Message #511

From: Mark Oram <markoram109@yahoo.co.uk>
Subject: Re: [MC4D] Magic120Cell Realized
Date: Mon, 12 May 2008 05:00:39 +0000

I know what you mean, Jay, about the dodecahedron not being an ‘immediately obvious’ Platonic solid. (I think that is what you were getting at?) I had felt the same myself when I first came across them. Maybe this suprise arises as the triangle and square can tile an infinite 2-D plane, wheras the pentagon can not. Strange, then, that the Pentagon CAN ‘tile’ a 3-D solid quite happily!
 
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?
— On Sat, 10/5/08, Jay Berkenbilt <ejb@ql.org> wrote:
From: Jay Berkenbilt <ejb@ql.org>Subject: Re: [MC4D] Magic120Cell RealizedTo: 4D_Cubing@yahoogroups.comDate: Saturday, 10 May, 2008, 3:57 PM

I have to add my voice to the rest in expression of awe at thispuzzle. It’s been years since I’ve even done mc4d – my life hasgotten busier. One day maybe I’ll try it, and I’m sure I’lleventually play around with it just to see what it feels like. Aswith many of the other participants on this list, I have always had aspecial affinity for the 120-cell. It always seemed to me that itsort of snuck in to the regular polyhedron list, just barely fitting,kind of like the pentagon just barely being able to be the face shapeof one of the platonic solids. :-) Do I recall correctly that thispolyhedron 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 thelast few steps as localized solutions. Each twist affects such asmall number of pieces that the constraints don’t play a big roleuntil the end. It seems that each twist would necessarily alterpieces on the 12 adjacent cells.I don’t find it surprising that five random twists would result insome interacting pieces. The first twist affects pieces on 12 of
the120 cells, not including the cell twisted. In order for the secondtwist to not interact with any pieces, it must be on a cell that isneither 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 notsure how many cells that is. If you managed to get one, there areeven fewer places for the third twist. It seems to me that the numberof twists after which there is some guaranteed interaction must bevery small….maybe three or four? I could probably work it out, butI imagine others on this list could do it faster. My "math chops" maybe good compared to the general population, but not compared to manyof the readers of this list. :-)Anyway, the 120 cell puzzle is a work of beauty!–Jay

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