# Message #1217

From: Matthew <damienturtle@hotmail.co.uk>

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

Date: Wed, 27 Oct 2010 22:56:11 -0000

I really wish I wasn’t just a second year maths student and knew more geometry, unfortunately a lot of that went over my head (I’ll understand it one day). I can still mess around with it and get a feel of how it works though, even if I don’t know much of the theory behind it. One question I do have, how many hexagons per cell? I’m curious but I find it pretty hard to count them manually.

Another first step towards a masterpiece Andrey, and probably the craziest puzzle we have yet seen, so well done as usual :).

Matt

— In 4D_Cubing@yahoogroups.com, "Andrey" <andreyastrelin@…> wrote:

>

> 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

> > >

> > >

> > >

> > >

> >

>