# Message #1226

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

Subject: Re: MHT633 v0.1 uploaded

Date: Thu, 28 Oct 2010 18:55:14 -0000

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

>

> @Matt

>

> > One question I do have, how many hexagons per cell? I’m curious but I find

> > it pretty hard to count them manually.

>

>

> Not only are there an infinite number of cells in this tessellation, but

> every one of these cells has an infinite number of hexagons! (The

> identification of cells mentioned above also serves to turn this aspect of

> the puzzle into a manageable finite situation.) The cells are a {6,3}

> tiling, the same as you’d see on a bathroom floor, only going on forever.

> This tiling is curved when living in hyperbolic space though, and the

> result is that the view from within hyperbolic space is such that only a

> small number of the hexagonal facets can be seen - the rest curve out of

> view. With sticker shrink, I suppose it would be possible to display a much

> larger number of the facets, since that allows one to see somewhat through

> the cell.

>

> Hope that’s helpful and not too much rambling…

Ah, thanks for that. I knew {6,3} was an infinite tiling (and that {6,3,3} is an infinite tiling of cells), but I wasn’t sure if anything funny happened here which changed that (the cells look reasonably finite, but they behaved strangely enough that I knew they could be infinite). I don’t mind rambling, it would be hard to explain all this using too much detail! I now just need to slowly get my head around infinite tilings infinitely tesselated in a hyperbolic space.

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

>

> Matthew,

> I think that understanding of this geometry for second year maths student is not more more difficult than for PhD specialist in Computer Algebra. Hyperbolic geometry never was in my field of research, and half of the math for this puzzle I’ve developed from the scratch during my vacation at Madeira (and first pictures with coloring of {6,3} tilings were washed by the tidal wave).

> As Roice said, each cell is infinite. But is has periodic coloring, and numbers of _different_ 2C stickers in one face are the folloing:

> 8 Colors - 7 stickers

> 12 Colors - 9 stickers

> 20 Colors (a) - 16 stickers

> 20 Colors (b) - 12 stickers

> 28 Colors - 13 stickers

> 32 Colors (a) - 21 stickers

> 32 Colors (b) - 31 stickers

It’s not that I can’t understand it, it’s that I don’t know the theory yet, but after two or three years I should be in a far better position to understand all this once I have been taught more of the concepts involved. And I’m reasonably familiar with the MagicTile program (I’ve solved the Klein’s Quartic {7,3}) so I have a rough understanding of hyperbolic space. I’m going to have fun learning a few things about it from what I can find on the internet :). Yet again I’m made aware of geometry I didn’t know that I knew nothing about until after someone programmed a puzzle with that geometry! I look forward to later versions so I can see what solving is like.

Matt