Message #1219

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: MHT633 v0.1 uploaded
Date: Thu, 28 Oct 2010 05:21:40 -0000

Melinda,
Yes, some puzzles are not very large. 8 Color version has 28 2C pieces, 56 - 3C and 28 - 4C, that is close to 3^4 cube. Problem is in the high connectivity of the model: every two faces of 8Color have common 2C piece! The same is true for 32 Colors (b) (but that puzzle has about 2000 pieces, like 3^7 or 120-cell).
You can see other sides of each cell if you ctrl-click some its point (and it jumps to the center of the screen) and then use left-drag to fly around this point. But you always will see only a part of the surface. And there is no "3D projection" of hyperbolic space! Yes, I had to calculate some virtual Cartesian coordinates (depending on the camera position) to upload the model in DirectX, but with only one reason - to use 3d viewer features (like z-buffer and lighting computations).
For navigation controls I’ll say that now they are almost the same as in MC4D. Your center of projection has the same role as my camera
"point of view". It is some point of 4D-sphere/H3 space, it’s always shown in the center of the screen, and left-dragging rotates camera
around this center keeping the distance to it. Right-dragging(up/down) moves camera and center along the direction of view. Shift-left-dragging keeps camera in place but turns it so that center moves to another point of space (it’s funny that I didn’t know about this feature of MC4D until today - but yes, it’s there and it’s the same as in MHT). So I don’t see reasons to change these controls :) Three different "zoomings" in MHT may lead to some confusion - you may move close to center (and objects that were on sides of camera will appear behind it), you may narrow view angle (in this case objects
on the side will remain on the side and you’ll see them sometimes when rotate camera around POV) and you may play with FishEye slider - in combination with wide angle you will see objects on the back side of you :)

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

Roice,
Now I show all stickers inside the fixed distance from some point: for version 0.11 this distance is 2.4 (where 1 is the distance between centers of 2C) - and there are 4750 stickers arranged in 80 faces in this ball. In the next version I’ll increase ball radius to 2.9 (that’ll give 12400 stickers in 224 faces). I thought about the explicit control of this radius (with some fixed positions), but it’s not so easy.
To select position of the ball center use "Area center" slider. On the left side it makes the area centered in the camera position (it’s good for strong FishEye view and for frequent shift-left-drag movement) and on the right side area is centered at the point of view (=rotation center). If you have slider in this position, area will not be recalculatied during left-drag rotations, because center remains the same.
I know about error of new puzzle start - it can be fixed with one line of code when I’ll find the proper place for it )))
Second image is the view from inside of the cell. Of cource there you are inside of the central sticker, but from inside it’s transparent, so you don’t see its faces. I say - why not? It’s the only position from where you can see all stickers of the cell (but they hide the rest of space :( ) And you can see that face is really infinite :)

And don’t wait much from sticker shirinking: you will see more of 1C, but I’m not sure about stickers on the back side. And probably view with small stickers will be a little nonrealistic…

Good luck!

Andrey