# Message #1210

From: Andrey <andreyastrelin@yahoo.com>

Subject: Re: [MC4D] MHT633 v0.1 uploaded

Date: Wed, 27 Oct 2010 17:56:59 -0000

Thank you, Roice!

I knew that you’ll recognize it quickly (may be even without running the progam, just by name :) ). After failing of search of periodic paintings of {5,3,4} and {4,3,5} I found the note about "11 H3 honeycombs which have infinite (Euclidean) cells and/or vertex figures" and took a better look in them.

It was evident that {6,3,3} and {4,4,3} can be derived from set of orispheres such that each of them tangents infinitely many of others and tangent points make {3,6} or {4,4} lattice in the orisphere (that is congruent to Euclidean plane). So I took my favorite model (half-space), orisphere z=1 and tangent points a+b*i or a+b*j (where i^2+1=0, j^2-j+1=0 when we consider (x,y) plane as C={x+i*y}). For each point there was sphere with center (a+b*j,1/2) and radius 1/2. These spheres make first level of the tesselation.

To build next levels I had to describe movements of H3. It was easy: this group is equivalent to the group of Möbius transformations of C ((a*w+b)/(c*w+d)). To preserve the tesselation we should take a,b,c,d from CZ=Z[j]/(j^2-j+1). So we can show that orispheres may be indexed by "rational" numbers CQ=Q[j]/(j^2-j+1)=CZ/CZ.

Periodic colorings of CQ may be derived from colorings of {6,3}. Each of them is defined by some ideal IZ=Z[x]/{c0,c1*x+c2,x^2-x+1}. If we take fractions a/b "coprime" a,b from IZ and call a1/b1 and a2/b2 equivalent only when a1=a2*x^k, b1=b2*x^k (where x is the element from definition of IZ), we’ll get coloring of CQ :) Not very easy, and I haven’t tried to write strict description of all this.

Sticker shrinking is not implemented - now I took fixed coordinates of blocks’ verticies. It’s one of more things to do.

(b) puzzles use some reduction of paintings described here. I’ve not tested them, may be there are some wrong points in the model.

Autorotation and autosliding - may be… Are you sure that it will help more than moving of the mouse?

Good luck!

Andrey

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> Wow Andrey, this is amazing!

>

> I didn’t even know a {6,3,3} tessellation was attainable. I glossed

> over such possibilities after reading on Wikipedia that there were only 4

> hyperbolic honeycombs. But going back now, I see that claim is "constrained

> by the existence of the regular polyhedra {p,q},{q,r}". So in this case,

> the "regular polyhedron" {p,q} is an infinite hexagonal tiling! (am I

> understanding right?) I’m looking forward to studying this more.

>

> I’d also be curious to hear of any new found knowledge about how you

> determined allowable coloring sets.

>

> I’ve only had a few minutes to play with it, but here are a couple quick

> comments:

>

> - the sticker size slider isn’t working for me, and I wasn’t having much

> luck trying to edit this setting in the settings file either.

> - both of the (b) puzzles crash the program for me, and I have to delete the

> settings file to get the program to start again after that.

> - I would love it if you could implement the auto-spinning as in MC4D. I so

> want to set the thing in motion and watch it for a while to try to

> understand the space better. I’m facing the frustrating feeling people must

> get when they want more from something I make! Care to open source your

> code? :D

>

> Truly Fantastic Andrey!

>

> Roice

>

>

> On Wed, Oct 27, 2010 at 5:19 AM, Andrey <andreyastrelin@…> wrote:

>

> > Guess what is it ;)

> >

> >

> > http://groups.yahoo.com/group/4D_Cubing/photos/album/1962624577/pic/452190950/view?picmode=large&mode=tn&order=title&start=1&dir=asc

> >

> > Program is here:

> >

> > http://games.groups.yahoo.com/group/4D_Cubing/files/MC7D/mht633.zip

> >

> > It is not complete - Save/Load, animation and macros are not implemented,

> > and not tested at all. But there is Help window (for clicks and navigation)

> > - on Ctrl-F1 key. For colors editing and highlighting by mask use Ctrl-Right

> > click on the sticker.

> >

> > My first impression - solving is impossible even for small puzzles:)

> >

> > Good luck )))

> >

> > Andrey

> >

> >

> >

> > ————————————

> >

> > Yahoo! Groups Links

> >

> >

> >

> >

>