Message #1910

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: {7,3} vertex and edge turning puzzles
Date: Tue, 08 Nov 2011 17:35:22 -0600

> The FT {7,3} is to the Rubik’s cube as the VT {7,3} is to the Dino Cube
> (with stationary corners), as the ET {7,3} is to the helicopter cube (a
> better analog is TomZ’s Curvy Copter).
>

Oh cool, makes me wonder if I should update the naming of these to reflect
these analogies. I welcome opinions (I’m also not up to speed on the
colorful names, and would appreciate a list if people did want them
displayed).

>
> Comparing the depth of cuts in ET{7,3} and ET{3,7}, I wonder what’s the
> unit of depth. If the length of a side is the unit, then they have similar
> depth. But I think a better comparison is to count the number of circles
> that each circle intersects with. So I would say ET {7,3}’s cuts are much
> shallower and therefore it has much less pieces. The same for the VT
> puzzles.
>

Interesting. My previous comment was comparing cut depth based on the
length of an edge being the unit. On the ET{3,7}, the circle radius is 0.5
times the edge length. On the ET{7,3}, I made it 0.7 times the edge
length, so the latter is deeper in this sense. But the edge lengths are
not the same, and so this is maybe not the right way of looking at it.

I checked the circle radii using the distance metric in hyperbolic
geometry, and the situation is indeed the opposite from that perspective.
The ET{7,3} radius is about 0.40, and the ET{3,7} radius about 0.56. When
I looked at the puzzles again, it was obvious that the circles look bigger
in the ET{3,7}.

Consider the distance metric perspective of cut depth in the spherical
world… For a sphere of unit radius, a Megaminx will be less deep-cut
than a Rubik’s Cube, because the {5,3} tiling for the dodecahedron must
have smaller polygons to fit on the sphere than the {4,3} tiling does. But
the puzzle difficulties are similar. Like you say, difficulty does seem to
be more a function of number of intersections between slicing circles
(rather than cut depth measured as an absolute distance in the relevant
geometry).

Btw, here is something that surprised me. {3,7} has a longer edge length
than {7,3}, so I wondered how the {3,5} and {5,3} compare. A guess might
be that the relationship is reversed because the geometries are different,
but the triangular {3,5} tiling wins out in spherical geometry too. I bet
it is the case that the edge length of {p,q} is always longer than
{q,p} when p < q.

>
> They are overall easy puzzles and I love them. I do solve complicated
> puzzles but I love the simple ones. Thanks.
>

You’re welcome :) Glad you enjoyed them!

Roice