Waterman polyhedra, was A055039 question

[wouter meeussen]

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.


----- Original Message ----- 
From: "Hugo Pfoertner" <all at abouthugo.de>
To: <seqfan at ext.jussieu.fr>
Sent: Thursday, May 25, 2006 10:51 AM
Subject: Waterman polyhedra, was A055039 question


> > David Wilson wrote:
> >
> > http://astronomy.swin.edu.au/~pbourke/geometry/waterman/index.html
> >
> > gives a list of integers that appears to coincide with A055039, and I
> > would be inclined to think it does.  It also gives a characterization
> > of those integers which does not appear on A055039.  Could someone
> > confirm this?
>
> It seems that most (none?) of the characterising numbers describing the
> Waterman polyhedra for root n seems to be in the OEIS. I tried
>
> 13,19,43,55,79,87,135,141,...  (number of sphere centers whose convex
> hull forms the polyhedron)
>
> without getting a match.
>
> So the following table might be a nice candidate for a transformation
> into a few sequences: (copied manually from P. Bourke's web pages, to be
> checked for typos)
>
> Properties of Waterman Polyhedra
>
> Root
>  |   spheres
>  |    |   vertices
>  |    |    |   faces
>  |    |    |    |   edges
>  |    |    |    |    |   volume  3*volume
>  1   13   12   14   24    6+2/3    20
>  2   19    6    8   12   10+2/3    32
>  3   43   24   26   48   45+1/3   136
>  4   55   12   14   24   53+1/3   160
>  5   79   24   14   36   81+1/3   244
>  6   87   32   42   72  116       348
>  7  135   48   26   72  172       516
>  8  141   54   68  120  200       600
>  9  177   36   38   72  248       744
> 10  201   24   14   36  256       768
> 11  225   48   50   96  338+2/3  1016
> 12  249   24   26   48  362+2/3  1088
> 13  321   72   74  144  494+2/3  1484
> 14  321   72   74  144  494+2/3  1484
> 15  369   48   26   72  542+2/3  1628
> 16  381   60   38   96  566+2/3  1700
> 17  429   48   62  108  697+1/3  2092
> 18  459   54   44   96  757+1/3  2272
> 19  531   72   74  144  869+1/3  2608
> 20  555   72   50  120  893+1/3  2680
> 21  603   72   74  144  973+1/3  2920
> 22  627   72   50  120 1013+1/3  3040
> 23  675   48   26   72 1045+1/3  3136
> 24  683   56   66  120 1144      3432
> 25  767  132  134  264 1332      3996
> 26  791   96   62  156 1364      4092
> 27  887  120  122  240 1540      4620
> 28  935   96   74  168 1572      4716
> 29  959   72   50  120 1596      4788
> 30  959   72   50  120 1596      4788
> 31 1055   96   50  144 1740      5220
> 32 1061  102   92  192 1808      5424
> 33 1157   96  122  216 2074+2/3  6224
> 34 1205   96   98  192 2122+2/3  6368
> 35 1253  120   74  192 2170+2/3  6512
> 36 1289   84  134  216 2306+2/3  6920
> 37 1409  120  134  252 2466+2/3  7400
> 38 1433  144  122  264 2506+2/3  7520
> 39 1481   96   74  168 2538+2/3  7616
> 40 1505   72   50  120 2562+2/3  7688
> 41 1553  120  146  264 2742+2/3  8228
> 42 1601   72   74  144 2774+2/3  8322
> 43 1721  168  170  336 3137+1/3  9412
> 44 1745  168  170  336 3169+1/3  9508
> 45 1865  120   98  216 3321+1/3  9964
> 46 1865  120   98  216 3321+1/3  9964
> 47 1961  144  122  264 3497+1/3 10492
> 48 1985  168  122  288 3529+1/3 10588
> 49 2093  108  134  240 3713+1/3 11140
> 50 2123  126  176  300 3829+1/3 11488
>
> Hugo Pfoertner
>