Abstract: This invention describes a 4-dimensional periodic table of elements (4D PT) based on the 4 known quantum numbers of the atom, namely, n (principal), l (azimuthal), m (magnetic) and s (spin), which determine the 4D Cartesian co-ordinates (n,l,m,s) of a 4-dimensional cubic lattice. Since the four quantum number combinations of each element are unique by Pauli's exclusion principle, each chemical element occupies a different vertex of this lattice and has a unique location in 4D space. The 4 quantum numbers are used as independent vectors of a 4-dimensional vector space so that quantum numbers can be added in the same manner as vectors are added. The 4D PT displays complementarity and the principle of zero cyclic sum in numbers associated with elements. The 4D PT of elements and their coordinates extend to chemical molecules and compounds by adding coordinates of individual elements into composite coordinates of molecules and compounds in a larger expansive PT.

Abstract: New designs for a sports ball comprising at least two polygonal panels and having an improved performance and uniformity. Each panel has doubly-curved edges that curve along and across the surface of the sphere. The panels are p-sided curved polygons, where p is an integer greater than 1. The single panels, in an imagined flattened state, have curved edges where each edge curves inwards, outwards or undulates in a wave-like manner. The edges are arranged so each individual panel is without mirror-symmetry and the edge curvatures are adjusted so the panel shape can be varied to achieve more uniform panel stiffness as well as economy in manufacturing. The ball also has a possible shape-induced spin due to the panel design and the overall rotational symmetry of the design.

Abstract: Expandable surfaces made from sheet materials with slits distributed on the surface of sheet material where the surfaces expand by application of force along or/and across the surface of sheet material. The unexpanded surfaces are flat sheets, or closed surfaces like cylinders, spheres, tubes, or custom-designed organic shapes marked with pre-formed or post-formed slit designs. The expanded surfaces can be single units or modules which can be attached to one another through various means. The sheet materials range from hard surfaces like metals, to softer materials like papers and plastics, or pliable materials like fabrics, rubbers, synthetic surfaces or bio-surfaces. The slits are arranged in patterns ranging from periodic, non-periodic to irregular designs. The slits can be straight, bent, curved or irregularly shaped with even or uneven spacing. Slitting can be achieved by digital cutting or punching devices like laser-cutting, water-jet cutting, digital punching, automated dies, etc.

Abstract: New designs for a sports ball comprising at least two polygonal panels and having an improved performance and uniformity. Each panel has doubly-curved edges that curve along and across the surface of the sphere. The panels are p-sided curved polygons, where p is an integer greater than 1. The single panels, in an imagined flattened state, have curved edges where each edge curves inwards, outwards or undulates in a wave-like manner. The edges are arranged so each individual panel is without mirror-symmetry and the edge curvatures are adjusted so the panel shape can be varied to achieve more uniform panel stiffness as well as economy in manufacturing. The ball also has a possible shape-induced spin due to the panel design and the overall rotational symmetry of the design.

Abstract: New designs for a sports ball comprising at least two polygonal panels and having an improved performance and uniformity. Each panel has doubly-curved edges that curve along and across the surface of the sphere. The panels are p-sided curved polygons, where p is an integer greater than 1. The single panels, in an imagined flattened state, have curved edges where each edge curves inwards, outwards or undulates in a wave-like manner. The edges are arranged so each individual panel is without mirror-symmetry and the edge curvatures are adjusted so the panel shape can be varied to achieve more uniform panel stiffness as well as economy in manufacturing. The ball also has a possible shape-induced spin due to the panel design and the overall rotational symmetry of the design.

Abstract: This invention deals with a morphological genome for design applications. This genome encodes all forms. It comprises a finite set of morphological genes, where each gene specifies a distinct group of morphological transformations defined by a group of independent topological, geometric or other parameters. The morph genes and their parameters are mapped within an integrated higher-dimensional framework with each parameter represented along an independent vector in higher-dimensional Euclidean space. Each distinct number associated with a parameter or a group of parameters is represented by a distinct point in this space referenced by its higher-dimensional Cartesian co-ordinates which represent the genetic code for the specific form being mapped. The morph genome can be used as an interactive design tool to generate known and new forms for applications in all design fields as well as for fabricating these forms when linked with digital fabrication devices within an integrated computational environment.

Abstract: New designs for a sports ball comprising at least two polygonal panels and having an improved performance and uniformity. Each panel has doubly-curved edges that curve along and across the surface of the sphere. The panels are p-sided curved polygons, where p is an integer greater than 1. The single panels, in an imagined flattened state, have curved edges where each edge curves inwards, outwards or undulates in a wave-like manner. The edges are arranged so each individual panel is without mirror-symmetry and the edge curvatures are adjusted so the panel shape can be varied to achieve more uniform panel stiffness as well as economy in manufacturing. The ball also has a possible shape-induced spin due to the panel design and the overall rotational symmetry of the design.

Abstract: This invention deals with an integrated morphological system, herein called a morphological genome (morph genome), for design applications. This is the genome that encodes all form and is similar in intent to the biological genome that encodes all living things by its genetic code. The morph genome comprises a finite set of morphological genes, each gene specifies a distinct group of morphological transformations, each group of transformations is defined by a group of independent topological, geometric or other parameters. The morph genes and their parameters are mapped within an integrated higher-dimensional framework, with each parameter being represented along a vector in higher-dimensional Euclidean space. Each distinct number associated with a parameter or a group of parameters is represented by a distinct point in this space and is referenced by the higher-dimensional Cartesian co-ordinates of that point. This space uses a combination of discrete and continuous values for these parameters.

Abstract: Expandable surfaces made from sheet materials with slits distributed on the surface of sheet material where the surfaces expand by application of force along or/and across the surface of sheet material. The unexpanded surfaces are flat sheets, or closed surfaces like cylinders, spheres, tubes, or custom-designed organic shapes marked with pre-formed or post-formed slit designs. The expanded surfaces can be single units or modules which can be attached to one another through various means. The sheet materials range from hard surfaces like metals, to softer materials like papers and plastics, or pliable materials like fabrics, rubbers, synthetic surfaces or bio-surfaces. The slits are arranged in patterns ranging from periodic, non-periodic to irregular designs. The slits can be straight, bent, curved or irregularly shaped with even or uneven spacing. Slitting can be achieved by digital cutting or punching devices like laser-cutting, water-jet cutting, digital punching, automated dies, etc.

Abstract: A method for bending sheet metal includes introducing to the sheet metal thinned regions which are positioned either along or immediately adjacent to a bending line. These thinned regions allow the metal to be easily bent along the bending line using conventional hand tools or non-specialized machines. The thinned regions may be shaped as slots having a specific width, length, end shape, spacing from each adjacent slot, and depth into the metal sheet. According to one embodiment of the invention, each slot is cut through the entire thickness of the metal sheet. Other related embodiments require that the slots be only partially cut or etched thereby having a depth that is less than the thickness of the metal sheet. The thinned regions may be any appropriate shape as controlled by the shape of the bend, the type of metal, the thickness of the metal, the ductility of the metal, the angle of the bend, and the application of the metal (e.g., load bearing, etc).

Abstract: A method for bending sheet metal includes introducing to the sheet metal thinned regions which are positioned either along or immediately adjacent to a bending line. These thinned regions allow the metal to be easily bent along the bending line using conventional hand tools or non-specialized machines. The thinned regions may be shaped as slots having a specific width, length, end shape, spacing from each adjacent slot, and depth into the metal sheet.

Abstract: A class of undulated building structures constructed from sheets of suitable material and composed of sinuous curved faces meeting at constant angle at sinuous curved edges. The curved faces ensure a deformation-free (and hence a strain-free) surface, a critical feature for large-scale architectural constructions from sheet materials. The structures can be constructed by scoring a single sheet and folding the entire structure or from parts which are scored and folded. A variety of 2-dimensional scoring designs and a morphological technique for their derivation is disclosed. The technique enables the generation of symmetric and asymmetric 3-dimensional developable structures with combinations of convex and concave angle folds. The technique also provides the enabling feature for forming by various manufacturing methods. The structures can also be constructed from cast materials.

Abstract: A family of non-convex and convex tiles which can be tiled together to fill a planar surface in a periodic or non-periodic manner. The tiles are derived from planar space frames composed of a plurality of regular p-sided polygonal nodes coupled by a plurality of struts. p is any odd number greater than three and an even number greater than six. The nodes and struts, along with the areas bounded by them, make up a tiling system. In addition, the lines joining the along the center lines of the struts define a large family of convex and non-convex tiles. The convex tiles include zonogons, and the non-convex tiles include tiles with one or more concave vertices including singly-concave, bi-concave (doubly-concave), multiply-concave and S-shaped tiles. The tiles can be converted to 3-dimensional space-filling blocks. When these blocks are hollow and inter-connected, architectural environments are possible.

Abstract: Nine different classes of polyhedral structures based on surface subdivisions of the infinite class of plane-faced zonohedra. The structures fit together periodically or non-periodically into a variety of orderly and irregular-looking space structures including polyhedra with plane or curved faces, single-, double-, multi-layered and multi-directional space-filling building systems, plane-faced and curved space labyrinths. The surfaces of curved faces could be continuously curved surfaces, or facetted polygonal surfaces or minimal surfaces. The curves surfaces could be flexible nets in tension used by themselves or in combination with space frames, or stiff shell-like surfaces, and the edges could be hinged or fixed to others. The plane faces could be panels or folded plates, and the straight edges could be converted into struts, and the vertices into nodes.

Abstract: A family of convex and non-convex tiles which can be tiled together to fill a planar surface in a periodic or non-periodic manner. The tiles are derived from planar space frames composed of a plurality of regular p-sided polygonal nodes coupled by a plurality of struts. p is any odd number greater than three and an even number greater than four. The nodes and struts, along with the areas bounded by them, make up a tiling system. In addition, the lines joining the along the center lines of the struts define a large family of convex and non-convex tiles. The convex tiles include zonogons, and the non-convex tiles include tiles with one or more concave vertices. The latter comprise singly-concave, bi-concave and S-shaped tiles. The tiles can be converted to 3-dimensional space-filling blocks. When these blocks are hollow and inter-connected, architectural environments are possible.

Abstract: A family of space structures having subdivided faces, where such faces are subdivided into rhombii in non-periodic arrangements. The rhombii are derived from regular planar stars with n vectors, and the source space structures are composed of regular polygons. The family includes: globally symmetric structures where the fundamental region is subdivided non-periodically, or globally asymmetric structures composed of regular polygons which are subdivided non-periodically or asymmetrically. The rhombii can be further subdivided periodically or non-periodically. The family further includes all regular polyhedra in the plane-faced and curve-faced states, regular tessellations, various curved polygons, cylinders and toroids, curved space labyrinths, and regular structures in higher-dimensional and hyperbolic space. The structures can be isolated structures or grouped to fill space.

Abstract: A family of space frame systems composed of a plurality of nodes coupled by struts and derived from a family of non-regular polyhedra by joining the center of these polyhedra to their faces, edges and vertices. The space frames permit periodic, non-periodic, random and irregular building configurations. The building system can be combined with panels, tensile and membranes systems, or can be converted into plate systems or nodeless space frame systems. The spaces and configurations defined by the building system include single-layered, double-layered, multi-layered configurations, non-layered and multi-directional configurations, polyhedral packings and space-fillings, infinite polyhedra, and various 3-dimensional projections of n-dimensional polytopes for architectural environments. The n-dimensional polytopes include the infinite classes of hyper-cubes and hyper-cubic lattices, and a variety of 4-dimensional polytopes.

Abstract: Families of periodic space structures, configurations, labyrinths and space frames based on seven different known periodic space grids composed of one edge-length and even-sided polygons. The space structures have two different polyhedra corresponding to the vertices of the source grids. The space labyrinths are characterized by a continuous surface having two types of vertices and dividing space into two parts. The space frames have one type and length of strut and two different shapes of nodes. The structures have cross-disciplinary applications in the design and scientific fields. In architecture and design fields, applications for building systems one earth or outer space, architectural space layouts, environmental sculptures, playground structures, honeycomb structures, toys, puzzles, model-kits, etc. are possibilities.

Abstract: Families of node shapes based on prismatic symmetry for space frame constructions. The node shapes include various polyhedral, spherical, elipsoidal, cylindrical or saddle shaped nodes derived from polygonal prisms and its dual. The node shapes are determined by strut directions which are specified by various directions radiating from the center of a regular prism of any height. A plurality of such nodes is used in single-, double- or multi-layered space frames or space structures where the nodes are coupled by a plurality of struts in periodic or non-periodic arrays. The space frames are suitably triangulated for stability. Applications include a variety of architectural structures and enclosures for terrestrial or (outer) space environments. Suitable model-building kits, toys and puzzles are also possible based on the invention.

Abstract: This application discloses a tiling system for surfaces where the pattern of the tiling changes continuously from one portion of the tiling to another in an Escher-like metamorphoses with the difference the the metamorphoses are based on binary combinations of n transformations on the edges of the tile. Accordingly, the tiling is obtained from the n directions of the edges of an underlying zonohedron, a polyhedron derived as a projection of an n-dimensional cube. The zonohedron provides a hidden network for the continuous transformations of the tiles to one another. The derived designs utilize 3- and 4-sided polygons and have a variety of curved edges in and across the plane of the tile. The metamorphic designs provide visually attractive alternatives to periodic patterns used as architectural surfaces, walls, floors, ceilings, window screens and dividers, architectural space enclosures, visual art, textile designs and computer graphics amongst other varied applications.