Abstract:
An invention and method for producing ttetragonal prism shaped puzzle pieces that may have on their rectangular faces cubic projections. The pieces are grouped according to the volume of the tetragonal prism, which can have a stack of five, four, three, two or one cubic projection(s). The pieces are further differentiated according to the number of rectangular faces with cubic projections and their arrangement. A production method for making a wide variety of geometric or irregular objects consists of selecting pieces, repeated if necessary, and assembling the pieces into the form of a desired object. Then the pieces forming the periphery of the desired object are cut from the assembly of pieces. This provides the base pattern for the pieces necessary to make the desired object. Thereafter, mechanisation of the pieces allows for mass production of a wide variety of three-dimensional puzzles with interior pieces and detailed exterior form.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This is a PCT National Stage Application. Priority is claimed from foreign (Spain) application ES  200201373  U (WIPO WO 03101559), May 29, 2002. 
     
    
     BACKGROUND  
       [0002]     The present version of the invention is generally related to pieces for three-dimensional construction games. Construction games, jigsaws and 3-D puzzles are included within this category.  
         [0003]     Traditionally in this field there are jigsaw puzzle pieces which are constructed by assembling pieces that in turn can be used to produce geometric shapes: cubes, stars, pyramids, spheres, etc (normally geometric or regular shapes). There are also “3D-puzzles”, which consist of pieces similar to those of flat puzzles, but sturdier and, when assembled, reproduce the surface of a three-dimensional figure. Generally, these puzzles have no pieces on the inside. There are also puzzle pieces that belong to so called “construction games”, which can fit together and can lead to reproductions of three-dimensional figures (irregular or geometric), with a certain approximation of their outline.  
         [0004]     These puzzles and their related pieces do not have interior pieces as well as exterior ones. Also, these puzzles do not provide for representation of a constructed irregular object with precision in the outline and shape.  
       SUMMARY  
       [0005]     The present version of the invention is directed to pieces and a method capable of reproducing by means of assembling three-dimensional pieces, a great variety of three-dimensional shapes (not just geometric) with interior pieces as well as exterior ones and with precision in the outline of the constructed object.  
         [0006]     The disclosure describes a version of the invention made up of 27 pieces that are tetragonal prisms with cubic projections on the rectangular faces of each prism. When these pieces are chosen (using repetition of pieces, as necessary), assembled and fitted together, the pieces form multiple objects, like cubes, tetragonal prisms, and staggered pyramids.  
         [0007]     The disclosure also describes a manufacturing process for mechanizing the construction of the pieces to be assembled. First pieces are combined and put together to form the body of the desired shape. The pieces of the assembly are cut to reproduce the surface of the solid three-dimensional shape. The result is a three-dimensional puzzle (geometric or irregularly shaped) with i) three-dimensional pieces ii) a detailed outline of the desired exterior shape and iii) with interior pieces.  
         [0008]     The pieces are formed in the shape of tetragonal prisms with cubic projections on their rectangular faces. The volume of the tetragonal prism can be used to differentiate between different groupings of pieces. For example, the cubic extensions, or cubes, that form the backbone of the tetragonal prism (“cubeoid”) may have a volume equivalent to the superposition of 5, 4, 3, 2, or 1 cubic extension(s) (i.e., cubes).  
         [0009]     Within these five groups, the pieces are differentiated by the number of cubic projections (cubes) on the available rectangular faces of the cubeoid. There can be cubes on 4 of the rectangular faces of the cubeoid; on 3 of the rectangular faces of the cubeoid; on 2 of the adjacent faces of the cubeoid; on 2 of the non-adjacent faces of the cubeoid; on 1 of the faces of cubeoid or none. The cubic extensions (i.e., cubes) are placed symmetrically on the different rectangular faces in a way so that the cubic extensions (i.e., cubes) are all situated at the same height on the tetragonal prism.  
         [0010]     This means that the cubic projections of one piece fit together with the empty spaces between the cubic projections of another piece, and so when the pieces are put together, the pieces form compact and cohesive shapes. 
     
    
     DETAILED DESCRIPTION OF THE DRAWINGS  
       [0011]     These and other features, aspects, and advantages of the present version of the invention will become better understood with regard to the following description, appended claims, and accompanying drawings where:  
         [0012]      FIG. 1  is a tetragonal prism ( 31 ) that is a cubeoid of five cubes and is designated as being the 1 piece.  
         [0013]      FIG. 2  is a tetragonal prism ( 32 ) of equal volume to piece  1 , with two cubic projections ( 33  and  34 ) situated in one of its rectangular faces and is designated as the 2 piece.  
         [0014]      FIG. 3  is a tetragonal prism ( 35 ) of equal volume to pieces  1  and  2  with two pairs of cubic projections situated respectively in two of its adjacent rectangular faces ( 36 ,  37 ,  39  and  39 ), and is designated as being the 3 piece.  
         [0015]      FIG. 4  is a tetragonal prism ( 40 ) of equal volume to shapes  1 ,  2  and  3 , with three pairs of cubic projections situated respectively in three of its rectangular faces ( 41 ,  42 ,  43 ,  44 ,  45 , and  46 ), and is designated in the described version of the invention as being the 4 piece.  
         [0016]      FIG. 5  is a tetragonal prism ( 47 ) of equal volume to shapes  1 ,  2 ,  3  and  4 , with two pairs of cubic projections ( 48 ,  49 ,  50  and  51 ) situated respectively side by side two of its non-adjacent rectangular faces, and is designated in the described version of the invention as being the 5 piece  
         [0017]      FIG. 6  is a tetragonal prism ( 52 ) of equal volume to shapes  1 ,  2 ,  3 ,  4  and  5 , with four pairs of cubic projections ( 53 ,  54 ,  55 ,  56 ;  57 ,  58 ,  59  and  60 ) situated respectively on its four rectangular faces, and is designated in the described version of the invention as being the 6 piece  
         [0018]      FIG. 7  is a tetragonal prism ( 61 ) of equal volume to the superposition of four cubic projections, and is designated in the described version of the invention as being the 7 piece  
         [0019]      FIG. 8  is a tetragonal prism ( 62 ) of equal volume to shape  7  with two cubic projections ( 63  and  64 ) situated on one of its rectangular faces and is designated in the described version of the invention as being the 8 piece  
         [0020]      FIG. 9  is a tetragonal prism ( 65 ) of equal volume to shapes  7  and  8 , with two pairs of cubic projections ( 66  and  67 ;  68  and  69 ) situated respectively on two of its adjacent rectangular faces, and is designated in the described version of the invention as being the 9 piece  
         [0021]      FIG. 10  is a tetragonal prism ( 70 ) of equal volume to shapes  7 ,  8  and  9 , with two pairs of cubic projections ( 71  and  72 ;  73  and  74 ) situated respectively on two of its non-adjacent rectangular faces, and is designated in the described version of the invention as being the 10 piece  
         [0022]      FIG. 11  is a tetragonal prism ( 75 ) of equal volume to shapes  7 ,  8 ,  9  and  10 , with three pairs of cubic projections ( 76  and  77 ,  78  and  79 ,  80  and  51 ), situated respectively on three of its rectangular faces, and is designated in the described version of the invention as being the 11 piece  
         [0023]      FIG. 12  is a tetragonal prism ( 82 ) of equal volume to shapes  7 ,  8 ,  9 ,  10  and  11 , with a pair of cubic projections ( 83 ,  84 ,  85 ,  86 ,  87 ,  88 ,  89  and  90 ) on each of its four rectangular faces, and is designated in the described version of the invention as being the 12 piece  
         [0024]      FIG. 13  is a tetragonal prism ( 91 ) of equal volume to the superposition of three cubic projections, and is designated in the described version of the invention as being the 13 piece  
         [0025]      FIG. 14  is a tetragonal prism ( 92 ) of equal volume to shape  13  with two cubic projections ( 93  and  94 ) situated in one of its rectangular faces, and is designated in the described version of the invention as being the 14 piece  
         [0026]      FIG. 15  is a tetragonal prism ( 95 ) of equal volume to shapes  13  and  14 , with two pairs of cubic projections ( 96 ,  97 ,  98  and  99 ) situated respectively on two of its adjacent rectangular faces, and is designated in the described version of the invention as being the 15 piece  
         [0027]      FIG. 16  is a tetragonal prism ( 100 ) of equal volume to shapes  13 ,  14  and  15 , with two pairs of cubic projections ( 101 ,  102 ,  103  and  104 ) situated respectively on two of its non-adjacent rectangular faces, and is designated in the described version of the invention as being the 16 piece  
         [0028]      FIG. 17  is a tetragonal prism ( 112 ) of equal volume to shapes  13 ,  14 ,  15  and  16 , with three pairs of cubic projections ( 106 ,  107 ,  108 ,  109 ,  110  and  111 ) situated respectively on three of its rectangular faces, and is designated in the described version of the invention as being the 17 piece  
         [0029]      FIG. 18  is a tetragonal prism ( 112 ) of equal volume to shapes  13 ,  14 ,  15 ,  16  and  17 , with four pairs of cubic projections ( 113 ,  114 ,  115 ,  116 ,  117 ,  118 ,  119  and  120 ) situated respectively on its four rectangular faces, and is designated in the described version of the invention as being the 18 piece  
         [0030]      FIG. 19  is a tetragonal prism ( 1121 ) of equal volume to shapes  13 ,  14 ,  15 ,  16 ,  17  and  18 , with four cubic projections ( 122 ,  123 ,  124  and  125 ) situated respectively on its four rectangular faces, and is designated in the described version of the invention as being the 19 piece  
         [0031]      FIG. 20  is a tetragonal prism ( 126 ) of equal volume to shape  19 , with three cubic projections ( 127 ,  128  and  129 ) situated respectively on three of its rectangular faces, and is designated in the described version of the invention as being the 20 piece  
         [0032]      FIG. 21  is a tetragonal prism ( 130 ) of equal volume to shapes  19  and  20 , with two cubic projections ( 131  and  132 ) situated respectively on two of its adjacent rectangular faces, and is designated in the described version of the invention as being the 21 piece  
         [0033]      FIG. 22  is a tetragonal prism ( 133 ) of equal volume to shapes  19 ,  20  and  21 , with two cubic projections ( 134 ,  135 ) situated respectively on two of its non-adjacent faces, and is designated in the described version of the invention as being the 22 piece  
         [0034]      FIG. 23  is a tetragonal prism ( 136 ) of equal volume to shapes  19 ,  20 ,  21  and  22 , with a cubic projection ( 137 ) situated on one of its rectangular faces, and is designated in the described version of the invention as being this the 23 piece  
         [0035]      FIG. 24  is a tetragonal prism ( 142 ) of equal volume to the superposition of two cubic projections, and is designated in the described version of the invention as being this the 24 piece  
         [0036]      FIG. 25  is a tetragonal prism ( 142 ) of equal volume to shape  24  with a cubic projection ( 143 ) situated on one of his rectangular faces, and is designated in the described version of the invention as being the 25 piece  
         [0037]      FIG. 26  is a tetragonal prism ( 139 ) of equal volume to shapes  24  and  25 , with two cubic projections ( 140  and  141 ) situated respectively on two of its adjacent rectangular faces, and is designated in the described version of the invention as being the 26 piece  
         [0038]      FIG. 27  is a cube ( 144 ) of equal volume to a cubic projection, and is designated in the described version of the invention as being the 27 piece  
         [0039]      FIG. 28  is a cube made from the assembling of various pieces: six 2 pieces, eight 3 pieces, nine 4 pieces, eight 5 pieces, two 6 pieces, seven 8 pieces, fifteen 9 pieces, two 10 pieces, six 11 pieces, two 12 pieces, nine 14 pieces, seven 15 pieces, five 16 pieces, one 17 piece, two 24 pieces, two 25 pieces and one 26 piece.  
         [0040]      FIG. 29 : The six faces of the same cube in  FIG. 28  laid out flat  
         [0041]      FIG. 30 : The result of processing the pieces from the same cube of  FIG. 28  or  FIG. 29  to produce the form of a triangular based prism.  
     
    
     DETAILED DESCRIPTION  
       [0042]     The current version of the described version of the invention is 27 pieces that are tetragonal prisms with cubic projections on the rectangular faces of each prism. When these pieces are assembled and fitted together, they form multiple shapes, such as cubes, tetragonal prisms, and staggered pyramids.  
         [0043]     The 27 pieces that are tetragonal prisms have one to thirteen equal cubes. Each of the tetragonal prisms has a backbone of one or more equal cubes, hereinafter called a “cubeoid.” The cubeoid of a tetragonal prism is one of the subsets of five equal cubes: 
        a) a stack of five equal cubes ( 1 ,  2 ,  3 ,  4 ,  5  and  6 );     b) a stack of four equal cubes ( 7 ,  8 ,  9 ,  10 ,  11  and  12 );     c) a stack of three equal cubes ( 13 ,  14 ,  15 ,  16 ,  17 ,  18 ,  19 ,  20 ,  21 ,  22  and  23 );     d) a stack of two equal cubes ( 24 ,  25  and  26 ) or;     e) one ( 27 ) cube.        
 
         [0049]     Furthermore, the tetragonal prism is further categorized into one of six possible sub-groups defined by the possible arrangement of cubes on the four sides (excluding the top and bottom side) of the cubeoid: 
        a) all four sides have additional cubes ( 6 ,  12 ,  18 ,  19 ),     b) three sides have additional cubes ( 4 ,  11 ,  17 ,  20 ),     c) two adjacent sides have additional cubes ( 3 ,  9 ,  15 ,  21 ,  26 ),     d) two non-adjacent sides have additional cubes ( 5 ,  10 ,  16 ,  22 ),     e) one side has additional cubes ( 2 ,  8 ,  14 ,  23 ,  25 ), or     f) no side has additional cubes ( 1 ,  7 ,  13 ,  24 ,  27 ).        
 
         [0056]     Another way to describe the sub-grouping of pieces is that within the five primary groups, the pieces are differentiated by the number of rectangular faces on the tetragonal prism that have cubic projections. The number of faces that have cubic projections can be 4 (pieces  6 ,  12 ,  18 ,  19 ); 3 (pieces  4 ,  11 ,  17 ,  20 ); 2 adjacent faces (pieces  3 ,  9 ,  15 ,  21 ,  26 ); 2 non-adjacent (pieces  5 ,  10 ,  16 ,  22 ); 1 (pieces  2 ,  8 ,  14 ,  23 ,  25 ); or none ( 1 ,  7 ,  13 ,  24 ,  27 ).  
         [0057]     The cubic projections, or cubes, on the rectangular faces of the cubeoid are placed symmetrically. This means that the cubic projections on the different rectangular faces are all situated at the same height on the tetragonal prism.  
         [0058]     A tetragonal prism with two cubes on any of the rectangular sides of the cubeoid has space equal to the volume of a single cube between the two cubes; and therefore, a tetragonal prism with two cubes on any one side of the cubeoid can have the two cubes at the height of the second and fourth cubes of the cubeoid ( 1 ,  2 ,  3 ,  4 ,  5  and  6 ); or at the height of the first and third cube in the cubeoid ( 7 ,  8 ,  9 ,  10 ,  11 ,  12 ,  14 ,  15 ,  16 ,  17 ,  18 );  
         [0059]     When the tetragonal prism has one cube on a side of the cubeoid, the cube can be placed at the height of the second cube in the cubeoid ( 19 ,  20 ,  21 ,  22 , and  23 ) or at the height of the first cube in the cubeoid ( 25 ,  26 ). When the cubeoid comprises two cubes, the tetragonal prism can have one cube on one side of the tetragonal prism ( 25 ) or two cubes on two adjacent sides of the tetragonal prism, each cube on one side at the height of the first cube in the cubeoid ( 26 ). Or the cubeoid can comprise one cube ( FIG. 27 ).  
         [0060]     The cubic projections of one piece fit together in the empty spaces between the cubic projections of another piece, and so when the pieces are put together, the pieces form compact and cohesive shapes.  
         [0061]     One example of putting the pieces together is a cube ( FIG. 28 ), the volume of which is equivalent to 9×9×9 cubic projections made by assembling the group of pieces, with repetition of pieces: the 2 piece, the 3 piece, the 4 piece, the 5 piece, the 6 piece, the 7 piece, the 8 piece, the 9 piece, the 10 piece, the 11 piece, the 12 piece, the 14 piece, the 15 piece, the 16 piece, the 17 piece, the 23 piece, the 24 piece, the 25 piece and the 26 piece.  
         [0062]     Other examples include: 
        a. Two 26 pieces can make a cube of 2×2×2.     b. A 16 piece, a 17 piece and an 18 piece can make a cube of 3×3×3.     c. Two 11 pieces, one 12 piece, one 10 piece and four 8 pieces can make a cube of 4×4×4.     d. Nine 2 pieces, one 3 piece, two 5 pieces, one 6 pieces and two 4 pieces can make a cube of 5×5×5 . . .     e. One 10 piece, one 15 piece, six 2 pieces, two 5 pieces, four 4 pieces, two 3 pieces, two 6 pieces, three 8 pieces, four 14 pieces, one 16 piece, one 22 piece and one 25 piece can make a cube of 6×6×6        
 
         [0068]     Once put together, the pieces reproduce a three-dimensional shape. It is possible to form many different three-dimensional shapes from the pieces. The three-dimensional shapes may have interior and exterior pieces. So it is possible to reproduce a great variety of three-dimensional shapes which might be, for example, cubes, tetragonal prisms, or staggered pyramids. It is also possible to reproduce a great variety of three-dimensional objects, geometric or irregular, after mechanization of some of the constituent pieces.  
         [0069]     Piece  1  is a tetragonal prism ( 31 ) comprising a stack of five equal cubes.  
         [0070]     Piece  2  is a tetragonal prism ( 32 ) comprising a stack of five equal cubes with equal cubes placed on one of the rectangular faces of the said stack at the height of the second cube of the said stack ( 34 ) and at the height of the fourth cube ( 33 ) of the said stack.  
         [0071]     Piece  3  is a tetragonal prism ( 35 ) comprising a stack of five equal cubes with equal cubes placed on two of the adjacent rectangular faces of said stack at the height of the second cube in said stack ( 37  and  39 ) and the fourth cube in said stack ( 36  and  38 ).  
         [0072]     Piece  4  is a tetragonal prism ( 40 ) comprising a stack of five equal cubes with three pairs of equal cubes placed on three of rectangular faces of said stack at the height of the second cube in said stack ( 42 ,  44  and  46 ) and the fourth cube in said stack ( 41 ,  43  and  45 ).  
         [0073]     Piece  5  is a tetragonal prism ( 47 ) comprising a stack of five equal cubes with two pairs of equal cubes placed on two of the non-adjacent rectangular faces of said stack at the height of the second cube in said stack ( 49  and  51 ) and the fourth cube in said stack ( 48  and  50 ).  
         [0074]     Piece  6  is a tetragonal prism ( 52 ) comprising a stack of five equal cubes with four pairs of equal cubes on four rectangular faces of said stack at the height of the second cube in said stack ( 54 ,  56 ,  58  and  60 ) and the fourth cube in said stack ( 53 ,  55 ,  57  and  59 ).  
         [0075]     Piece  7  is a tetragonal prism comprising a stack of four equal cubes ( 61 ).  
         [0076]     Piece  8  is a tetragonal prism ( 62 ) comprising a stack of four equal cubes with two equal cubes on one of the rectangular faces of said stack at the height of the first cube in said stack ( 64 ) and the third cube in said stack ( 63 ).  
         [0077]     Piece  9  is a tetragonal prism ( 65 ) comprising a stack of four equal cubes with two pairs of equal cubes on two adjacent rectangular faces of said stack at the height of the first cube in said stack ( 67  and  69 ) and the third cube in said stack ( 66  and  68 ).  
         [0078]     Piece  10  is a tetragonal prism ( 70 ) comprising a stack of four equal cubes with two pairs of equal cubes placed on two non-adjacent rectangular faces of said stack at the height of the first cube in said stack ( 72  and  74 ) and the third cube in said stack ( 71  and  73 ).  
         [0079]     Piece  11  piece is a tetragonal prism ( 75 ) comprising a stack of four equal cubes with three pairs of equal cubes placed on three of the rectangular faces of said stack at the height of the first cube of said stack ( 77 ,  79  and  81 ) and the third cube of said stack ( 76 ,  78  and  80 ).  
         [0080]     Piece  12  piece is a tetragonal prism ( 82 ) comprising a stack of four equal cubes with four pairs of equal cubes placed on the four rectangular faces of said stack at the height of the first cube in said stack ( 84 ,  86 ,  88  and  90 ) and the third cube of said stack ( 83 ,  85 ,  87  and  89 ).  
         [0081]     Piece  13  piece is a tetragonal prism ( 91 ) comprising a stack of three equal cubes.  
         [0082]     Piece  14  is a tetragonal prism ( 92 ) comprising a stack of three equal cubes with two equal cubes placed on one of the rectangular faces of said stack at the height of the first cube in said stack ( 94 ) and the third cube in said stack ( 93 ).  
         [0083]     Piece  15  is a tetragonal prism ( 95 ) comprising a stack of three equal cubes with two pairs of equal cubes placed on two of the adjacent rectangular faces of said stack at the height of the first cube in said stack ( 97  and  99 ) and the third cube in said stack ( 96  and  98 ).  
         [0084]     Piece  16  is a tetragonal prism ( 100 ) comprising a stack of three equal cubes with two pairs of equal cubes placed on two of the non-adjacent rectangular faces of said stack at the height of the first cube in said stack ( 102  and  104 ) and the third cube of said stack ( 101  and  103 ).  
         [0085]     Piece  17  is a tetragonal prism ( 105 ) comprising a stack of three equal cubes with three pairs of equal cubes placed on three of the rectangular faces of said stack at the height of the first cube in the said stack ( 107 ,  109  and  111 ) and the third cube in the said stack ( 106 ,  108  and  110 ).  
         [0086]     Piece  18  is a tetragonal prism ( 112 ) comprising a stack of three equal cubes with four pairs of equal cubes placed on the four rectangular faces of said stack at the height of the first cube in the said stack ( 114 ,  116 ,  118  and  120 ) and the third cube in said stack ( 113 ,  115 ,  117  and  119 ).  
         [0087]     Piece  19  is a tetragonal prism ( 121 ) comprising a stack of three equal cubes with four equal cubes placed on the four rectangular faces of said stack at the height of the second cube in said stack ( 122 ,  123 ,  124  and  125 ).  
         [0088]     Piece  20  is a tetragonal prism ( 126 ) comprising a stack of three equal cubes with three equal cubes placed on three of the rectangular faces of said stack at the height of the second cube in said stack ( 127 ,  128  and  129 ).  
         [0089]     Piece  21  is a tetragonal prism ( 130 ) comprising a stack of three equal cubes with two equal cubes placed on two of the adjacent rectangular faces of said stack at the height of the second cube in said stack ( 131  and  132 ).  
         [0090]     Piece  22  is a tetragonal prism ( 133 ) comprising a stack of three equal cubes with two equal cubes placed on two of the non-adjacent rectangular faces at the height of the second cube in said stack ( 134  and  135 ).  
         [0091]     Piece  23  is a tetragonal prism ( 136 ) comprising a stack of three equal cubes with one equal cube placed on one of the rectangular faces of said stack at the height of the second cube in said stack ( 137 ).  
         [0092]     Piece  24  is a tetragonal prism ( 138 ) comprising a stack of two equal cubes.  
         [0093]     Piece  25  is a tetragonal prism ( 142 ) comprising a stack of two equal cubes with one equal cube ( 143 ) placed on one of the rectangular faces of said stack at the height of the first cube in said stack.  
         [0094]     Piece  26  is a tetragonal prism ( 139 ) comprising a stack of two equal cubes with two equal cubes ( 140  and  141 ) placed on two of the adjacent rectangular faces of said stack at the height of the first cube in said stack.  
         [0095]     Piece  27  is a tetragonal prism ( 144 ) with a volume of one equal cube.  
         [0096]     Various materials may be used, such as plastic and wood.  
         [0097]     The pieces may be painted as well, with the colour of the paint being used to indicate the location of the puzzle piece. The exterior surface may be painted to match the colour and texture of a desired object, such as a building or car. The interior pieces may be painted with different colours, including one colour for each direction of the three-dimensional space.  
         [0098]     The described version of the invention can further reproduce precise objects following a simple method. First, the 27 pieces can be made by way of moulding or machine tools. For example, machine tools are capable of height precision and producing complex geometry, (for example, in numerical control). Or the manufacturing process could make moulds of the 27 pieces, or a subset of the 27 pieces, that would be used to form the pieces of the selected object.  
         [0099]     Upon completion of making the selected pieces, the selected pieces are assembled to match the desired object or an approximation. Once the results have been obtained by assembling the groups of said pieces, with repetitions of pieces, a desired shape may be obtained by cutting the pieces comprising the periphery of the assembled object. This reproduces a desired surface area for the desired three-dimensional object and the desired volume, integrated in the total volume, of the object ( FIG. 29 ).  
         [0100]     For example, suppose that one has a 9×9×9 cube ( FIG. 28 ) and that one wants to make a prism with a triangular base from the pieces described. One can accomplish this as shown in  FIG. 30  by cutting the pieces situated in the vertical plane of the diagonal of the base of the cube, as if one were cutting the cube along the plane which unites a top vertical edge with the opposite diagonal edge along the cube&#39;s base.