Message #1961

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Another {7,3} puzzle
Date: Tue, 13 Dec 2011 19:39:49 -0600

On Tue, Dec 13, 2011 at 12:23 AM, schuma wrote:

> Roice, thanks for making the beautiful {6,3} and {5,3}!

You’re welcome :) I’m glad the work I did to make things configurable is
paying off a bit now. These both only took only a few minutes to
configure. I think I’ll send an email soon describing how to do this.

> It’s an optical illusion that the face circle appears larger. I think it’s
> because of the hexagons, and the fact that the face circle is one color
> whereas the others are filled by two or three colors.

The optical illusion got me too! A consequence of your observation is that
all the small pieces are now the same size. It would therefore be possible
to extend this puzzle so that edges could twist through 1/6th a turn
instead of 1/2 turn. Such twists could end up locking adjacent vertex
twists, but this would extend the orbits of the small pieces (any of them
could go to any other). But honestly, the thought of trying to code this
extension scares me - it’s nice to not have to worry about tracking things
like locked twists.

> (2) In {5,3}, if the circles are perfect circles, there should be some
> tiny pieces. Am I right? One can make the vertex circles tangent to each
> other, and then make the face circles the right size to be tangent to the
> vertex circles. But when he/she adds the edge circles, these circles not
> necessarily pass through the tangent points. By zooming it in, it seems
> like the adjacent edge circles intersect within the face circles by a
> little bit. If it’s true, I appreciate eliminating the tiny pieces to keep
> this puzzle neat.

Wow, this I did not expect, but you are right! Since the {5,3} doesn’t fit
together perfectly, I immediately suspected the {7,3} does not either, and
sure enough, that is true as well. In both cases, the thickness I used for
the slices had the effect of removing those tiny pieces (the slice
thickness is configurable, so this doesn’t have to be the case). I like
them better without the tiny pieces as well, but this diminishes the
elegance of these puzzles a bit, excepting the {6,3} of course.

> (3) I noticed the geometric issue of {5,3} because I was trying to draw
> the corresponding cube puzzle ({4,3}). And I found that to prevent the
> small pieces in {4,3}, one has to make some circles into ellipses. And the
> result is not that neat.
>
> Another solution to make a nice {4,3} of this kind is to change the size
> of the face circles so that they are tangent to the edges. The puzzle would
> be like this one (Gelatinbrain 3.5.2):
>
I made the {4,3} FEV puzzle you suggest, but it does have tiny pieces in
MagicTile (due to interaction between edge and vertex circles). I haven’t
added to the program yet, because I’d like your opinion between two
options, one leaving the tiny pieces
in<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/1173747084/view?picmode=original&mode=tn&order=ordinal&start=1&dir=asc>,
and one where the slice thickness is increased enough to avoid
them<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/115232229/view?picmode=original&mode=tn&order=ordinal&start=1&dir=asc>.
Or maybe something else would be better. Let me know what you think.

Roice