Abstract:
Two three-dimensional shapes are described. The shapes may be used together both for the working of space-relational puzzles and for the working of many and various tile-like designs.

Description:
BACKGROUND AND SUMMARY 
     This invention relates to puzzle pieces which have either a nature to tile non-periodic geometric designs against a flat surface or to spatially interrelate as an integrated and coherent three-dimensional structure. 
     The simpler of the piece shapes comprises and conforms closely to a polyhedron having six surfaces, one of which surfaces is a concave polygon of five equal sides. Each of the other five surfaces borders a different one of the five sides and together make five dihedral angles with the polygon. Four of these dihedrals measure about 63 degrees each. The fifth is about half of that measure and is located opposite the two divergent sides of the polygon. 
     The other piece shape is in ways similar to the first and conforms closely to a lengthier polyhedron of two more surfaces than the earlier polyhedron. The six surfaces described for that earlier polyhedron are represented in the second polyhedron, more or less preserving all angular relationships. The second polyhedron does differ though in that the surfaces which preserve the corresponding fifth dihedral angle are, by comparison, expanded lengthwise and extended widthwise at midsections while the two surfaces corresponding opposite of this angle are seen to be non-adjacent to include between them instead the seventh and eighth surfaces, these latter four surfaces making, furthermore, parallel pairs of surfaces. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     The invention will be understood more fully from the following description, with reference to the accompanying drawings wherein: 
     FIG. 1 illustrates the geometry of the larger, or bottom, surface of the simpler piece shape; 
     FIGS. 2a-c illustrate the shape of the simpler piece, with FIG. 2a an end view, FIG. 2b a rotated view, FIG. 2c a plan view of the shape; 
     FIGS. 3a-c illustrate the shape of the second piece, using similar type views as were used in FIGS. 2a-c; 
     FIG. 4 illustrates a geometric design which can be duplicated by the puzzle pieces; 
     FIG. 5 illustrates a three-dimensional puzzle which can be assembled also using the puzzle pieces. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     FIG. 1 illustrates the concave polygon F6 and its five equal sides 1,2,3,4 and 5 which comprise the back, or bottom surface F6 of polyhedron V FIGS. 2a-2c, to which polyhedron shape closely conforms the shape of the simpler puzzle piece herein described. The divergent sides (2 and 3) of the polygon impose on diagonals D1 and D2 from opposite side 5. In FIG. 2c the four surfaces F1, F2, F3 and F4 each hold 63 degree dihedral relationships with the surface F6 (FIG. 2b) while at edge 5 the dihedral of surface F5 with F6 compares to about half the measure. Altogether this substantially locates the remaining surface adjacencies, or edges, J1,J2,J3,J4 and J5. For better ease of identification of that latter dihedral angle associated with edge 5 it may be referred to herein as simply &#34;the fifth dihedral angle&#34;. 
     Polyhedron V is represented in polyhedron W of FIGS. 3a-3c through corresponding surfaces G1,G2,G3,G4,G5 and G6 for which exists an equivalent set of dihedral angles as were described for polyhedron V. The surface adjacencies, or edges, 21,22,23,24,25,K1,K3,K4,K5,K6 and K7 of polyhedron W corrrespond respectively to adjacencies, or edges, 1,2,3,4,5,J1,J3,J4,J5,J6 and J7 of polyhedron V. Polyhedron W further includes seventh and eighth surfaces G7 and G8 between surfaces G3 and G2 to make respective pairs of parallel surfaces plus also new adjacencies or edges 26,27 and 28. The surfaces G5 and G6 of polyhedron W are, in comparison to their counterparts in polyhedron V, expanded uniformly lengthwise and extended widthwise to make agreeable edges 29,30,31 and 32 for the inclusion, or rather perhaps insertion, of those seventh and eighth surfaces G7 and G8 of polyhedron W. The two polyhedrons as such are then equivalent overlooking the described expansion, extension, and insertion upon the midsection of polyhedron W. 
     FIG. 4 is of a somewhat suggestive, tile-like design substantially duplicable using puzzle pieces of the shapes previously described, and is also substantially one of many different possible designs which might occur in the course of play activity with puzzle pieces of these types. The method involved here places the largest surface of each piece, corresponding to either surface F6 or G6 of previous illustration, upon a useable tabletop or the like. Substantial angular compatabilities of these surfaces and equivalent edge lengths allow them to closely interlay about one another for characteristically non-periodic and often highly differentiable geometric designs. 
     FIG. 5 illustrates a more or less integrated and coherent assemblage comprising a substantial denominator in an endless series of possible three-dimensional puzzles which could occur also out of play activity with the described puzzle pieces. Removed from combination to reveal its orientation is piece V.