Waterman polyhedra, was A055039 question

[Hugo Pfoertner]

wouter meeussen wrote:
> 
> hi All,
> 
> I agree with the data presented,
> and offer the Mma lines (based on the ConvexHull3D package & demo-file on
> http://users.pandora.be/Wouter.Meeussen/ConvexHull3D.m ):
> 
> base[n_Integer]:=Flatten[Table[If[EvenQ[x+y+z]&&x^2+y^2+z^2<=2n,w[x,y,z],{}]
> ,{x,0,2n},{y,x,2n},{z,y,2n}]]/.w->List;
> fac=Flatten[Outer[List,{-1,1},{-1,1},{-1,1}],2];
> setK[n_Integer]:=Union[Flatten[Outer[Times,fac,Flatten[Permutations/@base[n]
> ,1],1],1]];
> ra={0};Table[po1=setK[n];Length[po1];po=Select[po1,#.#>=Min[ra]&];ch3D=Conve
> xHull3D[po];ra=#.#&/@Part[po,Union@@ch3D];{n,Length[po1],Length[Union@@ch3D]
> ,Length[ch3D],1/2*Plus@@Length/@ch3D,3HullVolume[po,ch3D]},{n,1,50}]
> 
> just in case anyone with Mma onboard gets 'playfull and new inspiration'.
> 
> 2cts worth,
> 
> W.

I'll include your Mmca code in the first of the 5 sequences that I will
submit as A119869 -- A119873.

Using Mark Newbold's wonderful CCPOLY Java Applet,
http://dogfeathers.com/java/ccppoly.html

I checked the data again and created a table for the Waterman polyhedra
family where the center of the enclosing sphere is at an empty lattice
point instead of filled lattice point. The Java applet will allow also
other centers, but I hesitate to drown Neil by an avalanche of too many
sequences.

Attacched is the table for the "void centered Waterman polyhedra". Can
you modify your Mmca program and check? I'll submit the 5 columns as
A119874 ... A119878.

Hugo Pfoertner

----------------
Properties of Waterman Polyhedra of Void Center type
(center 6 in Mark Newbold's Java Applet)
Radius=sqrt(1+2*N)

 N
 |   spheres
 |    |   vertices
 |    |    |   faces
 |    |    |    |   edges
 |    |    |    |    |   volume  3*volume
 0    6    6    8   12    1+1/3    10
 1   14    8    6   12    8        24
 2   38   24   14   36   32        96
 3   38   24   14   36   32        96
 4   68   30   56   84   90+2/3   272
 5   92   24   26   48  114+2/3   344
 6  116   24   14   36  134+2/3   404
 7  116   24   14   36  134+2/3   404
 8  164   48   50   96  237+1/3   712
 9  188   48   62  108  281+1/3   844
10  236   48   26   72  329+1/3   988
11  236   48   26   72  329+1/3   988
12  266   30   32   60  385+1/3  1156
13  298   56  102  156  500      1500
14  370   72   50  120  580      1740
15  370   72   50  120  580      1740
16  418   48   38   84  664      1992
17  466   48   26   72  696      2088
18  490   72   74  144  772      2316
19  490   72   74  144  772      2316
20  586   96  134  228 1010+2/3  3032
21  610   96  134  228 1042+2/3  3128
22  682   72   50  120 1114+2/3  3344
23  682   72   50  120 1114+2/3  3344
24  736   54   68  120 1230+2/3  3692
25  784   96   62  156 1322+2/3  3968
26  856   72   74  144 1402+2/3  4206
27  856   72   74  144 1402+2/3  4206
28  904   96  122  216 1589+1/3  4768
29  976   72   50  120 1677+1/3  5032
30 1048  120  134  252 1809+1/3  5428
31 1048  120  134  252 1809+1/3  5428
32 1144   96   50  144 1965+1/3  5896
33 1168  120  146  264 2061+1/3  6184
34 1264   96   74  168 2149+1/3  6448
35 1264   96   74  168 2149+1/3  6448
36 1312   96  110  204 2257+1/3  6772
37 1368   80  114  192 2428      7284
38 1464  144  146  288 2604      7812
39 1464  144  146  288 2604      7812
40 1566  102  140  240 2832      8496
41 1638  120   74  192 2888      8664
42 1686  144  134  276 2996      8988
43 1686  144  134  276 2996      8988
44 1830  144  170  312 3260      9780
45 1878  168  122  288 3300      9900
46 1926   96   74  168 3332      9996
47 1926   96   74  168 3332      9996
48 1974   96   98  192 3526+2/3 10580
49 2046   96  158  252 3762+2/3 11288
50 2214  168  194  360 4070+2/3 12212
51 2214  168  194  360 4070+2/3 12212
52 2310  192  146  336 4182+2/3 12548
53 2382  168  122  288 4318+2/3 12956
54 2454  120   74  192 4350+2/3 13052
55 2454  120   74  192 4350+2/3 13052
56 2550  144  134  276 4602+2/3 13808
57 2598  120   86  204 4650+2/3 13952
58 2718  168  122  288 4818+2/3 14456
59 2718  168  122  288 4818+2/3 14456
60 2796  126  164  288 5169+1/3 15508