Message #2142
From: schuma <mananself@gmail.com>
Subject: Re: Simulator of Twisty Star (the Compound of Five Tetrahedra)
Date: Fri, 11 May 2012 21:31:15 -0000
A very interesting question about this puzzle is "how many distinct solved states are there?"
Since there are five tetrahedra with five colors, if all the tetrahedra are numbered, there are 5!=120 ways to color them. But let’s not number them, but use the common convention that two coloring schemes are identical if a global rotation can change one to another. Mirroring is not allowed because this shape is chiral and thus asymmetric with respect to mirroring. Since the shape has 60 rotational symmetries, the number of distinct coloring schemes is down to 2. One still need to show the existence of a sequence of valid moves to get from one coloring scheme to another. I just did that by solving the puzzle from one scheme to the other. So I’ve just confirmed that "there are two distinct states for Twisty Star".
The two states can be described in this way. If you look at the center of puzzle in the default view, there are five petals just like a flower. From magenta, going clockwise, the colors are (magenta, yellow, green, red, blue). Because these five petals are corners of five different tetrahedra, the coloring of a "flower" determines the coloring of the whole puzzle. If you look at the puzzles from other angle, you can find other 11 "flowers" with different color sequences. In fact, for any even permutation of (magenta, yellow, green, red, blue), you can always find a flower and a starting color such that the clockwise color sequence is that permutation. But for any odd permutation, it’s not there. All odd permutations are on the other solved state.
So from the original state, if you want to do a three cycle to three colors, like magenta -> yellow -> green -> magenta, you don’t have to twist the puzzle. Global rotation suffices. If you want to swap two colors, you need to do a lot of twists.
— In 4D_Cubing@yahoogroups.com, "schuma" <mananself@…> wrote:
>
> Hi everyone,
>
> The compound of five tetrahedra is a geometric object that interests me for a long time. As a freshman, I carefully drew it using compass and straightedge on my notebook. I’ve always been intrigued by its relation to the dodecahedron, the chirality, and its pretty shape. In short, it’s my geometric crush.
>
> So far I haven’t seen a twisty puzzle based on this shape. Recently, inspired by Leslie Le’s Super Star, I decided to write a simulator. That’s definitely a good motivation for me to learn Java applet programming. It’s done and can be found here:
>
> http://people.bu.edu/nanma/TwistyStar/TwistyStar.html
>
> You may need to upgrade JRE to see it. I call it "Twisty Star" because it’s a twisty puzzle, and also the shape looks twisted. Some screenshots:
>
> [http://twistypuzzles.com/forum/download/file.php?id=30776]
> [http://twistypuzzles.com/forum/download/file.php?id=30777]
>
> It’s a compound of five tetrahedra intersecting with each other. The 20 vertices coincide with the vertices of a dodecahedron. The five tetrahedra are colored by five colors.
>
> It can be twisted around 20 vertices. Since the cuts are right above the faces of the tetrahedra, it can be regarded as "face-turning" as well as "vertex-turning". In other words, the dual of this solid is its mirror. So it’s almost self-dual. Therefore there’s a one to one correspondence between the vertices and the faces. Mathematically it’s related to the face-turning icosahedra.
>
> The vertex to twist is labeled by a small circle around it, and the moving region is highlighted. Even with these assistances, it’s not easy to see how it turns. Sometimes with only one twist away from the solved state, I just can’t find the twist. I haven’t solved it yet. For this color scheme, I guess it has multiple solved states, meaning that one can, for example, swap the red tetrahedron with the blue one.
>
> I think it’s possible to build a physical version. I posted this simulator on the Twistypuzzles forum [http://twistypuzzles.com/forum/viewtopic.php?f=1&p=281801] and request the builders over there to consider it. Maybe someone will make it someday.
>
> I’d like to thank Melinda, Roice, Jeremy and Brandon for their feedback.
>
> Please let me know if you see more glitches.
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> Have fun solving it!
>
> – schuma
>