows a perspective view of linkage  30  which is comprised of eight links  31 , 32 , 33 , 34 , 35 , 36 , 37  and  38  which are pivotally connected end to end. Each link terminates with two gear ends. Gear  41  is pivotally attached to link  31 . Likewise gears  42 , 43 , 44 , 45 , 46 , 47  and  48  are respectively pivotally attached to links  32 , 33 , 34 , 35 , 36 , 37  and  38 . 
   The gear ends of link  31  engage with gears  48  and  42 . Likewise the gear ends of all the links engage with the gears which are pivotally attached to their neighboring links. 
     FIG. 6  shows linkage  30  in plan view. Links  31  and  33  are pivotally connected to link  32  and their respective gear ends are engaged with gear  42 . The dashed lines passing through the end pivots of links  31  and  33  form an angle  51  between them. Similarly the dashed lines passing through link pairs  33 - 35 ,  35 - 37  and  37 - 31  form angles  52 ,  53  and  54  respectively. 
     FIG. 7  shows linkage  30  in the same view as  FIG. 6 , with angles  55 , 56 , 57 , 58  being formed by the dashed lines passing through link pairs  32 - 34 ,  34 - 36 ,  36 - 38  and  38 - 32  respectively. 
     FIG. 8  shows linkage  30  in a different position. The angle  51  formed by the dashed lines passing through links  31  and  33  is unchanged from  FIG. 6 . Likewise, the similarly formed angles  52 , 53  and  54  are unchanged from those formed in  FIG. 6 . 
     FIG. 9  shows linkage  30  in a folded position. The four angles  51 , 52 , 53  and  54  respectively formed between link pairs  31 - 33 ,  33 - 35 ,  35 - 37  and  37 - 31  are unchanged from those formed in  FIGS. 6 and 7 . 
     FIG. 10  shows linkage  30  in the same view as  FIG. 9 . The four angles  51 ,  52 ,  53  and  54  respectively formed between link pairs  32 - 34 ,  34 - 36 ,  36 - 38  and  38 - 32  are unchanged from those formed in  FIGS. 6 and 7 . 
   Thus linkage  30  demonstrates a key feature of the invention: the relative angle between two links that are each connected to a common link between them, and that are synchronized by a gear that is connected to said common link, will form a constant and unchanging angle for any given position of the linkage. 
   In  FIG. 11 , an assembly  80  is shown in exploded view. Assembly  80  is comprised of eight links having a polygonal profile, one layer comprised of links  61 , 63   65  and  67 ; a second layer comprised of links  62 , 64 , 66  and  68 . Additionally eight gears  71 , 72 , 73 , 74 , 75 , 76 , 77  and  78  are pivotally connected respectively to the eight links. 
     FIG. 12  shows assembly  80  in a closed position whereby links  61 ,  63 ,  65  and  67  form a continuous surface.  FIG. 13  shows assembly  80  in a partially open position and  FIG. 14  shows  80  in its fully opened position. 
     FIG. 15  shows a assembly  80  in plan view. An image of a triangle  81  has been printed on links  61 ,  63 ,  65  and  67 .  FIG. 16  and  FIG. 17  show assembly  80  in a partially opened and fully opened position respectively. 
     FIG. 18  shows assembly  80  in a second partially opened position whereby the links have been rotated past their fully opened position.  FIG. 19  shows assembly  80  in its second closed position. A second image of a square  82  has been printed on links  61 ,  63 ,  65  and  67 . Thus, assembly  80  is shown to have the capability to “flip” between two separate images. 
     FIG. 20  shows an assembly  100  comprised of six links  101 ,  102 , 103 , 104 , 105  and  106  which are pivotally connected end to end. Three gears  112 ,  114  and  116  are pivotally attached to links  102 , 104  and  106  respectively. Angles  91 , 92  and  93  are shown formed between link-pairs  105 - 101 ,  101 - 103  and  103 - 105  respectively. 
     FIG. 21  shows assembly  100  in a partially opened position. The angles between link pairs  105 - 101 ,  101 - 103  and  103 - 105  are respectively  91 , 92  and  93  being unchanged from  FIG. 20 . 
     FIG. 22  shows assembly  100  in a fully opened position. The angles between link pairs  105 - 101 ,  101 - 103  and  103 - 105  are respectively  91 , 92  and  93  being unchanged from  FIGS. 20 and 21 . 
     FIG. 23  shows assembly  100  in exploded view with three covering panels  122 ,  124  and  126  shown.  FIG. 24  shows these covering panels attached to links  102 ,  104  and  106  respectively such that a ring is formed when  100  is in its fully opened position. Said ring has certain flying characteristics for straight sustained flight when thrown in a spinning motion. 
     FIG. 25  shows assembly  100  with its attached covering panels shown in a partially opened position.  FIG. 26  shows assembly  100  in its fully closed position such that it forms the profile of a three pronged shape. Said shape has certain flying characteristics similar to a boomerang when thrown with a spin such that it flies in a loop returning to the thrower. 
     FIG. 27  shows a perspective view of mechanical assembly  120  consisting of three links  122 ,  124  and  126  which are pivotally connected to each other end-to-end. Link  122  has two ends  121  and  123  each of which have an attached pulley. Similarly, the ends of link  126  has two ends  125  and  127  ends each of which have an attached pulley. 
   Belt  130  engages pulley ends  123  and  125 .  FIG. 28  shows a plan view of assembly  120 . The two dashed lines shown passing through the end pivots of links  122  and  126  form an angle  128  between them. 
     FIG. 29  shows assembly  120  in a different position, links  122  and  126  having been rotated relative to link  124 . Links  122  and  126  form an angle  128  between them, said angle being unchanged from the angle shown in  FIG. 28 . The reason that this angle remains unchanged is because belt  130  synchronizes the relative motion of links  122  and  126 . 
     FIG. 30  shows assembly  120  in a third position. The relative angle  128  between links  122  and  126  remains unchanged. 
     FIG. 31  shows a loop assembly  200  comprised of sixteen links, each having two pulley ends, every other link being connected via a belt such that a constant angle  210  is formed between every other link.  FIG. 32  shows assembly  200  in a different position where the angle  210  between every second link is unchanged. 
     FIG. 33  shows assembly  200  in its fully opened position, the angle  210  between every second link remaining unchanged. 
     FIG. 34  shows a perspective view of mechanical linkage assembly  320  consisting of three links  322 ,  324  and  326  which are pivotally connected to each other end-to-end. Link  322  has two ends  321  and  323  each of which have an attached bevel gear. Similarly the ends of link  326  has two ends  325  and  327  ends each of which have a geared an attached bevel gear. 
   Gear assembly  330  comprised of two bevel gears fixed to a common shaft is pivotally connected to link  324  and engages bevel gear ends  323  and  325 .  FIG. 35  shows a plan view of assembly  320 . The two dashed lines shown passing through the end pivots of links  322  and  326  form an angle  328  between them. 
     FIG. 36  shows assembly  320  in a different position, links  322  and  326  having been rotated relative to link  324 . Links  322  and  326  form an angle  328  between them, said angle being unchanged from the angle shown in  FIG. 35 . The reason that this angle remains unchanged is because gear assembly  330  synchronizes the relative motion of links  322  and  326 . 
     FIG. 37  shows assembly  320  in a third position. The relative angle  328  between links  322  and  326  remains unchanged. 
     FIG. 38  shows a linkage or loop assembly  400  comprised of six links, each having two bevel gear ends, every other link being connected via a bevel gear assembly such that a constant angle  410  is formed between every other link. 
     FIG. 39  shows assembly  200  in a different position where the angle  410  between every second link is unchanged. 
     FIG. 40  shows an assembly  500  which is comprised of four links  515 ,  525 ,  535  and  545  each having two gear ends, four links  510 ,  520 ,  530  and  540  having a triangular profile and three pivots each. Four gears  512 ,  522 ,  532  and  542  are pivotally attached to links  510 ,  520 ,  530  and  540  respectively. Assembly  500  further includes a central link  505  which is pivotally connected to the third pivot each of links  510 ,  520 ,  530  and  540 . Central link  505  serves to assist in the synchronization of assembly  500 . 
   An angle  550  is formed between links  515  and  525 . Similarly, angles  560 ,  570  and  580  are formed between link-pairs  525 ,  535 ;  535 ,  545 ; and  545 ,  515  respectively. 
     FIG. 41  shows assembly  500  in a different position. The angle  550  formed between links  515  and  525  is unchanged from  FIG. 40 . Similarly, angles  560 ,  570  and  580  are formed between link-pairs  525 ,  535 ;  535 ,  545 ; and  545 ,  515 , and are unchanged from  FIG. 40 . 
     FIG. 42  shows assembly  500  in another position. Angles  550 ,  560 ,  570  and  580  are formed between link-pairs  515 ,  525 ;  525 ,  535 ,  535 ,  545 ; and  545 , 515 , and are unchanged from  FIGS. 40 and 41 . 
   It will thus be seen that the objects set forth above, among those made apparent from the proceeding description, are efficiently attained, and, in since certain changes may be made in the construction of the inventive structure without departing from the spirit and scope of the invention, it is intended that all matter contained in the above description and shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense. 
   It is also to be understood that the following claims are intended to cover all of the general and specific features of the invention described herein and all statements of the scope of the invention which, as a matter of language, might be said to fall there between.

Summary:
A ring linkage is hereby disclosed that is comprised of at least six links, each link having at least two pivots located proximate to their ends, said links being arranged in a loop whereby each link is pivotally attached via its end pivots to two neighboring links. 
     The motion of the linkage is synchronized by a multiplicity of mechanical elements that serve to synchronize the relative rotation of the links in the assembly such that when a given link rotates by an angle, every second link rotates by the same angle. These synchronizing elements may be either gears, cables or belts, thus the relative angle between the every second link in the ring linkage (as defined by lines passing through their respective end pivots) remains constant and unchanging even as the position of the linkage is changed.