Abstract: A building system composed of space-filling saddle zonohedra which fit together periodically and non-periodically into a variety of orderly and irregular looking space structures. The saddle zonohedra are composed of convex or non-convex saddle polygons and are derived from the infinite class of plane-faced zonohedra. The surfaces of the saddle polygons could be continuously curved surfaces or faceted polygonal surfaces derived from minimal surfaces. The surved surfaces could be flexible membranes or nets in tension used by themselves or in combination with space frames, or stiff shell-like surfaces, and the edges of the space frames could be hinged or fixed to others. Applications of the system include architectural and environmental structures, fixed or retractible space frames, decks or platforms, educational kits or toys, curved tiles, and saddle polyhedra nodes for space frames.

Abstract: A building system composed of space-filling convex and non-convex saddle zonogons which fit together periodically and non-periodically into a variety of orderly and irregular looking space structures including saddle zonohedra, single-, double- and multi-layered structures, multi-directional space-filling structures and curved space labyrinths. The saddle zonogons are derived from the infinite class of plane zonogons, and the saddle zonohedra from the infinite class of plane-faced zonohedra. The surfaces of the saddle polygons could be continuously curved surfaces of faceted polygonal surfaces derived from minimal surfaces. The curved surfaces could be flexible membranes or 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.

Abstract: A family of space frames formed from one or more parallel, planar layers, with each layer comprising a plurality of prismatic nodes arranged in a spaced, non-periodic or periodic, planar array and a plurality of struts coupling adjacent nodes together. Each node has top and bottom surfaces of the same configuration and p side surfaces interconnecting the top and bottom surfaces or any suitable shape derived from this configuration. P is any odd number greater than three or any even number greater than four. Adjacent layers are coupled via interconnecting struts that are perpendicular to each layer or tilted relative to each layer. Successive layers are the same or different. The space frames can also define, locate, and map out tiled surfaces or block-filled volumes.

Abstract: Building structures based on polygonal members with coplanar straight edges and icosahedral symmetry. The building structures comprise a set of ten elementary polygonal members, including four triangular members and six parallelogram-shaped members, that are combined to form tetrahedral, octahedral and half-octahedral or trucated tetrahedral, cuboctahedral and truncated octahedral or rhombohedral and parallelpiped building members that in turn fill a three-dimensional space periodically or non-periodically. The orientation of the building members is such that all edges are parallel to the fifteen two-fold axes of rotation of icosahedral symmetry.

Abstract: Crescent-shaped, polygonal tiles useful for covering walls, floors, ceilings, streets or paths and for producing toys, games and structures. The tiles each have a substantially convex outer edge, a substantially concave inner edge, and at least seven sides forming the edges. In some instances, these sides are straight and of equal length, and in other instances they are not straight but enclose the same area as enclosed by the straight sides of equal length. The tiles, either by themselves or in combination with others, interconnect to fill a plane. Either by themselves or in combination with others, the tiles form mosaics with periodic or non-periodic patterns.