n-fold Hyperbolic Paraboloids and Related Structures

Abstract

Utilizing the three different hyperbolic paraboloids which divide each of the defining tetrahedra (DT) in half, new geometric objects have been created which have unique stacking and interlocking characteristics and are inherently strong and rigid due to their triangular and hyperbolic paraboloid faces. These geometric objects can be utilized to build aesthetic and utilitarian components such as beams, trusses, packaging foams, toys, repeating cellular structures and others. The n-fold hyperbolic paraboloids are new geometric objects. The “n-” in the title stands for any integer greater than or equal to three. The fourfold hyperbolic paraboloid (FIG. 2 C) has the special attribute of being space filling. Like the cube the fourfold hyperbolic paraboloids can be continuously stacked so that there is no unenclosed volume between them. All of the n-fold hyperbolic paraboloids have unique stacking and interlocking attributes. The interlocking and stacking characteristics of these objects result from the saddle shaped compound curvature of the DT hyperbolic paraboloids. Three new three faced geometric objects or trihedrons (FIGS. 3 A 3 B, and 3 G) have been created from each DT, each has two DT isosceles triangular faces with the third face being one of the three hyperbolic paraboloids which divide the DT in half. These trihedrons have stacking, interlocking, and strength and rigidity characteristics similar to the n-fold hyperbolic paraboloids. Thickening the surfaces of the three hyperbolic paraboloids which divide the DT in half (FIGS. 4 B, 4 E, and 4 G) results in new thin geometric objects specifically defined by the DT. These new thin geometric objects can be used to build repeating cellular structures which effectively harness material properties to result in inherent strength and rigidity due to the hyperbolic paraboloid shape of the cell walls. They can also be applied to appurtenances of other geometric objects such as square bars, spheres, etc such that these other geometric objects can be joined in an interlocking fashion

1 . A family of geometric objects (e.g.  FIGS. 2A ,  2 C,  2 F, and  2 H) comprising sections of n hyperbolic paraboloid surfaces arranged symmetrically about an axial axis with curvature of said hyperbolic paraboloid surfaces defined by the arrangement of the four L1 edges of the defining tetrahedron wherein said n is any integer equal to or greater than three wherein said sections of n hyperbolic paraboloid surfaces can be extended to intersect at 2n edges each of length and position predetermined by said L1 edges of said defining tetrahedrons wherein said n sections of hyperbolic paraboloid surfaces can be extended to result in said hyperbolic surfaces intersecting at two axial and said n circumferential vertices whereby multiple units of said geometric objects being stackable in an interlocking fashion. 
     
     
         2 . The geometric object of  claim 1  wherein the interior of said geometric objects is solid, is a void, or is composed of a frame or cellular structure. 
     
     
         3 . The geometric object of  claim 1  wherein said sections of n hyperbolic paraboloid surfaces have been extended to intersect at said 2n edges of length and position predetermined by said L1 edges of said defining tetrahedrons wherein said sections of n hyperbolic paraboloid surfaces have been extended to intersect at two axial and n circumferential vertices. 
     
     
         4 . The geometric object of  claim 3  wherein said 2n L1 edges and/or said axial vertices, and/or said n circumferential vertices, and/or said sections of n hyperbolic surfaces have been modified to provide means for attaching said geometric object to adjacent said geometric objects or other structural components. 
     
     
         5 . A geometric object ( FIG. 3A ) comprising two planar defining tetrahedron isosceles triangular faces and a third face consisting of a section of a hyperbolic paraboloid surface with the curvature of said section of a hyperbolic paraboloid surface being defined by the arrangement of the four L1 edges of said defining tetrahedron wherein said section of a hyperbolic paraboloid surface and said two planar defining tetrahedron isosceles triangular faces can be extended to said four L1 edges of length and position predetermined by said defining tetrahedron wherein said two planar isosceles triangular face extension results in a fifth edge which is an L2 edge of said defining tetrahedron and said fifth edge is not a part of said section of a hyperbolic paraboloid surface whereby multiple units of said geometric object are stackable in an interlocking fashion at the interface of said hyperbolic paraboloid surfaces and stackable at the interfaces of said two planar defining tetrahedron isosceles triangular faces. 
     
     
         6 . A geometric object ( FIG. 3B ) comprising two planar defining tetrahedron isosceles triangular faces and a third face consisting of a section of a right handed hyperbolic paraboloid surface with the curvature of said section of a right handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of said defining tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a right handed sense wherein said two planar defining tetrahedron isosceles triangular faces and said section of a right handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron wherein the extension of said two planar defining tetrahedron isosceles triangular faces results in a fifth edge wherein said fifth edge which is an L1 edge of length and position predetermined by said defining tetrahedron and said fifth edge is not a part of said section of a right handed hyperbolic paraboloid surface whereby multiple units of said geometric object are stackable in an interlocking fashion at the interface of said right handed hyperbolic paraboloid surfaces and stackable at the interfaces of said two planar defining tetrahedron isosceles triangular faces. 
     
     
         7 . A geometric object ( FIG. 3C ) comprising two planar defining tetrahedron isosceles triangular faces and a third face consisting of a section of a left handed hyperbolic paraboloid surface with the curvature of said section of a left handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of said defining tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a left handed sense wherein said two planar defining tetrahedron isosceles triangular faces and said section of a left handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron wherein the extension of said two planar defining tetrahedron isosceles triangular faces results in a fifth edge wherein said fifth edge which is an L1 edge of length and position predetermined by said defining tetrahedron and said fifth edge is not a part of said section of a left handed hyperbolic paraboloid surface whereby multiple units of said geometric object are stackable in an interlocking fashion at the interface of said left handed hyperbolic paraboloid surfaces and stackable at the interfaces of said two planar defining tetrahedron isosceles triangular faces. 
     
     
         8 . The geometric object of  claims 5 ,  6 , and  7  wherein the interior of said geometric object is solid or is composed of a frame or cellular structure. 
     
     
         9 . The geometric object of  claims 5 ,  6 , and  7  ( FIGS. 3A ,  3 B,  3 C) wherein said two planar defining tetrahedron isosceles triangular faces and said section of hyperbolic paraboloids have been extended to intersect at five edges of length and position predetermined by said defining tetrahedron. 
     
     
         10 . The geometric object of  claims 5 ,  6 , and  7  ( FIGS. 3A ,  3 B,  3 C) wherein said L1 edges and/or said L2 edges and/or said sections of hyperbolic paraboloids and/or said two planar defining tetrahedron isosceles triangular faces have been modified for design purposes e.g. to provide means for attaching said geometric objects to adjacent said geometric objects or other structural components. 
     
     
         11 . A geometric object ( FIG. 4B ) comprising a section of a thickened hyperbolic paraboloid surface with the curvature of said section of a thickened hyperbolic paraboloid surface predetermined by the arrangement of the four L1 edges of the defining tetrahedron wherein said section of a thickened hyperbolic paraboloid surface can be extended to said four L1 edges of length and position predetermined by said defining tetrahedron whereby multiple units of said sections of thickened hyperbolic paraboloid surfaces can be connected at their edges and/or corners. 
     
     
         12 . A geometric object ( FIG. 4E ) comprising a section of a thickened right handed hyperbolic paraboloid surface with the curvature of said section of a thickened right handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of the defining tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a right handed sense wherein said section of a thickened right handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron whereby multiple units of said sections of thickened right handed hyperbolic paraboloid surfaces can be connected at their edges and/or corners. 
     
     
         13 . A geometric object ( FIG. 4G ) comprising a section of a thickened left handed hyperbolic paraboloid surface with the curvature of said section of a thickened left handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of the space filling tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a left handed sense wherein said section of a thickened left handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron whereby multiple units of said sections of thickened left handed hyperbolic paraboloid surfaces can be connected at their edges and/or corners. 
     
     
         14 . The geometric objects of  claims 11 ,  12 , and  13  ( FIGS. 4B ,  4 E,  4 G) wherein said sections of thickened hyperbolic surfaces have been extended to said L2 and said L1 edges of length and position predetermined by said defining tetrahedron. 
     
     
         15 . The geometric objects of  claims 11 ,  12 , and  13  ( FIGS. 4B ,  4 E,  4 G) wherein said L1 edges and/or said L2 edges and/or corners and/or said sections of thickened hyperbolic paraboloid surfaces have been modified for design purposes e.g. to provide means for attaching said geometric objects to adjacent said geometric objects or other structural components.