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 Post subject: Simulator of “Twisty Star” (the Compound of Five Tetrahedra)
PostPosted: Tue May 08, 2012 4:30 pm 
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Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
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:
Attachment:
Solved.png
Solved.png [ 20.49 KiB | Viewed 3390 times ]

Attachment:
Scrambled.png
Scrambled.png [ 20.89 KiB | Viewed 3390 times ]

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.

Puzzle builders, I have a request for you. Do you think it’s feasible to build a physical version of it? I think it’s a beautiful object, and not very hard to solve. At least I want to own one.

I’d like to thank Melinda, Roice, Jeremy and Brandon for their feedback.

Please let me know if you see more glitches.

Have fun solving it!

-- schuma

ps. I found the notebook I mentioned. Here's what I drew 14 years ago. It turns out to be the chiral twin of the shape in the applet.
Attachment:
photo_resized.jpg
photo_resized.jpg [ 39.83 KiB | Viewed 3230 times ]


Last edited by schuma on Wed May 09, 2012 3:10 am, edited 1 time in total.

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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Tue May 08, 2012 5:08 pm 
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Hey Nan, it's great to see this and congratulations on your first major twisty puzzle program!

The puzzle seems really hard. I haven't even identified all of the pieces. I have pure commutators for several of them but no real solving strategy yet.

In dragging around the puzzle to change the view, at certain views you draw a polygon that should be behind another. I'm not sure how you're doing your back-face culling / z-ordering but this might not be a bug in your algorithm but rather the result of floating point roundoff error. This is really common with 64bit floating point and certain trig operations. Don Hatch has written about some common pitfalls and ways around them here http://www.plunk.org/~hatch/rightway.php. I hate floating point. When I do need to use it for precise stuff I usually use GMP http://gmplib.org/. Java has BigDecimal http://docs.oracle.com/javase/6/docs/api/java/math/BigDecimal.html but I'm not sure if it supports transcendental functions.

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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Tue May 08, 2012 5:20 pm 
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bmenrigh wrote:
In dragging around the puzzle to change the view, at certain views you draw a polygon that should be behind another. I'm not sure how you're doing your back-face culling / z-ordering but this might not be a bug in your algorithm but rather the result of floating point roundoff error.


Yeah, this is a known bug. I know the occlusion fails from the beginning. I'm ordering the polygons with respect to the z coordinates of their centers, and draw them from the farthest one to the closest one. It's known as the painter's algorithm [http://en.wikipedia.org/wiki/Painter's_algorithm], and it's known that it fails in some cases. The geometry of this puzzle is actually complicated because it's not convex. Because I know the algorithm fails in some certain configurations, I have manually fine tuned the centers of the polygons a little bit, so the reference centers are no longer the barycenters any more, just to alleviate the issues. But the issue still visible, especially during animation.

Maybe I'll learn some better algorithms and fix this issue in the future.


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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Tue May 08, 2012 8:01 pm 
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Schuma, congrats on your first puzzle simulator! This is a pretty puzzle indeed, and even if there may be bugs, I think it functions quite well.

The compound of 5 tetrahedra holds a special place for me too, as it was Tom Hull's origami model of this shape that got me into designing origami woven polyhedra compounds of my own, which has been a hobby of mine for many years now. If you are interested in making an origami FIT yourself, he has diagrams for it on his website here: http://mars.wne.edu/~thull/fit.html

With regards to solving this puzzle, I find it visually very confusing, but I have worked out the following solve order with commutators that may not be optimal:
Corners (20 3-sticker pieces): [1,1]
Centers (60 pieces): [5,1] (I feel like this is pretty bad, but I didn't find a shorter one)
Corner-Edges (60 pieces): [3,1]

Those are some strange piece-types. I wonder what the puzzle might look like if made as a GB dodecahedron in the 1.2.x series. Also, even after having worked out a solve method, I think an actual solve would be very confusing/difficult.

I doubt a physical version of this is possible because of how the Corner-Edges would have basically nothing to hold onto I think, but maybe an actual puzzle designer will have a better idea (they always surprise me).


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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Tue May 08, 2012 8:08 pm 
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DKwan wrote:
Schuma, congrats on your first puzzle simulator!

Thank you! Actually I made a 2D twisty puzzle years ago. [http://twistypuzzles.com/~sandy/forum/viewtopic.php?f=1&t=15234] But this one means a lot more for me.


DKwan wrote:
Those are some strange piece-types. I wonder what the puzzle might look like if made as a GB dodecahedron in the 1.2.x series.


I have just analyzed this puzzle. All the pieces here can be found in GB 1.2.4. The interesting thing is that on GB 1.2.4 the smallest triangles are in two mirrored (chiral) orbits. But on Twisty Star, only one orbit exists. You have to go to the other chiral form of Twisty Star to find the other orbit!


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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Tue May 08, 2012 11:58 pm 
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I have adjusted the parameters of the painter's algorithm, so that the glitch that appears in the default view is gone.

Attachment:
Image 000.png
Image 000.png [ 2.66 KiB | Viewed 3252 times ]


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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Thu May 10, 2012 2:13 am 
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I've updated the simulator, adding several features, including a "mirror" button, using background color to indicate whether the puzzle is solved or not, a sticker filter to display certain types of pieces, detecting whether a type of pieces is solved or not, etc. With these features, I found the solving feature much comfortable. For example, since the solving status of a certain type is shown explicitly, if I make a mistake undoing setup, I can immediately see it and thus undo some moves.

This is a screenshot showing only two types of pieces.
Attachment:
Image 000.png
Image 000.png [ 28.14 KiB | Viewed 3192 times ]


Last edited by schuma on Thu May 10, 2012 2:56 am, edited 1 time in total.

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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Thu May 10, 2012 2:53 am 
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Thanks for making this Schuma, it is really beautiful, and visually difficult! I have no time to try a solve at the moment, but it is on my list :) . Those new features will make it more accessible I think.

Cheers,
Burgo.

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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Fri May 11, 2012 4:30 pm 
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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.


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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Fri May 11, 2012 9:43 pm 
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ADD OIL! :lol: :lol:
It is beautiful.
The puzzle's chinese name made me laugh yesterday.You call it "拧巴星" :D :mrgreen: :lol: :P

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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Fri May 11, 2012 9:59 pm 
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csgg wrote:
The puzzle's chinese name made me laugh yesterday.You call it "拧巴星" :D :mrgreen: :lol: :P


Thanks. Even "twisty" is not a proper English word. And I don't know how to translate it into Chinese. 拧巴 is the best one I can come up with. It sounds even funnier than the English name.


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 Post subject: Re: Simulator of “Twisty Star” (the Compound of Five Tetrahe
PostPosted: Fri May 11, 2012 11:39 pm 
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schuma wrote:
A very interesting question about this puzzle is "how many distinct solved states are there?"

The permutations and chirality of the corners with this question can be similarly modeled using the faces of a 5-color dogic. There are 4 distinct ways to evenly distribute the colored faces of a 5-color dogic, corresponding to the same 2 color-schemes and their chiral pairs as you have described for this puzzle.

I have often wondered why people don't consider those as the "true" solved states of the 5-color dogic rather than the (in my opinion) ugly color schemes with the same-colored faces in adjacent groups. Of course swapping the faces of a 5-color dogic is much easier than swapping around the corners of the 5-tetrahedra puzzle! =P


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