# Message #542

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] higher dimensional book recommendations

Date: Wed, 23 Jul 2008 23:21:13 -0500

Yeah, I really like the 7x7x7. It was pricey for sure (which is why I held

off on the 6^3), but it is quite an amazing product and works better than

the 5^3 I have. I’ve only solved it once so far and this first try took

hours!

On the topic of the

"hyper-Gigaminx<http://www.google.com/search?hl=en&q=gigaminx>"

and related puzzles, I wanted to mention that there is a corners only

variant of Megaminx (even number of pieces per side) called the

Impossiball<http://en.wikipedia.org/wiki/Impossiball>.

It doesn’t look so much like a Megaminx, but it is equivalent. I’ve never

handled one myself, but on the wiki page I saw that the twisting does

involve portions of the puzzle getting pushed outward because the slicing

doesn’t happen on single planes. Maybe this is a little impure in a sense,

but since it is physically constructable, we are almost obliged to imagine a

4D version as well. And although I’m not aware of a 4-to-a-side Megaminx

having been constructed yet, maybe it is possible to think about other

even-sided versions of that and M120C too. I haven’t thought enough about

it to be sure though.

Nelson and I were discussing some of these variants off-line and faked

pictures by messing with the settings. He named the 4D Impossiball

"Inconceivaball" :) Also, he surprised me with a picture of a 4D Pyraminx

Crystal<http://www.mefferts.com/products/details.php?lang=en&category=13&id=171>…I

hadn’t even known this 3D puzzle existed. It seems there are enough

permutation problems out there to keep one busy for a very long time! I

uploaded the screenshots Nelson and I did to the yahoo group photo area (click

here <http://games.ph.groups.yahoo.com/group/4D_Cubing/photos/browse/40b9> if

you’d like to see).

cya,

Roice

P.S. Thanks for info about the hypersphere chapter in Nonplussed, which is

now on my Amazon wishlist for that reason! Week’s book has a chapter on the

hypersphere as well and I learned new things from it, but I also was aware

of it missing some discussion (he described great-spheres but did not go

into how it is possible to slice a hypersphere into two sections using

tori).

On 7/22/08, David Smith <djs314djs314@yahoo.com> wrote:

>

> Hello All,

>

>

>

> Roice, thanks for the book recommendations! I also appreciate the info

>

> on V-Cubes. I am going to get both the 6x6 and 7x7. I would also love

>

> to have the entire collection, once they come out with cubes up to 11x11

>

> in size.

>

>

>

> Melinda, thanks for the Scientific American article, and for your work on

>

> infinite regular polyhedra and the related links! I also prefer

>

> to read about the results of great mathematicians and scientists, but I too

>

> like to read about the people themselves. I’ll have to get these books

>

> eventually, but I have many books on my reading list right now, including

>

> both Hardy and Wright’s classic *An Introduction to the Theory of Numbers*

>

> and the Feynman Lectures.

>

>

>

> Roice and I have been in communication for some time now, and he

>

> recommended I update the group with my math results on Magic120Cell.

>

> I have updated my paper on the number of reachable positions of

> Magic120Cell

>

> which was originally a post on this group, and Roice has very kindly agreed

>

> to host it and my future papers on his website. I have also solved the

>

> problem of the number of Magic120Cell programs with exactly/at most k

>

> colors. These results can be found on my website,

>

>

>

> http://mathproofs.bravehost.com/.

>

>

>

> It occurred to me today that both Magic120Cell and the Megaminx can

>

> have any odd number of pieces per edge, and after I complete my papers

>

> on the nxnxn Rubik’s Cube and the Magic120Cell coloring problem,

>

> I will try my hand at finding a formula for the number of different

> positions

>

> of a Magic120Cell program with any number of pieces per edge.

>

>

>

> I hope my work on Magic120Cell will be interesting to some of you.

>

> The only book recommendation I have would be *Nonplussed!* by

>

> Julian Havil. Only one chapter would be appropriate for this group,

>

> namely Hyperdimensions, Chapter 12. It contains the most

>

> complete discussion of the n-dimensional sphere I have ever seen.

>

> It requires knowledge of calculus, but is well worth the effort!

>

>

>

> All the Best,

>

>

>

> David

>

> — On *Tue, 7/22/08, Melinda Green <melinda@superliminal.com>* wrote:

>

> From: Melinda Green <melinda@superliminal.com>

> Subject: Re: [MC4D] higher dimensional book recommendations

> To: 4D_Cubing@yahoogroups.com

> Date: Tuesday, July 22, 2008, 1:45 AM

>

> I haven’t read Jeffrey Weeks’ book though he did coauthor my favorite

> Scientific American article of all time titled "Is Space Finite?"<http://cosmos.phy.tufts.edu/~zirbel/ast21/sciam/IsSpaceFinite.pdf>which is related to the work I’ve done

> cataloguing infinite regular polyhedra<http://www.superliminal.com/geometry/infinite/infinite.htm>.

> Of particular interest to members of our list might be his familiar games

> you can play in such tiled spaces<http://www.geometrygames.org/TorusGames>which includes a nice implementation of chess in toroidal space. I managed

> to beat it, but only because it plays deterministically. If you try that, be

> sure to switch from the "fundamental domain" mode to "tiling" mode. We

> exchanged a few emails a long time ago and I found him to be a very nice and

> approachable guy.

>

> Professor Coxeter <http://en.wikipedia.org/wiki/Coxeter> is most

> definitely a giant of mathematics. His book Regular Polytopes<http://www.amazon.com/exec/obidos/ASIN/0486614808/>is quite possibly the definitive work on the subject. It’s very dense

> reading but is a great reference work to have around if only for the tables

> of 4D vertices at the end. I consider it the bible of polyhedra. I was told

> that he was a member of a polyhedra mailing list that I was on for several

> years along with John Conway<http://en.wikipedia.org/wiki/John_Horton_Conway>,

> Mangus Wenninger <http://en.wikipedia.org/wiki/Magnus_Wenninger> and

> others. He never posted there but if he read the list then he probably read

> some of my posts.

>

> I’m normally much more interested in scientist’s works than I am of the

> people themselves but I love a good personal story too and will have to read

> Coxeter’s. My favorite so far is The Man Who Loved Only Numbers: The Story

> of Paul Erdos and the Search for Mathematical Truth<http://www.amazon.com/exec/obidos/ASIN/0786884061/qid=997660173/sr=1-1/ref=sc_b_1/107-3481249-3609337>.

> A really fun read about an amazing man.

>

> -Melinda

>

> Roice Nelson wrote:

>

> I’ve finished a couple books recently that I highly enjoyed and are

> apropos to the group.

>

> The Shape of Space<http://www.amazon.com/Shape-Space-Pure-Applied-Mathematics/dp/0824707095/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1216142339&sr=8-1>by Jeffrey Weeks

>

> This does not require a deep math background - it is described as being at

> a high school level, but I really learned a ton and enjoyed it immensely.

> It is chock-full of dimensional analogy, interesting abstractions, and very

> fun to read with big, easy text and lots of pictures! It has also generated

> a number of thoughts for possible additional Rubik analogues in my mind.

> Briefly describing, the flexibility of topology opens up whole new worlds

> here, and if you abstract the original cube as just a 6-cell of faces on a

> topological sphere, all of a sudden there a veritable infinite number of new

> puzzles one could make. I’ve discussed possibly coding with my brother a 3D

> puzzle based on cell divisions of hexagons on a topological torus (e.g. a

> 12-cell is one option we did some sketches of; btw, the hexagonal

> tiling turns out to be important because 3 cells still meet at each

> vertex). In the presentation we envision, the faces would have to stretch

> and deform when twisting due to the non-uniform curvature of a torus, but we

> hypercubists definitely don’t care about such appearances on our screen ;)

>

> King of Infinite Space: Donald Coxter, the Man Who Saved Geometry<http://www.amazon.com/King-Infinite-Space-Coxeter-Geometry/dp/0802714994/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1216142369&sr=1-1>by Siobhan Roberts

>

> This is a biography of Donald Coxeter, a new intellectual hero of mine

> after reading it. I really love the genre of mathematical/ scientific

> biographies, and this is a good one. The book is much more history than

> math, with plenty of enjoyable anecdotal stories about Coxeter and his peers

> (Hardy, Einstein, Von Neumann, etc.). Overall it is an engaging, sweet

> portrait of someone enthralled with polytopes for his entire life.

>

>

>

>