Waterman polyhedra, was A055039 question

[Hugo Pfoertner]

> 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