Lattice Packings with finite many regular Polyhedra

lattice  tetrahedral/diamond  simple cubic  body-centered cubic  face-centered cubic
Voroni-cell truncated tetrahedron  cubus  truncated octahedron  rhombic dodecahedron
neighbours  4  6  8  12
angle  arccos( \frac{1}{3}) = 109.471^{∘}  π/2 = 90^{∘}  arccos(\frac{1}{3}) = 70.529^{∘}  arccos(\frac{1}{2}) = π/3 = 60^{∘}
connection  even vertices  faces  vertices  edges
n  rotate  free  rotate  free  rotate  free  rotate  hexa  free
1  1  1  1  1  1  1  1  1  1
2  1  1  1  1  1  1  1  1  1
3  1  1  2  2  3  3  5  4  4
4  4  3  8  7  14  12  28  25  20
5  10  7  29  23  88  61  225  210  131
6  39  24  166  112  686  407  2274  2209  1211
7  160  88  1023  607  5966  3226  24955  24651  12734
8  726  385  6922  3811  54722  28335  286143  284768  144158
9  3344  1713  48311  25413  516454  262091 3367443 3360995 1687737
10  16004  8112  346543 178083 4970445 2501168 40358811 40328652 20196788
11  77323  38869 2522522 1279537 48527372 24328920
12  379117  190128 18598427 9371094
13  1874540  938357
14  9347557  4676750
15  46921847 23466650
lattice  ∗ diamond  simple ∗  body-centered ∗ face-centered ∗  dual face-centered ∗
Voroni-cell † tetrahedron  cubus  tru. octahedron ♢ dodecahedron  tetrahedron/octahedron
n  translate  translate  translate  translate  translate rotate  free
1  1  1  1  1  2  2  2
2  4  3  4  6  8  1  1
3  12  15  28  50  40  4  4
4  44  86  216  475  214  14  11
5  182  534  1790  4881  1272  65  44
6  796  3481  15587  52835  7944  350  203
7  3612  23502  140746  593382  51532  2211  1186
8  16786  162913  1305920  6849415  343334 14403  7419
9  79426  1152870  12374069  80757819  2333233 97599 49444
10  381250  8294738  119223556  968400940  16108598 671731 337660
11  1851832  60494549  1164465225  11773656517  112657388
12  9086022  446205905  11502924648  144791296055  796429099
13  44970688  3322769321  114721053058 1797935761182 5682158819
14  224279188  24946773111 1153539900783  40860361625
15  1126038472  188625900446
16  5687027204 1435074454755
17  28873452032 10977812452428
18  147281732692
19  754443210294
n → ∞ C 5.53^{n} n^{−3/2} C 8.345^{n} n^{−3/2} C 11.18^{n} n^{−3/2} C 13.93^{n} n^{−3/2}  C 7.98^{n} n^{−3/2}
† pyramidal capped truncated  ^{♢} rhombic  ^{∗} cubic

The heading translate means that the symmetries are only the translations of the lattice, rotate means that the symmetries are the motion-group of the lattice, i.e. its translations and rotations, and free means that the symmetry-group is the motion-group together with a reflection — this identifies mirror-images. Further we could add a fourth variante hexa, the motion-group together with a half-rotation about its 3-fold axes. In the case of the poly-rhombic-dodecahedra this results in counting the so called polyspheres in the fcc-lattice. For the cubic variantes, the motion-group is the hexaeder-group which is isomorphic to S_{4}. See also:

http://www.johnrausch.com/PuzzlingWorld/chap18.htm .  Last updated: 1999-02-24 11:40 UT+1

lattice  triangular prismatic  hexagonal prismatic  hexagonal closed packed
Voroni-cell  triangular prism  hexagonal prism  trapezoid dodecahedron
neighbours  5  8  12
n  translate  rotate  free  translate  rotate  free  translate  rotate hexa free
1  1  1  1  1  1  1  1  1  1  1
2  4  2  2  4  2  2  9  2  1  2
3  19  3  3  24  5  5  89  19  6 10
4  90  13  11  168  23  20  906  154  84
5  449  43  30  1274  123  89  9578  1620  813
6  2354  221  137  10165  911  569  104904  17448  8788
7  12833  1095  606  84074  7134  4002  1183916  197280  98655
8  71965  6123  3243  714282  60020  31899  13694247 2280658
9  412175  34477  17681  6196308 517412 266736  161612441 26928717
10  2399829 200672 101718  54654830 4558747 2315811 1938888373
11  14159039 1180635 593931  488687023  23579618581
12  84459563 7042181 3532385  4419237539
13  508488575  40347456681
14  3085710689  371398127581
15  18854472081
16  115901568520
17  716268850779
n → ∞  C 6.74^{n} n^{−3/2}  C 10.23^{n} n^{−3/2}  C 13.93^{n} n^{−3/2}

Lattice Packings with finite many regular Polygons

lattice  triangular  square  hexagonal
neighb.  6  4  3
angle  π/3 = 60^{∘}  π/2 = 90^{∘}  2π/3 = 120^{∘}
n  translate  rotate  free  translate  rotate  free  translate  rotate  free
1  1  1  1  1  1  1  1  1  1
2  3  1  1  2  1  1  3  1  1
3  6  1  1  6  2  2  11  3  3
4  14  4  3  19  7  5  44  10  7
5  36  6  4  63  18  12  186  33  22
6  94  19  12  216  60  35  814  147  82
7  250  43  24  760  196  108  3652  620  333
8  675  120  66  2725  704  369  16689  2821  1448
9  1838  307  160  9910  2500  1285  77359  12942  6572
10  5053  866  448  36446  9189  4655  362671  60639  30490
11  14016  2336  1186  135268  33896  17073  1716033 286190 143552
12  39169  6588  3334  505861 126759  63600  8182213 1364621 683101
13  110194  18373  9235  1903890 476270 238591  39267086 6545430 3274826
14  311751  52119  26166  7204874 1802312 901971  189492795
15  886160 147700  73983  27394666 6849777 3426576  918837374
16  2529260 422016 211297  104592937  4474080844
17  7244862 1207477 604107  400795844  21866153748
18  20818498 3471067 1736328  1540820542  107217298977
19  59994514 9999135 5000593  5940738676  527266673134
20  173338962  22964779660  2599804551168
21  501994070  88983512783  12849503756579
22  1456891547  345532572678  63646233127758
23  4236446214  1344372335524
24  12341035217  5239988770268
25  36009329450  20457802016011
26  105229462401
27  307942754342
n → ∞  C 3.036^{n} n^{−1}  C 4.0625^{n} n^{−1}  C 5.1831^{n} n^{−1}