Message #514

From: Jay Berkenbilt <ejb@ql.org>
Subject: Re: [MC4D] Magic120Cell Realized
Date: Tue, 13 May 2008 22:11:02 -0400

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. Reading
my original post, I see I used the word "polyhedron" all over the
place when I meant polytope or polychoron. I know that the
icosohedron and dodecahedron are duals, but I was just not remembering
whether the 120-cell had a dual.

I realize now that the 120-cell’s dual is the 600-cell and that it’s
the 24-cell with octohedron cells that’s self-dual. (I looked on
wikipedia.) I remembered that one of the six platonic 4-topes was
self-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@yahoogroups.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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