Greetings. It would appear that even 15 is suboptimal, since Alexey Nigin has subsequently found a 12-faced golyhedron! https://yadi.sk/d/dmqoxL8kR6fSC A remarkable property of this golyhedron -- which must necessarily be possessed by an 11-faced golyhedron, if one exists -- is that it is `monolithic'. In particular, it is possible to build it from unit cubes with no adhesive such that the structure does not collapse under gravity. Equivalently, it can be easily built in layers by a 3D printer. 1 1 8 1 5 5 Monolithic polyhedra can be described by an configuration of numbers* giving the height (in cubes) of each `tower'. The configuration above is a description of Alexey's 12-faced golyhedron. (* formally, a finitely-supported function from Z^2 to the non-negative integers) This makes exhaustively searching for 11-faced golyhedra suddenly tractable, since monolithic polyhedra are easy to represent. I expect that it will not be long before we either prove Alexey's golyhedron optimal or find an 11-faced example. Here is the current timeline of golyhedra discovered so far: 2014-04-29: 32 (Goucher) 2014-05-09: 15 (Nigin) 2014-05-16: 19* (Friedman) 2014-05-24: 12 (Nigin) * majorised by an earlier but independent result. Sincerely, Adam P. Goucher Sent: Friday, May 16, 2014 at 2:48 PM From: "Erich Friedman" <efriedma@stetson.edu> To: "Adam P. Goucher" <apgoucher@gmx.com> Subject: Re: [math-fun] Golyhedra adam thanks for the update. i'm going to guess 15 is optimal. erich On May 16, 2014, at 9:36 AM, Adam P. Goucher wrote:
Erich,
Very nice. You seem to have also had the excellent idea of using only one club foot; Alexey Nigin found a 15-face golyhedron using a 4-by-2 club foot and a similar method:
http://cp4space.wordpress.com/2014/05/11/golyhedron-update/
This means that the minimum number of faces is either 11, 12 or 15.
Both your golyhedron and Alexey's are considerably better than my 32-face behemoth.
Sincerely,
Adam P. Goucher
----- Original Message ----- From: Erich Friedman Sent: 05/16/14 01:42 AM To: Adam P. Goucher, Ed Pegg, alexanderbellos@googlemail.com Subject: Re: [math-fun] Golyhedra
adam
i found a golyhedron with 19 faces. george sicherman deserves credit for giving me the good idea of a 2x3 club foot. and you deserve credit for your snake idea. there is no golyhedron of the same form with 15 faces, but i haven't checked all different club feet. or there may be some other construction possible for smaller n.
a picture is enclosed. i would post this to math-fun, but i can't post for some reason. feel free to forward it if you think people care.
erich