Message #515
From: Mark Oram <markoram109@yahoo.co.uk>
Subject: Re: [MC4D] Magic120Cell Realized
Date: Wed, 14 May 2008 15:52:28 +0000
Jay,
Thanks for the update, and for amending my original post. I certianly didn’t know about the duality (?) of the 24- and 120-cells; and thank-you for the wikipedia research and for extending to my understanding.
Mark.
— On Wed, 14/5/08, Jay Berkenbilt <ejb@ql.org> wrote:
From: Jay Berkenbilt <ejb@ql.org>Subject: Re: [MC4D] Magic120Cell RealizedTo: 4D_Cubing@yahoogroups.comDate: Wednesday, 14 May, 2008, 3:11 AM
Mark Oram <markoram109@ yahoo.co. uk> wrote:> I fear the dual of the dodecahedron is in fact the icosohedron; while> the cube and octahedron are similarly dual to each other. The> tetrahedron IS its own dual however: possibly this is where your> recollection came from?My whole thing was about the 120-cell, not the dodecahedron. Readingmy original post, I see I used the word "polyhedron" all over theplace when I meant polytope or polychoron. I know that theicosohedron and dodecahedron are duals, but I was just not rememberingwhether the 120-cell had a dual.I realize now that the 120-cell’s dual is the 600-cell and that it’sthe 24-cell with octohedron cells that’s self-dual. (I looked onwikipedia.) I remembered that one of the six platonic 4-topes wasself-dual.It’s been too many years since
I’ve really played with these. :-)–Jay>> — On Sat, 10/5/08, Jay Berkenbilt <ejb@ql.org> wrote:>> From: Jay Berkenbilt <ejb@ql.org>> Subject: Re: [MC4D] Magic120Cell Realized> To: 4D_Cubing@yahoogrou ps.com> Date: Saturday, 10 May, 2008, 3:57 PM>> I have to add my voice to the rest in expression of awe at this> puzzle. It’s been years since I’ve even done mc4d – my life has> gotten busier. One day maybe I’ll try it, and I’m sure I’ll> eventually play around with it just to see what it feels like. As> with many of the other participants on this list, I have always> had a> special affinity for the 120-cell. It always seemed to me
that it> sort of snuck in to the regular polyhedron list, just barely> fitting,> kind of like the pentagon just barely being able to be the face> shape> of one of the platonic solids. :-) Do I recall correctly that this> polyhedron is its own dual?>> > Honestly, the reason I wasn’t planning on working through a> solution> > was that I am a bit scared of the sheer number of pieces! I just> > finished up the final parts that I felt were needed for it to be> > solvable today, and I actually haven’t even figured out a single> > sequence yet. So as of this evening, I only have the thoughts> about> > it we’ve discussed in the past, which is that it will be easier> in> > some ways than MC4D because of the larger space to sequester> pieces,> > but that it will be a big effort in time. Also, I think I
am> ready> > for a bit of a rest and was too excited to share to let it sit> on a> > shelf. Sarah will be happy to get my attention back now too> since> > I’ve been spending a lot of time on it lately :)>> My recollection of solving the megaminx is that you can do all but> the> last few steps as localized solutions. Each twist affects such a> small number of pieces that the constraints don’t play a big role> until the end. It seems that each twist would necessarily alter> pieces on the 12 adjacent cells.>> I don’t find it surprising that five random twists would result in> some interacting pieces. The first twist affects pieces on 12 of> the> 120 cells, not including the cell twisted. In order for the second> twist to not interact with any pieces, it must be on a cell that> is> neither any of the 12
affected faces nor adjacent to any of them> (except that it could be another twist of the first face). I’m not> sure how many cells that is. If you managed to get one, there are> even fewer places for the third twist. It seems to me that the> number> of twists after which there is some guaranteed interaction must be> very small….maybe three or four? I could probably work it out,> but> I imagine others on this list could do it faster. My "math chops"> may> be good compared to the general population, but not compared to> many> of the readers of this list. :-)>> Anyway, the 120 cell puzzle is a work of beauty!>> –Jay>>
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