Message #1956

From: schuma <mananself@gmail.com>
Subject: Re: Another {7,3} puzzle
Date: Tue, 13 Dec 2011 06:23:11 -0000

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

I just solved them, and here are some interesting things about them:

(1) Melinda said, in {6,3}, "It’s slightly disappointing that the face pieces aren’t exactly the same size as the others but I think it looks the most attractive."

This is very funny. The face circles do seem to be larger than other circles. But if you think about it… they are actually exactly the same size. The geometry is like placing six quarters (vertex circles) around another quarter (face circle). <cf. http://etc.usf.edu/clipart/42900/42920/circle-26_42920.htm> One can verify the equality of the sizes by drawing some horizontal lines in a screenshot.

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.

(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.

(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):

<http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/hexa_fe1.gif>

I like this configuration since all the circles have the same radius.

Note that GB 3.5.2 is not an FEV turning puzzle. In GB 3.5.2 vertices are not turning and edges can turn by 90 degrees (by allowing deformation). I’m not describing it very well. One may want to check GB to see how it works. I think it would be aesthetically better to make it into a FEV turning puzzle. I will suggest it to Gelatinbrain soon.

Nan

— In 4D_Cubing@yahoogroups.com, Melinda Green <melinda@…> wrote: