# Message #541

From: David Smith <djs314djs314@yahoo.com>

Subject: Re: [MC4D] higher dimensional book recommendations

Date: Tue, 22 Jul 2008 18:54:47 -0700

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?" which is related to the work I’ve done cataloguing infinite regular polyhedra. Of particular interest to members of our list might be his familiar games you can play in such tiled spaces 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 is most definitely a giant of mathematics. His book Regular Polytopes 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, Mangus 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. 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 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 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.