Message #1912

From: schuma <mananself@gmail.com>
Subject: Re: {7,3} vertex and edge turning puzzles
Date: Wed, 09 Nov 2011 01:59:22 -0000

Maybe another possibility is to use ceased heptagons, just like this one:

<http://westy31.home.xs4all.nl/Geometry/KleinHoles.jpg>

This photo is from a long webpage by Gerard Westendorp. If you go to <http://westy31.home.xs4all.nl/Geometry/Geometry.html#constant> and scroll UPWARD a little bit, you can find the discussion about it where he considered different ways to "glue" the cardboard model. One way results in Klein Quartic. Another way results in the dual of the {3,7} IRP that we have in MagicTile. There are even more ways resulting in other {7,3} with other connectivity. I need to catch up on math but it looks interesting.

Some experience about solving IRP puzzles:

I’ve solved three simple IRP puzzles. For each of them, I solved it in two ways: the "show as IRP" option true and false. I found solving in the IRP view more challenging than in the Poincare disk (PD) view, because (1) in PD I can see everything but in IRP I can only see about half; (2) in IRP the pieces with best visibility are on boundary, but some of them are truncated; (3) when I’m making a turn sometimes the face that I need to click on is facing the other direction. Sometimes I have to carefully adjust the viewpoint and click on some inside faces. Because of the above reasons, I always first look at the puzzle with the IRP view off to study it, before turning on the IRP.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:
>
> That’s a really creative idea. When I first saw your suggestion, I had the
> same question as Nan.
>
> MagicTile can’t support this with only via configuration at the moment. It
> will take some code work, but I like the thought of trying to handle it in
> the future.
>
> Like the {3,7} puzzles, my bet is this IRP puzzle will not have the same
> connection pattern between heptagonal faces that the current KQ puzzle has.
>
> A further speculation is that although there might not be a {7,3} IRP that
> fits into *R**3*, maybe there is one that fits into *R**4 *or something
> else. In any case, I wouldn’t be surprised if there is a different set of
> 2-dimensional IRPs that can work in 4-dimensional space, in a similar way
> to the ones Melinda has enumerated for 3 dimensions.
>
> Roice
>
>
> P.S. Short implementation thoughts, probably just for me…
>
> A longer term goal is to support truncated tilings (giving puzzles based on
> uniform tilings, etc.). I’m not sure how it is going to evolve exactly, but
> one approach would be to still have one texture mapped to each polygon in
> the underlying regular tiling. It’d just be that the texture now contained
> portions from multiple tiles instead of just one (e.g., a soccer ball
> puzzle would have 32 faces, but only 20 textures). The {7,3} IRP could
> potentially fit well into that piece of work. What I’m thinking is that the
> code would handle this as a {3,7} tiling that is truncated all the way to
> its dual.
>
> Maybe the right solution for truncated tilings is still one texture per
> face though, in which case this IRP could be harder to do…
>
>
> On Tue, Nov 8, 2011 at 1:03 AM, Melinda Green <melinda@…>wrote:
>
> > No, I was inquiring into the possibility of such a puzzle. I meant to
> > send privately to Roice because I didn’t want to pressure him but I
> > screwed up.
> >
> > I’m pretty sure there isn’t a true {7,3} IRP though I hope that I am
> > wrong. I could however imagine a MagicTile version in which a {7,3}
> > texture could be mapped onto the VT {3,7} IRP surface to approximate
> > one. Seems doable though the real way to do this sort of thing might be
> > to map it onto a minimal curvature surface with the same topology. The
> > IRPs are interesting because they can be constructed using flat
> > polygonal faces but there are all sorts of crazy puzzles that become
> > possible without that constraint.
> >
> > -Melinda
> >
> > On 11/7/2011 10:50 PM, schuma wrote:
> > > Is there a {7,3} IRP?
> > >
> > > — In 4D_Cubing@yahoogroups.com, Melinda Green<melinda@> wrote:
> > >> How about a FT {7,3} IRP?
> >
>