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 Post subject: We need icosahedra
PostPosted: Fri Jul 23, 2010 7:53 pm 
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so many puzzle are coming out these days.

why is there no icosahedron?

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 Post subject: Re: We need icosahedra
PostPosted: Fri Jul 23, 2010 7:59 pm 
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or trigonal prism?

pentagonal prism?

and more balls?

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 Post subject: Re: We need icosahedra
PostPosted: Fri Jul 23, 2010 8:04 pm 
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dont fret, I will be coming out with one seriously awesome icosahedron soon.

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 Post subject: Re: We need icosahedra
PostPosted: Fri Jul 23, 2010 8:15 pm 
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do you mean releasing to the market? or have you finished a custom puzzle?

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 Post subject: Re: We need icosahedra
PostPosted: Fri Jul 23, 2010 8:52 pm 
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Me too, I'm also making one-just a custom

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 Post subject: Re: We need icosahedra
PostPosted: Fri Jul 23, 2010 9:06 pm 
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guys! I am talking about mass production products.

Have they completely forgotten about this shape? or do they simply fear to fail miserable because of the Dogic's epicness?

A 3x3x3 shapemod wouln't be too much.
or Skewb.
or a re-run of dogic!
or..
The Magic Icosahedron (Lee Tut's Icosaminx)

to me those are just too obvious...

:roll:

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 Post subject: Re: We need icosahedra
PostPosted: Fri Jul 23, 2010 11:57 pm 
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Doubleyou wrote:
so many puzzle are coming out these days.

why is there no icosahedron?

Jason and I have each made the "most natural" twisty icosahedron, Gelatinbrain 2.1.1:

Image

Image

Plus Jason has built many other variations in his "Radiolarian" family. Also I have an inside scoop that another icosahedron is probably coming soon, from somebody other than Jason or me!

I had always planned to put my icosahedron up for sale on Shapeways at some point. But it needs a few tweaks before I can do that, and honestly I kind of lost motivation when Jason's appeared just before mine. And other projects have taken priority. But I do expect to get back to it at some point, and make it available.


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 1:00 am 
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Bhearn, can you tell us if yours or any of Jason's FTI can jumble. In theory they can, but I'm not sure if your mechanism allows it. I'm pretty sure all of the FTI so far have a deep enough cut.

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 1:12 am 
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bhearn wrote:
Also I have an inside scoop that another icosahedron is probably coming soon, from somebody other than Jason or me!
Here is the face-turning icosahedron that Bob refers to. It is still work in progress, and it may take a couple of months to turn from virtual to real. Thanks to Jason and Bob for the inspiration!

Oskar
Attachment:
Icosaix - view 2.jpg
Icosaix - view 2.jpg [ 141.7 KiB | Viewed 6550 times ]

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 3:24 am 
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I was wondering why there wasn't any Face Turning Icosahedron that looked like a Tutt's Icosahedron, as it's a Magic to a Face Turning Octahedron, as to a Tutt's Icosahedron to a FTI.

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 4:03 am 
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There is a chance that an icosahedron will get mass produced too! :D Just look what happened with the Gigaminx and Teraminx. I wanted a Gigaminx so badly when I first saw one, and I thought I probably never would get one, but then all of a sudden they get mass produced :D

I'm in an optimistic mood today, for sure :lol: But this might happen. Someone could realize that a mass produced icosahedron would sell well, and make one. All they need is enough people to request it.


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 6:14 am 
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Either way, the FTI in any form will be a beast to solve. Each corner will have five possible orientations, with clockwise/anticlockwise twists always adding up to multiples of 5, resulting in 5^11 orientations and 12!/2 positions for corners alone. It is really the corners that make a Rubik's cube so danged hard to solve without algorithms. While it is technically possible for a FTO corner to be physically placed into four orientations, the fact that there is an even number of faces to a corner prohibits this from taking place, allowing only two possible orientations to exist.

For corners, there are 5^11 orientations and 12!/2 positions.

For edges, there are 2^29 orientations and 30!/2 positions.

For centers, there are 60 triangles in 20 colors of 3 each, so 60! / (3!^20).

All moves result in three-cycles only, so parity is always even.

Puzzle is not deep enough to jumble, so no voodoo mathematics going on while calculating the figures.


So the total distinct number of possible permutations is:
5^11 * 12!/2 * 2^29 * 30!/2 * 60!/(3!^20) = 1.8950767844810268594904722128969 * 10^123

Yes, this puzzle qualifies for the "greater than one googol" club :mrgreen:

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:17 am 
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GuiltyBystander wrote:
Bhearn, can you tell us if yours or any of Jason's FTI can jumble. In theory they can, but I'm not sure if your mechanism allows it. I'm pretty sure all of the FTI so far have a deep enough cut.

OH MY GOD! IT JUMBLES!!!!!!

How did I not know this??? And here I had any number of people playing with it at IPP. Nobody had any idea that it jumbled.

OK, this ups the priority on tweaking it for sale on Shapeways. :D :D :D

Photos shortly.


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:18 am 
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great news! shapeways!!! YAY!!!

LOL at you didnt know it jumbles...

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:20 am 
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bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!


LOL Great news Bob! Can't wait for the photos!

:)


Pantazis

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:29 am 
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Doubleyou wrote:
or trigonal prism?

pentagonal prism?

and more balls?


Are you kidding me? Look at my shapeways shop I have a Triangular Prism on the way, and not so long ago, Clauswe made a pentagonal prism ball with my files...

:) Greg :)

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:53 am 
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...again... I am talking about production puzzles only.

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:54 am 
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I guess these pics would be a little clearer if the puzzle were solved, but hopefully you get the idea. :)

I still cannot get over the fact that I designed and built a puzzle that jumbles, and didn't even know it until 6 months after building it! Thanks so much, GuiltyBystander, for pointing this out to me. I'm going to be grinning from ear to ear all day.

You all know those "does it blend?" videos? I can just see it... "Does it jumble? ... twist twist twist... "YES! IT JUMBLES!"

Image

Image


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 9:57 am 
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bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!
Bob,

Are you sure? Jumbling has a quite strict definition. So let's test your design against the definition.
1) Is your puzzle bandaged in any way?
2) If so, unbandage it.
3) Is your puzzle still bandaged, then go to step 2)
4) Did you enter in an infinite unbandaging loop at step 3)?
5) If so, then your puzzle jumbles, otherwise, it doesn't.

Given the regular geometry of the icosahedron, I am skeptical about it being able to jumble. On the other hand, your picture is very convincing. Wow, what is happening here?

Oskar

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 10:08 am 
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The fact that your FTI is shape-shifting(jumbling?) is very surprising to me. I have designed a few FTI's (where the depth of cuts change) and I don't believe they shapeshift. Did you apply some sort of non-traditional mechanism?

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 10:42 am 
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If you look at the corners of a dodecahedron (the axes in icosahedral puzzles), you see that there are 3 corners directly adjacent to any given face, and there are 6 corners adjacent to those 3. Those 6 are all equidistant to the first one, but if you take the distances going around in either direction, the distances between pairs of them are either 1 or the golden ratio. So, if an icosahedral puzzle has deep enough cuts to intersect these 6 faces as this puzzle does, there is an angle between 0 and 120 degrees that aligns 3 of the six with the other 3 on the part that isn't turning, but that angle isn't 60 degrees (it is about 44.4775 degrees) and I am almost sure that it is irrational, that should mean that it jumbles.
Edit: This is basically the relevant part from the above link, as I found out after noticing it.
Also, wasn't there an inventor of the edge turning cube that made it before finding out about the jumbling as well?

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Last edited by contrabass on Sat Jul 24, 2010 10:46 am, edited 1 time in total.

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 10:44 am 
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Oskar wrote:
bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!
Bob,

Are you sure?

I haven't read the linked thread in detail yet (just got on a train), but by observation, I would be extremely surprised if there were any way to unbandage it.

A very interesting question now is whether YOURS will jumble. Evidently the natural icosahedral geometry jumbles, which is quite surprising. But you have altered that geometry due to the non-planar surface cuts, and to mitigate the self-intersection problem. So I am very curious!


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 10:50 am 
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explained nicely here:
http://twistypuzzles.com/forum/viewtopi ... &start=100

It's the 'extremely long post' by Allagem...
Bhearn, can you please post pictures of the puzzle jumbling? I'd love to see it (:
PS also read my post a few posts below, it partially describes the situation...

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 11:27 am 
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bhearn wrote:
A very interesting question now is whether YOURS will jumble. Evidently the natural icosahedral geometry jumbles, which is quite surprising. But you have altered that geometry due to the non-planar surface cuts, and to mitigate the self-intersection problem. So I am very curious!
Bob,

I am not sure. Here are some pictures trying to replicate with my virtual model what you did with your physical model. According to Bram, jumbling is hard to prevent.

Oskar
Attachment:
Icosaix - view 6.jpg
Icosaix - view 6.jpg [ 142.79 KiB | Viewed 6413 times ]

Attachment:
Icosaix - view 7.jpg
Icosaix - view 7.jpg [ 138.63 KiB | Viewed 6413 times ]

Attachment:
Icosaix - view 8.jpg
Icosaix - view 8.jpg [ 141.4 KiB | Viewed 6413 times ]

Attachment:
Icosaix - view 9.jpg
Icosaix - view 9.jpg [ 98.07 KiB | Viewed 6413 times ]

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 12:15 pm 
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bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!

How did I not know this??? And here I had any number of people playing with it at IPP. Nobody had any idea that it jumbled.

lol, nice. Didn't one of the helicopter cube inventors have a similar experience? Do you think you can direct Jason to this thread and get him to post pics of his jumbling Radiolarians?

Have you tried a full :scrambled: :arrow: :solved: yet? I know you've solved it several times without jumbling. Does the FTI have orbits like the helicopter cube does? I'm curious what, if anything, this means for solving it. You'll have to do jumbling moves triplets much like you have to do them in pairs on the helicopter cube right? I can't wait for your insight.

Oskar wrote:
Given the regular geometry of the icosahedron, I am skeptical about it being able to jumble. On the other hand, your picture is very convincing. Wow, what is happening here?

I was quite surprised when I read about it too. This post shows the jumbling of FTI pretty well. I think the only part missing for a proof is the exact jumbling angle. He approximates the angle at 44.4775121859 but that's not an irrational number. Can anyone do the math to find this angle? I'm not exactly sure how to go about it.

Here's a jaap sphere where you can see the tale-tell signs of jumbling. It doesn't show the angle so it's not exactly a proof either.

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 1:13 pm 
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An exact value for the angle is ArcCos((3sqrt(5)-1)/8). Of the top of my head, I can't prove that this is irrational, but it looks like it and Mathematica isn't simplifying it.

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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 1:39 pm 
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bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!

How did I not know this??? And here I had any number of people playing with it at IPP. Nobody had any idea that it jumbled.

you're welcome. 8-)


contrabass wrote:
An exact value for the angle is ArcCos((3sqrt(5)-1)/8). Of the top of my head, I can't prove that this is irrational, but it looks like it and Mathematica isn't simplifying it.


Ah, I see I was racing Mathematica, no wonder you beat me :lol:

The exact angle is arccos((10s^4-6s^2-8)/(10s^4-6s^2+10)) where s is the golden ratio, s = (1+sqrt(5))/2

This indeed simplifies to arccos((3sqrt(5)-1)/8) which is irrational


By the way Oskar, Jared had that exact puzzle idea over a year ago: http://twistypuzzles.com/forum/viewtopic.php?f=1&t=13026


Also I know I said icosahedral puzzles only jumble if they are cut deep enough, but that was more of a safety margin for me so you guys don't criticize my remark :wink: The only way to NOT have an icosahedral puzzle jumble is too have cuts so shallow that only center pieces and edge pieces are present. The next piece you "run into" can jumble, as well as any pieces beyond that. This means a non-jumbling icosahedral puzzle would have to either cut into the core or use spherical cuts. If it's icosahedral-based and the cuts are planar and the mathematical core is intact, it jumbles 8-)

Peace,
Matt Galla

PS A significantly deeper cut icosahedral puzzle gains an additional jumbling situation- it's the exact same angle but for a slightly different reason. No icosahedral puzzle cut this deep exists yet as far as I know.


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 1:54 pm 
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bhearn wrote:
Oskar wrote:
bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!
Bob,

Are you sure?

I haven't read the linked thread in detail yet (just got on a train), but by observation, I would be extremely surprised if there were any way to unbandage it.

A very interesting question now is whether YOURS will jumble. Evidently the natural icosahedral geometry jumbles, which is quite surprising. But you have altered that geometry due to the non-planar surface cuts, and to mitigate the self-intersection problem. So I am very curious!


In general the simplest way to show that a puzzle can't be unbandaged is to show that the angle the faces need to be rotated to jumble is irrational. In general showing that a number is irrational is a very deep and interesting problem, but I'll take showing that the continued fraction expansion goes on for a while for all practical purposes.

For puzzles where the angle is rational I'm not so sure. Given that it turned out that Battle Gear can in fact be unbandaged, I'm reluctant to make claims about anything else lest I have to eat more crow.

I'm pretty sure that Oskar's will jumble as well. Jumbling is deeply embedded into the geometry of the puzzle, and is extremely hard to block even if you're trying to.

The flavor of this particular jumbling is very surprising and interesting, and different from anything we've seen before. It's particularly interesting that the normal rotations are so symmetric and that it only occurs if the cuts are deep enough.


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 7:27 pm 
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SEBUVER wrote:
I was wondering why there wasn't any Face Turning Icosahedron that looked like a Tutt's Icosahedron, as it's a Magic to a Face Turning Octahedron, as to a Tutt's Icosahedron to a FTI.

If you mean an FTI that looks like Oskar's does, then the reason is that technically, it doesn't work! (Read on...)

Allagem wrote:
By the way Oskar, Jared had that exact puzzle idea over a year ago: http://twistypuzzles.com/forum/viewtopic.php?f=1&t=13026

Right. I was also looking at this back then, and likely Jason was too. I gave up on trying to build that when I realized that the surface self intersects during rotation. I still wanted a way to avoid the tiny wing edges, just to simplify the mechanism, so next I tried this:

Image

This doesn't self intersect during turns, as long as the space behind the star edges is hollowed out. But the problem is, without the wing edges, there is nothing to hold the corners in. Try to make a mechanism for this, and the corners just fall out. Eventually I decided that the "canonical" FTI, Gelatinbrain 2.1.1, was the only way to go, tiny wing edges and all.

Oskar managed to solve both of these problems. For the self intersection, the raised, beveled surface helps, plus I believe there's a tiny amount of deformation required during a turn. To hold the corners in, the wing edges actually exist, but they are purely internal. I would not trust myself to design such a mechanism; I'd expect those pieces to catch in ways that would be hard to correct. But Oskar has shown me his design, and it looks good; also his track record speaks for itself.

stardust4ever wrote:
Either way, the FTI in any form will be a beast to solve. ...
For corners, there are 5^11 orientations and 12!/2 positions.
For edges, there are 2^29 orientations and 30!/2 positions.
For centers, there are 60 triangles in 20 colors of 3 each, so 60! / (3!^20).

... and don't forget the wing edges, at least in Jason's and my version (2.1.1)!

GuiltyBystander wrote:
Have you tried a full :scrambled: :arrow: :solved: yet? I know you've solved it several times without jumbling. Does the FTI have orbits like the helicopter cube does? I'm curious what, if anything, this means for solving it. You'll have to do jumbling moves triplets much like you have to do them in pairs on the helicopter cube right? I can't wait for your insight.

To tell the truth, I haven't actually solved it at all. I started solving it, and I'm pretty sure I figured out all the necessary transforms, but the puzzle is so catchy that I didn't bother to finish. It turns well when properly aligned, but otherwise catches easily on the little wing edges. (Thus, the needed Shapeways tweaks.)

However, no, there are no orbits like on the Helicopter cube. I don't expect that one will be able to reach otherwise inaccessible states by jumbling and then restoring the shape.

Allagem wrote:
bhearn wrote:
OH MY GOD! IT JUMBLES!!!!!!

How did I not know this??? And here I had any number of people playing with it at IPP. Nobody had any idea that it jumbled.

you're welcome. 8-)

Thank you very much for discovering this amazing fact. I only wish I had seen the original thread heading into IPP; it would have been great to show off a jumbling icosahedron!

Bob


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 Post subject: Re: We need icosahedra
PostPosted: Sat Jul 24, 2010 7:32 pm 
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Bram wrote:
I'm pretty sure that Oskar's will jumble as well. Jumbling is deeply embedded into the geometry of the puzzle, and is extremely hard to block even if you're trying to.

I'm inclined to agree. The one caveat is that, due to self intersection, there is a slight amount of deformation required during a turn, I believe. If this deformation is occurring at the jumbling angle, then that could change things.


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 Post subject: Re: We need icosahedra
PostPosted: Sun Jul 25, 2010 11:05 am 
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read the topic title. Your are going way off topic. enough bandage/jumble discussion already.. thanks :(

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 Post subject: Re: We need icosahedra
PostPosted: Mon Jul 26, 2010 12:13 pm 
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Okay, back on topic. I count 4 (5 if you count the cornerless radolarian) FTI that have been made so far and all were withing a 4 month span. All are very stunning puzzles and I wouldn't mind owning one of each.
Bob Hearn's Face-Turning Icosahedron
Jason's Radiolarian
Cornered Radiolarian
Jason's Circo-Radiolarian / Radiolarian II
Jason's Radiolarian 3

I think this is a nice set of puzzles. If don't count higher orders, you could say that there's more FTI than FTCubes :P . I'm sure we'll see deeper cut or higher order FTI's eventually, but give it time. It took Aleh almost 4 years to go from Brilic -> Starminx

One "simple" twisty puzzle I'd like to see just for completion is the helicopter octahedron. There's already several different helicopter cubes and face-turning rhombic dodecahedra. grigr looks like he has some plans, but nothing's been built yet as far as I know.

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 Post subject: Re: We need icosahedra
PostPosted: Tue Jul 27, 2010 10:56 pm 
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Yes, the jumbling ability is awesome, whether the winglet pieces are included in the puzzle or not.

So finally, we have face-turning versions of all five platonic solids. Four of these have been mass-produced. Four of them are non-jumbling.

Pyraminx, aka "Face Turning Tetrahedron" (also corner-turning as well)
Rubik's Cube, aka "Face Turning Cube"
"Face Turning Octahedron," aka puzzle of the same name.
Skewb Diamond, aka Junior version of "Face Turning Octahedron"
Megaminx, aka "Face Turning Dodecahedron"

and finally, (yet-to-be-mass-produced)
Jason's Radiolarian, aka "Face Turning Icosahedron"

I believe this challenges Oskar's Definition of a "Doctrinaire" Twisty Puzzle, because the FTI is based on Platonic Geometries, yet it does Jumble. So far, all mass-produced pure "twisty" puzzles, with the lone exception of Helicopter Cube, have geometries that have been either based on Platonic, dihedral, or 2D geometry. It would be difficult to argue that the FTI is not a doctrinaire Twisty Puzzle, as it is purely based on the geometry of a regular solid. It meets all other criteria for doctrinaire puzzles as well, such as all axes being uniform and intersecting at the zero point, etc, etc. The implications are astounding :shock:

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 Post subject: Re: We need icosahedra
PostPosted: Tue Jul 27, 2010 11:13 pm 
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That is really cool!

Strangely, this isn't the only Face-Turning puzzle that jumbles. The FTRD (face turning rhombic dodecahedron) jumbles. I know that the FTRD is very similar to the Helicopter Cube, but with slightly different cuts. Is the FTI similar to another puzzle with slightly different cuts?

-pi (Eitan)

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 Post subject: Re: We need icosahedra
PostPosted: Tue Jul 27, 2010 11:50 pm 
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stardust4ever wrote:
I believe this challenges Oskar's Definition of a "Doctrinaire" Twisty Puzzle, because the FTI is based on Platonic Geometries, yet it does Jumble. So far, all mass-produced pure "twisty" puzzles, with the lone exception of Helicopter Cube, have geometries that have been either based on Platonic, dihedral, or 2D geometry. It would be difficult to argue that the FTI is not a doctrinaire Twisty Puzzle, as it is purely based on the geometry of a regular solid. It meets all other criteria for doctrinaire puzzles as well, such as all axes being uniform and intersecting at the zero point, etc, etc. The implications are astounding :shock:
I believe the "Doctrinaire" definition most commonly use actually came from Bram. It shouldn't be hard to argue that FTI are non-doctrinaire. There's nothing in that definition that specifically mentions platonic solids, regularity of the axes, or intersecting at the zero point. Did Oskar make a different claim to what "doctrinaire" means?

pirsquared wrote:
Strangely, this isn't the only Face-Turning puzzle that jumbles. The FTRD (face turning rhombic dodecahedron) jumbles. I know that the FTRD is very similar to the Helicopter Cube, but with slightly different cuts. Is the FTI similar to another puzzle with slightly different cuts?
I don't know if this helps to visualize, but the FTI would be the same as a vertex turning dodecahedron because icosahedra and dodecahedra are duals of each other.

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 Post subject: Re: We need icosahedra
PostPosted: Thu Jul 29, 2010 7:51 am 
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pirsquared - are you talking about Rua and Taru? dont forget them. they jumble too.

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 Post subject: Re: We need icosahedra
PostPosted: Thu Jul 29, 2010 3:09 pm 
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GuiltyBystander wrote:
I don't know if this helps to visualize, but the FTI would be the same as a vertex turning dodecahedron because icosahedra and dodecahedra are duals of each other.

In that case, could the Dino Dodecahedron jumble if it were deeper cut?

Doubleyou wrote:
pirsquared - are you talking about Rua and Taru? dont forget them. they jumble too.

The Rua is a deeper-cut FTRD, and the Taru is cut even deeper. (I think...)

Actually, I just realized that it is the Curvy Copter II, not the Helicopter Cube, that is the cubic version of the FTRD. It has the same types of pieces, and probably solves similarly, except for the hidden centers which are exposed on the FTRD. I think I'm going to try making a dodecahedral version of the FTI.

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 Post subject: Re: We need icosahedra
PostPosted: Thu Jul 29, 2010 4:12 pm 
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pirsquared wrote:
In that case, could the Dino Dodecahedron jumble if it were deeper cut?
Yep. I think you'd have to go at least as deep as 1.2.3 on Gelatin Brain. Hmm, actually, you might be able to do it on 1.2.2. The dodecahedron shape on top of the FTI makes some pieces appear earlier/later than I expected. Anyways, here's where I think the jumbling happens on these vertex turning dodecahedra.
Attachment:
VTD.png
VTD.png [ 12.11 KiB | Viewed 5925 times ]

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