Message #153

From: Иван Тимофеев <>
Subject: Re: [MC4D] Hello
Date: Thu, 26 May 2005 08:30:27 +0400

Hello Liati and all!

It is great to get to know that new 4D-solver appears.
Forty solvers – we are the powerful community!

Liat Blatman wrote:
l> I also studied some group theory and found some good books in that context.

Very magnificent!
I’d like to look through that good books, if they connect to Rubiks cube.
Could you provide a link, preferably, as electronic document.

I assembled 3x3x3x3 almost year ago and it is the second time I find
the term "group theory" in our yahoogroup.
I have read that Erno Rubik created his greatest puzzle
trying to _visualize group theory_.
Yet, I didn’t find any instructions for studying groups using Rubiks cube.

In Maple math package there is a worksheet for 2x2x2 cube
"Group theory via Rubiks Cube"
Unfortunately, no visualization is implied.

What I found out myself?
What I call "series" as usual presents "commutant" in mathematics.
What I call "substitution before series" presents
"conjugate (adjoint?) commutant".

In cube one can see many famous finite groups.
Obviously, one-side rotations form cyclic C4 group.
Two neighbor-side 180 degree rotations form C12.
Two 180 degree central slice rotations are commutative and form
C2xC2 (Klein group). And so on.

All group theory notation is explained at Wolfram public
encyclopedia: Eric W. Weisstein. "Finite Group." From MathWorld – A
Wolfram Web Resource.
Also this encyclopedia has Rubik contest at

Best regards,
Ivan Timofeev