# 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