Notations for polyspheres: Torsten Sillke, FRA, 1998-06 -------------------------- Polyspheres can be simulated by edge-connected cubes. There is on distinction between these two models and that are the mirror images. If non-connected polyspheres are allowed, we will get further distinctions. The first case, where a distinction occurs [Cof91, Ch 18 p148] is the trishpere o o o (it lives in the hexagonal lattice. See piece 6 in the table below.) __ __ __/_/| /_/| /_/|_|/ |_|/ |_|/ __/_/| /_/| /_/|_|/ |_|/ |_|/ As edge-connected cubes or poly-rhombic-dodecahedra it has a mirror image, but as polyspheres there is an additional symmetry operation which is not a symmetry of the whole lattice. A rotation around an axis (+-1, +-1, +-1) with 60 degree will fix every third hexagonal sublattice orthogonal to this axis. For connected polyspheres this can only be the case if they live in a hexagonal lattice. Canonical notation: For each piece use the lexicographical smallest non-negative coordinates (over the reflection group of the cube). This gives the canonical coordinates. All coordinates for a polyspheres with n spheres are sorted lexicographically and numbered beginning by one. This gives a numbering for each n. Then we can make an overall numbering (see tables below). Mirror images are notated by a negative sign. Example: the set of 20 tetrasphere is (reflections allowed) 7..26 the set of 25 tetrasphere is (rotations only) 7..26, -10, -11, -19, -23, -24 the set of 28 tetra-edge-connected cubes is (rotations only) 7..26, -10, -11, -15, -16, -19, -23, -24, -26. Example packing: The following puzzles can be found in [Gor86]. I made a systematic study of polysphere tetrahedra and pyramid packings with like pieces upto octaspheres and found several others too. . . . 0 . . . . . 0 . 0 . . . . 7 . . a square pyramid . 1 . 4 . 2 . . 7 . 7 . . 9 . 10 trispheres no. 3, 1 . 5'. 6 . 2 5 . 4'. 6 5 . 6 9 2 solutions (4' <-> 5') . 1 . 4 . 2 . . 8 . 8 . . 9 . [Gor86, Sec. 10] . . 3 . 3 . . . . 8 . . Torsten Sillke 1994 . . . 3 . . . - . . . . 3 . . . . 4 . . . . - a truncated tetrahedron . 1 . . 3 . 3 . . 4 . 4 . . 2 . 4 tetraspheres no. 13, . . 2 . . 1 . 3 4 . 2 . . 1 . . unique solution, it snaps together . . . - . . 2 . . 1 . . - . . . [Gor86, Sec. 11] 6 . 4 . . 4 . 5 4 . 1 . . 4 . 0 box - face centered cubic packing . 6 . 0 5 . 5 . . 1 . 0 7 . 3 . 8 tetraspheres no. 13, 7 . 6 . . 5 . 0 7 . 1 . . 3 . 3 3 solutions, difficult . 6 . 2 7 . 2 . . 2 . 1 3 . 2 . [Gor86, Sec. 3] 3 . . . . 0 . . . . 0'. . . . 2 a tetrahedron . 1 . . 3 . 0 . . 3 . 2' . . 2 . 5 tetraspheres no. 15, . . 1 . . 1 . 0 3 . 4 . . 2 . . 2 solutions (0'<->2') . . . 1 . . 4 . . 4 . . 4 . . . [Gor86, Sec. 10] Wolfgang Schneider ca. 1970 Cube to Rhombic-dodecahedron: The edge-connected cube packings have density 1/2. If you dissect the unused cubes from there centers into six pyramids and add these to the used cubes you will get a dense rhombic-dodecahedra packing. The additional symmetry for planar polysheres: The reflection of a point (x,y,z) on the axis t*(1,1,1) is integral iff 3 | x+y+z. As for hexagonal planar polyspheres we can choose x+y+z=0 this operation is valid. Deriving the formular for the reflection R. Let e^T = (1,1,1) then the projection of a point 'a' on t*(1,1,1) is r = e (e^T a)/(e^T e) From this we get the reflection R as R(a) = 2 r - a = 2 e (e^T a)/(e^T e) - a = 2/3 e e^T a - a = (2/3 e e^T - I) a 1 [ -1 2 2 ] = - [ 2 -1 2 ] * a 3 [ 2 2 -1 ] References: Cof91: Stewart T. Coffin; The Puzzling Would of Polyhedral Dissections, Series: Recrations in Mathematics, No. 5, Oxford Univ. Press, New York, 1991 http://www.johnrausch.com/PuzzlingWorld/ (the online version) Gor86: Leonard Gordon; Some notes on ball-pyramid and related puzzles, privite publication, revised version, Chico CA, July 1986, 14p Wie96: Bernhard Wiezorke; Compendium of polysphere puzzles, privite publication, 2nd Ed. Aug. 1996 ---------------------------------------------------------------------------- Table: planar polyspheres, edge-connected cubes, or poly-rhombic-dodecahedra no1: 1 no2: 1-1 coordinates: 000 picture: o no1: 2 no2: 1-2 coordinates: 000 011 picture: o o no1: 3 o no2: 1-3 coordinates: 000 002 011 picture: o o no1: 4 no2: 2-3 coordinates: 000 011 022 picture: o o o no1: 5 o no2: 3-3 coordinates: 000 011 101 picture: o o no1: 6 o no2: 4-3 coordinates: 000 011 112 picture: o o no1: 7 o o no2: 1-4 coordinates: 000 002 011 013 picture: o o no1: 8 o no2: 2-4 coordinates: 000 002 011 020 picture: o o o no1: 13 o no2: 7-4 coordinates: 000 011 013 022 picture: o o o no1: 14 no2: 8-4 coordinates: 000 011 022 033 picture: o o o o no1: 15 o no2: 9-4 coordinates: 000 011 022 101 picture: o o o no1: 16 o no2: 10-4 coordinates: 000 011 022 123 picture: o o o no1: 18 o o no2: 12-4 coordinates: 000 011 101 112 picture: o o no1: 21 o o no2: 15-4 coordinates: 000 011 112 123 picture: o o no1: 22 o o no2: 16-4 coordinates: 000 011 112 202 picture: o o no1: 25 o o no2: 19-4 coordinates: 001 010 012 021 picture: o o no1: 26 o no2: 20-4 coordinates: 001 111 120 212 picture: o o o Table: planar polyspheres, edge-connected cubes, or poly-rhombic-dodecahedra - numberings, coordinates, orders of the symmetry groups, properties no1 no2 coordinates Ord_Ref Ord_Rot Property Ord_Mixed ---------------------------------------------------------------------------- 1 1-1 | 000 Ref 48 Rot 24 Prop HS Order 24 2 1-2 | 000 011 Ref 8 Rot 4 Prop HS Order 4 3 1-3 | 000 002 011 Ref 4 Rot 2 Prop S Order 2 4 2-3 | 000 011 022 Ref 8 Rot 4 Prop HS Order 4 5 3-3 | 000 011 101 Ref 6 Rot 3 Prop H Order 3 6 4-3 | 000 011 112 Ref 2 Rot 2 Prop H m Order -2 7 1-4 | 000 002 011 013 Ref 4 Rot 2 Prop S Order 2 8 2-4 | 000 002 011 020 Ref 4 Rot 2 Prop S Order 2 9 3-4 | 000 002 011 101 Ref 4 Rot 2 Prop Order 2 10 4-4 | 000 002 011 103 Ref 1 Rot 1 Prop M Order -1 11 5-4 | 000 002 011 110 Ref 1 Rot 1 Prop M Order -1 12 6-4 | 000 002 011 121 Ref 2 Rot 1 Prop Order 1 13 7-4 | 000 011 013 022 Ref 2 Rot 1 Prop S Order 1 14 8-4 | 000 011 022 033 Ref 8 Rot 4 Prop HS Order 4 15 9-4 | 000 011 022 101 Ref 1 Rot 1 Prop H m Order -1 16 10-4 | 000 011 022 123 Ref 1 Rot 1 Prop H m Order -1 17 11-4 | 000 011 101 110 Ref 24 Rot 12 Prop Order 12 18 12-4 | 000 011 101 112 Ref 4 Rot 2 Prop H Order 2 19 13-4 | 000 011 103 112 Ref 2 Rot 2 Prop M Order -2 20 14-4 | 000 011 112 121 Ref 2 Rot 1 Prop Order 1 21 15-4 | 000 011 112 123 Ref 2 Rot 1 Prop H Order 1 22 16-4 | 000 011 112 202 Ref 2 Rot 1 Prop H Order 1 23 17-4 | 000 011 112 211 Ref 1 Rot 1 Prop M Order -1 24 18-4 | 000 011 112 222 Ref 2 Rot 2 Prop M Order -2 25 19-4 | 001 010 012 021 Ref 16 Rot 8 Prop S Order 8 26 20-4 | 001 111 120 212 Ref 6 Rot 6 Prop H m Order -6 27 1-5 | 000 002 004 011 013 Ref 4 Rot 2 Prop S Order 2 28 2-5 | 000 002 004 011 103 Ref 2 Rot 2 Prop M Order -2 29 3-5 | 000 002 011 013 020 Ref 2 Rot 1 Prop S Order 1 30 4-5 | 000 002 011 013 022 Ref 2 Rot 1 Prop S Order 1 31 5-5 | 000 002 011 013 024 Ref 2 Rot 1 Prop S Order 1 32 6-5 | 000 002 011 013 101 Ref 1 Rot 1 Prop M Order -1 33 7-5 | 000 002 011 013 103 Ref 1 Rot 1 Prop M Order -1 34 8-5 | 000 002 011 013 114 Ref 1 Rot 1 Prop M Order -1 35 9-5 | 000 002 011 013 121 Ref 1 Rot 1 Prop M Order -1 36 10-5 | 000 002 011 013 123 Ref 1 Rot 1 Prop M Order -1 37 11-5 | 000 002 011 020 022 Ref 16 Rot 8 Prop S Order 8 38 12-5 | 000 002 011 020 101 Ref 1 Rot 1 Prop M Order -1 39 13-5 | 000 002 011 020 103 Ref 1 Rot 1 Prop M Order -1 40 14-5 | 000 002 011 020 112 Ref 1 Rot 1 Prop M Order -1 41 15-5 | 000 002 011 022 033 Ref 2 Rot 1 Prop S Order 1 42 16-5 | 000 002 011 022 103 Ref 1 Rot 1 Prop M Order -1 43 17-5 | 000 002 011 022 123 Ref 1 Rot 1 Prop M Order -1 44 18-5 | 000 002 011 031 121 Ref 2 Rot 1 Prop Order 1 45 19-5 | 000 002 011 101 110 Ref 2 Rot 1 Prop Order 1 46 20-5 | 000 002 011 101 121 Ref 2 Rot 1 Prop Order 1 47 21-5 | 000 002 011 103 110 Ref 1 Rot 1 Prop M Order -1 48 22-5 | 000 002 011 103 112 Ref 1 Rot 1 Prop M Order -1 49 23-5 | 000 002 011 103 114 Ref 1 Rot 1 Prop M Order -1 50 24-5 | 000 002 011 103 121 Ref 1 Rot 1 Prop M Order -1 51 25-5 | 000 002 011 103 202 Ref 1 Rot 1 Prop M Order -1 52 26-5 | 000 002 011 103 204 Ref 1 Rot 1 Prop M Order -1 53 27-5 | 000 002 011 103 213 Ref 1 Rot 1 Prop M Order -1 54 28-5 | 000 002 011 110 112 Ref 4 Rot 2 Prop Order 2 55 29-5 | 000 002 011 110 121 Ref 1 Rot 1 Prop M Order -1 56 30-5 | 000 002 011 110 200 Ref 2 Rot 1 Prop Order 1 57 31-5 | 000 002 011 110 211 Ref 1 Rot 1 Prop M Order -1 58 32-5 | 000 002 011 110 220 Ref 1 Rot 1 Prop M Order -1 59 33-5 | 000 002 011 112 123 Ref 1 Rot 1 Prop M Order -1 60 34-5 | 000 002 011 112 213 Ref 1 Rot 1 Prop M Order -1 61 35-5 | 000 002 011 121 130 Ref 1 Rot 1 Prop M Order -1 62 36-5 | 000 002 011 121 211 Ref 2 Rot 1 Prop Order 1 63 37-5 | 000 002 011 121 220 Ref 1 Rot 1 Prop M Order -1 64 38-5 | 000 002 011 121 231 Ref 2 Rot 1 Prop Order 1 65 39-5 | 000 002 110 112 121 Ref 2 Rot 1 Prop Order 1 66 40-5 | 000 004 011 013 022 Ref 4 Rot 2 Prop S Order 2 67 41-5 | 000 004 011 013 112 Ref 2 Rot 1 Prop Order 1 68 42-5 | 000 004 011 103 112 Ref 2 Rot 2 Prop M Order -2 69 43-5 | 000 011 013 022 031 Ref 4 Rot 2 Prop S Order 2 70 44-5 | 000 011 013 022 101 Ref 1 Rot 1 Prop M Order -1 71 45-5 | 000 011 013 022 103 Ref 1 Rot 1 Prop M Order -1 72 46-5 | 000 011 013 022 110 Ref 1 Rot 1 Prop M Order -1 73 47-5 | 000 011 013 022 112 Ref 1 Rot 1 Prop M Order -1 74 48-5 | 000 011 013 022 114 Ref 1 Rot 1 Prop M Order -1 75 49-5 | 000 011 013 022 121 Ref 1 Rot 1 Prop M Order -1 76 50-5 | 000 011 013 022 123 Ref 1 Rot 1 Prop M Order -1 77 51-5 | 000 011 013 022 132 Ref 1 Rot 1 Prop M Order -1 78 52-5 | 000 011 013 024 112 Ref 2 Rot 2 Prop M Order -2 79 53-5 | 000 011 013 112 121 Ref 1 Rot 1 Prop M Order -1 80 54-5 | 000 011 013 112 123 Ref 1 Rot 1 Prop M Order -1 81 55-5 | 000 011 013 112 222 Ref 1 Rot 1 Prop M Order -1 82 56-5 | 000 011 022 024 033 Ref 2 Rot 1 Prop S Order 1 83 57-5 | 000 011 022 033 044 Ref 8 Rot 4 Prop HS Order 4 84 58-5 | 000 011 022 033 101 Ref 1 Rot 1 Prop H m Order -1 85 59-5 | 000 011 022 033 112 Ref 2 Rot 1 Prop H Order 1 86 60-5 | 000 011 022 033 134 Ref 1 Rot 1 Prop H m Order -1 87 61-5 | 000 011 022 101 110 Ref 2 Rot 1 Prop Order 1 88 62-5 | 000 011 022 101 112 Ref 2 Rot 1 Prop H Order 1 89 63-5 | 000 011 022 101 121 Ref 2 Rot 2 Prop M Order -2 90 64-5 | 000 011 022 101 123 Ref 1 Rot 1 Prop H m Order -1 91 65-5 | 000 011 022 101 132 Ref 1 Rot 1 Prop M Order -1 92 66-5 | 000 011 022 101 202 Ref 2 Rot 1 Prop H Order 1 93 67-5 | 000 011 022 101 211 Ref 1 Rot 1 Prop M Order -1 94 68-5 | 000 011 022 103 112 Ref 1 Rot 1 Prop M Order -1 95 69-5 | 000 011 022 112 123 Ref 1 Rot 1 Prop H m Order -1 96 70-5 | 000 011 022 112 213 Ref 1 Rot 1 Prop H m Order -1 97 71-5 | 000 011 022 114 123 Ref 1 Rot 1 Prop M Order -1 98 72-5 | 000 011 022 123 132 Ref 2 Rot 1 Prop Order 1 99 73-5 | 000 011 022 123 134 Ref 1 Rot 1 Prop H m Order -1 100 74-5 | 000 011 022 123 213 Ref 1 Rot 1 Prop H m Order -1 101 75-5 | 000 011 022 123 222 Ref 1 Rot 1 Prop M Order -1 102 76-5 | 000 011 022 123 224 Ref 2 Rot 2 Prop H m Order -2 103 77-5 | 000 011 022 123 233 Ref 1 Rot 1 Prop M Order -1 104 78-5 | 000 011 024 112 123 Ref 1 Rot 1 Prop M Order -1 105 79-5 | 000 011 033 112 123 Ref 1 Rot 1 Prop H m Order -1 106 80-5 | 000 011 101 112 121 Ref 1 Rot 1 Prop M Order -1 107 81-5 | 000 011 101 112 222 Ref 2 Rot 1 Prop Order 1 108 82-5 | 000 011 101 121 211 Ref 2 Rot 1 Prop Order 1 109 83-5 | 000 011 101 121 220 Ref 1 Rot 1 Prop M Order -1 110 84-5 | 000 011 101 121 222 Ref 1 Rot 1 Prop M Order -1 111 85-5 | 000 011 103 112 202 Ref 1 Rot 1 Prop M Order -1 112 86-5 | 000 011 103 112 211 Ref 1 Rot 1 Prop M Order -1 113 87-5 | 000 011 103 112 222 Ref 1 Rot 1 Prop M Order -1 114 88-5 | 000 011 112 114 123 Ref 1 Rot 1 Prop M Order -1 115 89-5 | 000 011 112 114 213 Ref 1 Rot 1 Prop M Order -1 116 90-5 | 000 011 112 121 202 Ref 2 Rot 1 Prop Order 1 117 91-5 | 000 011 112 121 211 Ref 2 Rot 1 Prop Order 1 118 92-5 | 000 011 112 121 213 Ref 1 Rot 1 Prop M Order -1 119 93-5 | 000 011 112 123 132 Ref 1 Rot 1 Prop M Order -1 120 94-5 | 000 011 112 123 202 Ref 1 Rot 1 Prop H m Order -1 121 95-5 | 000 011 112 123 211 Ref 1 Rot 1 Prop M Order -1 122 96-5 | 000 011 112 123 213 Ref 1 Rot 1 Prop H m Order -1 123 97-5 | 000 011 112 123 222 Ref 1 Rot 1 Prop M Order -1 124 98-5 | 000 011 112 123 224 Ref 2 Rot 2 Prop H m Order -2 125 99-5 | 000 011 112 123 233 Ref 1 Rot 1 Prop M Order -1 126 100-5 | 000 011 112 132 222 Ref 1 Rot 1 Prop M Order -1 127 101-5 | 000 011 112 202 301 Ref 1 Rot 1 Prop M Order -1 128 102-5 | 000 011 112 204 213 Ref 2 Rot 2 Prop M Order -2 129 103-5 | 000 011 112 211 220 Ref 2 Rot 2 Prop M Order -2 130 104-5 | 000 011 112 211 301 Ref 1 Rot 1 Prop M Order -1 131 105-5 | 000 011 112 211 321 Ref 1 Rot 1 Prop M Order -1 132 106-5 | 000 011 112 213 222 Ref 1 Rot 1 Prop M Order -1 133 107-5 | 000 011 112 213 224 Ref 2 Rot 1 Prop H Order 1 134 108-5 | 000 011 112 213 303 Ref 2 Rot 1 Prop H Order 1 135 109-5 | 000 011 112 213 312 Ref 1 Rot 1 Prop M Order -1 136 110-5 | 000 011 112 213 323 Ref 2 Rot 2 Prop M Order -2 137 111-5 | 000 011 112 222 231 Ref 1 Rot 1 Prop M Order -1 138 112-5 | 000 011 112 222 233 Ref 2 Rot 2 Prop M Order -2 139 113-5 | 000 011 112 222 312 Ref 1 Rot 1 Prop M Order -1 140 114-5 | 000 011 112 222 321 Ref 1 Rot 1 Prop M Order -1 141 115-5 | 001 010 012 014 023 Ref 4 Rot 2 Prop S Order 2 142 116-5 | 001 010 012 021 111 Ref 8 Rot 4 Prop Order 4 143 117-5 | 001 010 012 021 113 Ref 2 Rot 1 Prop Order 1 144 118-5 | 001 010 012 023 032 Ref 4 Rot 2 Prop S Order 2 145 119-5 | 001 010 012 111 221 Ref 2 Rot 1 Prop Order 1 146 120-5 | 001 010 012 113 122 Ref 1 Rot 1 Prop M Order -1 147 121-5 | 001 010 012 120 131 Ref 1 Rot 1 Prop M Order -1 148 122-5 | 001 010 012 120 210 Ref 2 Rot 2 Prop M Order -2 149 123-5 | 001 010 012 120 221 Ref 1 Rot 1 Prop M Order -1 150 124-5 | 001 010 102 111 221 Ref 2 Rot 2 Prop H m Order -2 151 125-5 | 001 010 102 120 212 Ref 2 Rot 2 Prop H m Order -2 152 126-5 | 001 010 111 122 201 Ref 1 Rot 1 Prop M Order -1 153 127-5 | 001 010 111 122 212 Ref 2 Rot 2 Prop M Order -2 154 128-5 | 001 010 111 212 221 Ref 4 Rot 2 Prop H Order 2 155 129-5 | 001 021 111 210 212 Ref 8 Rot 4 Prop Order 4 156 130-5 | 001 111 113 120 212 Ref 1 Rot 1 Prop M Order -1 157 131-5 | 001 111 120 212 313 Ref 2 Rot 2 Prop H m Order -2 Symmetry Groups: Ord_ref: Order of the subgroup of the reflection group of the lattice, which fixes the polysphere. Ord_rot: Order of the subgroup of the rotation group of the lattice, which fixes the polysphere. Properties: property H: This polysphere is planar in the hexagonal lattice. property S: This polysphere is planar in the square lattice. property M: This polysphere (or edge-connected cubes) has a mirror image. <=> Ord_ref = Ord_rot AND NOT property-H property m: This edge-connected cubes has a mirror image. <=> Ord_ref = Ord_rot AND property-H Number of polyspheres, rhombic-dodecahedra n spheres/O_3 spheres/O_3+ rhombic-dodecahedra/O_3+ 1 1 1 1 2 1 1 1 3 4 4 5 4 20 25 28 5 131 210 225 6 1211 2209 2274 7 12734 24651 24955 8 144158 284768 286143 9 1687737 3360995 3367443 10 40328652 40358811 -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/