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 Post subject: Mixup Cube QuestionPosted: Tue Jun 18, 2013 12:28 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
The new WitEden & Oskar Mixup Cubes got me thinking about a question that I've had for a while now but I don't think I've ever asked it... its actually a series of question:

(1) Has anyone calculated the number of states that the normal Mixup 3x3x3 has?
(2) What about the number of states of a Super Mixup 3x3x3 (i.e. face center orientation is tracked)?
(3) Is the ratio of the number of states a Mixup 3x3x3 has in relation to the number of states a 3x3x3 has the same for both the normal and super version?

These next two question I think are known but I also think they depend on how the turns are defined.
(4) What is god's number for the 3x3x3? (specify your allowed turn set)
(5) What is god's number for the super 3x3x3? (using the same allowed turn set)

I think the answer to (4) is 20 for HTM and 26 for QTM. And I think I remember the solition to (5) is either 23 or 24 in HTM.

QTM = Quarter Turn Metric (Faces are only allowed to make 90 degree turns in either direction)
HTM = Half Turn Metric (Faces are allowed to make either 90 or 180 degree turns in either direction)

Personally the number I would be most interested in seeing would be a modified HTM which also allowed slice layer turns of either 90 or 180 degree turns in either direction) If anyone has these numbers I'd LOVE to see them.

Moving on:

(6) What is god's number for the normal Mixup 3x3x3?
(7) What is god's number for the Super Mixup 3x3x3?

And here I'd also like to use the modified HTM where we could have 45, 90, 135, or 180 degree slice layer turns.

The questions I'm really trying to get to here is:

(8) What is easier... the normal 3x3x3 or the normal Mixup 3x3x3?
(9) What is easier... the super 3x3x3 or the super Mixup 3x3x3?

and for the purposes of this question I define "easier" as meaning can be solved with the fewest number of turns. I stongly suspect that the Mixup versions of these puzzles will likely have smaller gods numbers then their 3x3x3 couterparts due to the additional moves (or degrees of freedom) available but I don't know if anyone has tried answer these questions or not.

Even if I'm correct that they have smaller god numbers it may be a harder problem to actually find the god number of these puzzles due to the larger number of states. So I'm far from certain if this is a question that is answerable at this stage.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Wed Jun 19, 2013 12:44 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
wwwmwww wrote:
(1) Has anyone calculated the number of states that the normal Mixup 3x3x3 has?
(2) What about the number of states of a Super Mixup 3x3x3 (i.e. face center orientation is tracked)?
(3) Is the ratio of the number of states a Mixup 3x3x3 has in relation to the number of states a 3x3x3 has the same for both the normal and super version?

Almost a year has gone since I touched GAP the last time but these questions could be answered easily with it.
All you have to do is the dirty work of defining the group:
96 numbers has the total puzzle.
32 numbers has each of the 9 permutations.

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 Post subject: Re: Mixup Cube QuestionPosted: Wed Jun 19, 2013 3:45 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
I've mentioned this before, but... a mixup cube shape-mod in the shape of a deltoidal icositetrahedron would be a lovely puzzle, and a "super" version too. If you got the angles right, though, it wouldn't be a shape-changer anymore.

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 Post subject: Re: Mixup Cube QuestionPosted: Wed Jun 19, 2013 4:15 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
I've mentioned this before, but... a mixup cube shape-mod in the shape of a deltoidal icositetrahedron would be a lovely puzzle, and a "super" version too. If you got the angles right, though, it wouldn't be a shape-changer anymore.
Oh... I love non-shape-changing puzzles. But I looked into trying to find a shape which would work for the Mixup over a year ago and at the time the best I could do was a sphere which didn't interest me all that much.

Does deltoidal icositetrahedron really work? At the face centers the angle in both directions needs to be 45 degrees. However at the edges the angle made between the edges that connect the edge vertex and the center vertex for a 45 degree angle but I very much doubt that the angle between the edges that connect the edge vertex and the corner vertex is also 45 degrees. If you try to change this angle you'll bend the kite shaped faces and form 2 triangular faces for each kite. And even if you do that the resulting shape of the edges and the face centers would be different.

Did you follow that? If not maybe this will help:

center-vertex = the vertex of a deltoidal icositetrahedron that would match the 3x3x3 face center position.
edge-vertex = the vertex of a deltoidal icositetrahedron that would match the 3x3x3 edge position.
corner-vertex = the vertex of a deltoidal icositetrahedron that would match the 3x3x3 corner position.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Wed Jun 19, 2013 4:30 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Hmm, I figured since it was the dual of the rhombicuboctahedron (which would work except for each piece having its own face!) it would probably work out. Now I'm not sure.

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 Post subject: Re: Mixup Cube QuestionPosted: Fri Aug 09, 2013 1:56 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Sorry for the bump, but... is there anyone who can help answer this question? Does the Deltoidal Icositetrahedron shape-mod of the Mixup Cube change shape or not?

I imagine if it does, it's subtle...

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 Post subject: Re: Mixup Cube QuestionPosted: Fri Aug 09, 2013 3:22 am

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
I think so. I remember in a previous thread somewhere pirsquared mentioned that there are 2 kinds of deltoidal icositetrahedra. One has 4 fold symmetry at all places wherever 4 polygons meet, while the other only has 4 fold where the angle opposite the obtuse one meet. Which one would suit our purposes here?

EDIT: I found the thread, it is the last post here.

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 Post subject: Re: Mixup Cube QuestionPosted: Fri Aug 09, 2013 1:32 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Thanks for the link! Looking at this post (the last one in the thread) makes me think that for the shape on the right - the cuboctahedron dual - the Mixup Cube version would not be shape changing, because on the face-turning puzzle shown, all 4-fold corner pieces are identical.

If anyone can render what a Mixup cube would look like in this shape, I'd be very grateful! Though I think it's not possible without gaps, so perhaps a Mixup Plus would be a better option. Perhaps someone with STL talent can create pieces that could glue onto an existing puzzle?

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 Post subject: Re: Mixup Cube QuestionPosted: Fri Aug 09, 2013 2:57 pm

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
wwwmwww wrote:
The new WitEden & Oskar Mixup Cubes got me thinking about a question that I've had for a while now but I don't think I've ever asked it... its actually a series of question:

(1) Has anyone calculated the number of states that the normal Mixup 3x3x3 has?
(2) What about the number of states of a Super Mixup 3x3x3 (i.e. face center orientation is tracked)?
(3) Is the ratio of the number of states a Mixup 3x3x3 has in relation to the number of states a 3x3x3 has the same for both the normal and super version?

We don't need GAP!!!! We can use math
I think it is easiest to calculate this assuming the moves available are quarter turns of the outside faces and 45 degree slice turns on all 3 axes. These allows for global reorientations so we must divide by 24 at the end.

There are 2 orbits of pieces on this puzzle: the corners and all of the other pieces. A quarter turn of a face rotates one center independently and flips the parity of both orbits. A 45 degree slice move flips the parity of the center/edge piece orbit only. So centers and edges can be rotated independently on this puzzle, but not without affecting the parity of the corners. Since we are allowing global reorientation we must note that a global reorientation preserves the parity of all orbits so no issues there. We know from 3x3x3 experience that a pure 3-cycle of corners is possible and a pure 3-cycle of edges is possible. Since the Mixup 3x3x3 allows edges and centers to intermix, this means a pure 3-cycle of centers is possible. So, every corner can be in one of 8 positions in one of 3 orientations. Odd permutations are possible but only with the restriction that rotations of the centers/edges are not independent. As usual, the orientation of the 8th corner is dependent on the previous 7. The edges and centers can each be in 18 positions in one of 4 orientations, but orientation is not visible on the 6 center pieces. Odd permutations are possible here. Finally there is at least one piece in the center/edge orbit that has indistinguishable orientation, we have no way of restricting the odd parities of the corners so in fact odd parities of corners are allowed without penalty. I believe this gives

( (8! * 3^8/3) * (18! * 4^12) )/24 = 3.947 * 10^29 positions

For the Super Mixup 3x3x3, the orientation of the 6 centers is visible. The orientation then of the 18th center/edge is dependent on the previous 17 and the parity of the corners. Note that a 180 degree face rotation now becomes relevant because it preserves parity of both orbits, but rotates a single center by 180 degrees so the rotation of the 18th center/edge actually has 2 choices instead of 4 or 1. The number then becomes

( (8! * 3^8/3) * (18! * 4^18/2) )/24 = 8.083 * 10^32 positions

The ratio of Mixup 3x3x3 positions to normal 3x3x3 positions is ( ( (8! * 3^8/3) * (18! * 4^12) )/24) / ( (8! * 3^8/3) * (12!/2 * 2^12/2) ) = (18!*2^24)/(24*12!*2^10) = (18*17*16*15*14*13*2^14)/24

= 9124577280

The ratio of Super Mixup 3x3x3 positions to Super 3x3x3 positions is ( ( (8! * 3^8/3) * (18! * 4^18/2) )/24) / ( (8! * 3^8/3) * (12!/2 * 2^12/2) * 4^6/2) = (18!*2^36)/(24*12!*2^22) = (18*17*16*15*14*13*2^14)/24

=9124577280

So yes the ratios are the same, which is actually a little surprising depending on how you look at it. For the 3x3x3 corner parity edge parity and center orientations are all linked so the 3x3x3 is constrained already regardless of center orientation. For the Mixup 3x3x3, the corner parity is linked to center/edge orientation only so without super stickers, it is possible to have 2 corners swapped. Adding the super stickers reveals the there was actually a center rotated. Doesn't it sound like there should be a difference in the ratios then? I had to think carefully about it for a little while but I think my math is correct and the ratios do remain the same. Adding super stickers to a normal 3x3x3 goes from a singly constrained system to a doubly constrained system while adding super stickers to a Mixup 3x3x3 goes from a unconstrained system to a singly constrained system. Does that make sense?

I have never played with a Mixup 3x3x3 nor do I own one, this is just me thinking about it - I hope I didn't make any mistakes

Peace,
Matt Galla

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 Post subject: Re: Mixup Cube QuestionPosted: Sat Aug 10, 2013 7:49 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
Does the Deltoidal Icositetrahedron shape-mod of the Mixup Cube change shape or not?
rubikcollector123 wrote:
EDIT: I found the thread, it is the last post here.
Here is the relevant picture from that thread:
Attachment:

Deltoidal Icositetrahedron Comparison.JPG [ 20.85 KiB | Viewed 2571 times ]

The shape we are interested in is the one on the right and it is clear the angles which lie in the xy, yz, and xz planes are 225 degree angles. The intersection of this shape and those planes form perfect octagons. However what I question is the angle toward us on that piece at the 45 degree point between the x and y planes (I'm assuming z is out of the page). That angle can't be part of a perfect octagon so I doubt very much that it too is 225 degrees. It may be close but I don't think that corner has 4 fold symmetry. If they do on Eitan's puzzle I suspect he has fudged it a bit. However I would LOVE to be proven wrong. At the moment I'm still recovering from a PC crash and I haven't yet gotten to the point of re-installing Solidworks so I can't say I'm certain.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sat Aug 10, 2013 8:28 pm

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
Why can't 225 degrees be part of a regular octagon?

The internal angles are 135 degrees so the external ones are 225. Right?

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 Post subject: Re: Mixup Cube QuestionPosted: Sat Aug 10, 2013 9:40 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
rubikcollector123 wrote:
Why can't 225 degrees be part of a regular octagon?
The whole point is that the plane which includes the line x=y and the z axis does NOT intersect this shape as a regular hexagon. Still irregular hexagons may contain the correct angle but I don't believe this one does.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sat Aug 10, 2013 10:18 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
I'm inclined to say the angles are identical because of the text in the post:

pirsquared wrote:
There are Type A vertices, where 4 long edges meet, and Type B vertices, where 2 long edges and 2 short edges meet. The angle to look at here is the angle between edges on opposite sides of the vertex; let's call the angle X.

In the picture below, the DI on the right is the one you get from a rhombicuboctahedron. On this shape, the angle is the same (225 degrees) on both types of 4-edge vertices. This was fantastic for the FTDI, since it resulted in the Type A and B corners being identical.

(Emphasis mine.)

Also, unless I'm mistaken, the FTDI was introduced before fudging was a "thing".

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 Post subject: Re: Mixup Cube QuestionPosted: Sat Aug 10, 2013 11:30 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
I'm inclined to say the angles are identical because of the text in the post:
Eithan may very well be correct. I don't think he is but I think I could check this once I get Solidworks up and running again... this is if that angle hasn't already been calculated elsewhere already.

The thing is there are two types of corners where 4 edges meet. The 6 that occur where the axes intersect the shape have 4 identical edges meeting. This gives them 4 fold symmetry automatically. The other 12 corners which relate to the edges of a cube have two of what I'll call the equatorial edges... these are the same as those on the other 6 corners and Eithan has already shown that there are enough degrees of freedom with this shape that you are allowed to adjust these angles such that they meet at the same angle everywhere. However once you have adjusted these angles to 135 degrees (or 225 degrees depending on how you look at it), the angle which the other type of edges intersect is fixed by the geometry.

So lets call the corners at the axes "Faces Pieces" and the other corners we'll call "Edge Pieces". There are 3 angles we need to look at:

Angle A is the angle at which equatorial edges meet at on the Face Pieces.
Angle B is the angle at which equatorial edges meet at on the Edge Pieces.
Angle C is the angle at which the "short edges" meet at on the Edge pieces.

Eithan has shown that angles A and B can be made equal. And I agree with this. Yes, they can be. Eithan's statement certainly also implies that angles B and C are equal when A and B are equal but he hasn't proven this and the angle C isn't even measured in his picture.

Hmmm... however I think I may be able to create this shape in POV-Ray and check something faster then it would take to install Solidworks. I can't easily measure angles in POV-Ray but I think it I can still pull out what I need. Give me a bit.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 12:17 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
All the 4-fold pieces in the FTDI are identical according to Eitan. What's more, they are all 90-degree rotationally symmetrical (they have to be since the equatorial-intersection ones are). I'm pretty sure that's all you need to demonstrate that the geometry is mixup-doctrinaire (though it's not a rigorous proof of course).

Sorry if I'm coming across rudely - it's not my intention. >_> It's entirely possible that I've missed something and I look forward to your analysis.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 12:46 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
Sorry if I'm coming across rudely - it's not my intention. >_> It's entirely possible that I've missed something and I look forward to your analysis.
No... not taken as rude at all. We've both been hand waving and its time for some math... or at least pretty pictures. Let's start with Eithan's picture again.
Attachment:

Deltoidal Icositetrahedron Comparison.JPG [ 20.85 KiB | Viewed 2505 times ]

Now let's focus on the shape on the right. If I can actually rotate the left Face Piece up and into the top-left Edge Piece position and things remain identical that tells me I should be able to remove the 4 front most surfaces and be able to extend all the other surfaces out and get a perfect cone with an octagonal base. If I did that in both direction I'd have a 16 sided dice which would look like this.
Attachment:

Deltoid1.png [ 4.43 KiB | Viewed 2505 times ]

And then to restore the front 4 faces I should be able to rotate this dice by 90 degree and take the intersection of the two. If I also rotated by 90 degrees along the other pretendicular direction I should get the same thing. In other words the third dice should add NO new faces to the intersection. So lets look at those 3 dice.
Attachment:

Deltoid2.png [ 25.04 KiB | Viewed 2505 times ]

At this point I already see indications this isn't working as planned as I see blue peaking out from between the yellow and red intersection which I shouldn't see. But let's proceed. Here is what the intersection looks like.
Attachment:

Deltoid3.png [ 16.32 KiB | Viewed 2505 times ]

And this geometry has one of the properties you were after. The Edge Pieces now finally have 4 fold symmetry and the angles A and B and C are all equal. However notice what we have done to get there. We now have 8 edges meeting at the center of the Face Pieces and only 4 edges meeting on the Edge Pieces. In other words they still are NOT identical. So in short we still have a shape changing Mixup Cube.

This doesn't give you angle C from Eithan's picture but its enough to prove that it is NOT 135 or 225 degrees. So Eithan's Edge Pieces either only have 180 degree rotational symmetry or if he believed they were identical to the Face Pieces and only designed those he may have fudged this design and not even realized it. You may very well be able to interchange these pieces and have everything function normally as I suspect the amount of fudging is very small.

Carl

P.S. Maybe this paragraph from above makes more sense now... What I expected to happen is exactly what did happen.
wwwmwww wrote:
Does deltoidal icositetrahedron really work? At the face centers the angle in both directions needs to be 45 degrees. However at the edges the angle made between the edges that connect the edge vertex and the center vertex for a 45 degree angle but I very much doubt that the angle between the edges that connect the edge vertex and the corner vertex is also 45 degrees. If you try to change this angle you'll bend the kite shaped faces and form 2 triangular faces for each kite. And even if you do that the resulting shape of the edges and the face centers would be different.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:24 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Allagem wrote:
I have never played with a Mixup 3x3x3 nor do I own one, this is just me thinking about it - I hope I didn't make any mistakes
Thanks Matt!!! You really should get yourseld a Mixup Cube... especially so now that they are mass produced. It really is a nice puzzle.

Now let me take your precise math and muddy it up a bit with some back of the napkin type calculations to see if I can tell if the God's Number for the Mixup Cube is higher or lower then that of the Rubik's Cube. Let's ignore the super versions for the moment.

First we know that the God's Number for the Rubik's Cube is 20. Knowing this let's hold the core stationary and see how many states we can reach in a single turn.

On each axis we can turn 2 faces to each of 3 new positions so that gives us access to 2*3*3 or 18 new states with a single turn. From each of those we have access to 17 new states as one takes us back to the start position. So knowing this I can estimate the number of states in a Rubik's Cube as 18*(17^(19)) or basically 4.3E+24. This is an over estimate by basically 5 orders of magnitude.

So lets do the same for the Mixup Cube and see what we get. On each axis we again have 2 independent layers or rotation and 1 dependent one. As the core of most Mixup Cubes is typically a Fused Cube lets hold the core stationary again which in this case means we are holding one of the corner pieces. So on each axis we can turn 1 face to 3 new postions or we can turn a slice layer to 7 new positions. So from the start state we have access to (3+7)*3 or 30 new states with a single turn. From each of those we have access to 29 new states as again one takes us back to the start position. So if the God's Number of the Mixup Cube was also 20 we could estimate its number of states as 30*(29^(19)) or about 1.831E+29. Looking at your math I see the correct answer is about 3.947E+29 so I'm short by about a factor of 2. This tells me quite clearly that the God's Number of the Mixup Cube is greater then 20. So my fear that the extra freedom provided by the Mixup Cube would produce a puzzle which could be solved in fewer turns appears to have been disproven. THANKS!!!

Allow me to take a shot in the dark and say that I guess that the God's Number for the Mixup Cube is 24. At least that is about where my back of the napkin type calculations take me.

Carl

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Last edited by wwwmwww on Sun Aug 11, 2013 1:26 am, edited 1 time in total.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:25 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
So it can't be done without fudging? Bummer.

So, then, I want to know - if you were to erroneously assume that this was not the case, and made a mixup puzzle with this shape anyway, unintentionally fudging it by making all the mixup pieces identical, just how non-planar would each "face" be?

I hope Eitan will post in this topic with his FTDI experience too.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:29 am

Joined: Mon Nov 30, 2009 1:03 pm
Looks non-shape-shifting to me. Or do I misunderstand the question?

Note that Mixup Cube has many "filler pieces" at the inside, which we could make visible at the outside. All holes would be filled, and no fudging is needed.

Oskar
Attachment:

Mixup Deltaoidal Icositetrahedron.jpg [ 96.89 KiB | Viewed 2494 times ]

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:40 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
So it can't be done without fudging? Bummer.
Well you could make it a sphere and not need fudging.
Jared wrote:
So, then, I want to know - if you were to erroneously assume that this was not the case, and made a mixup puzzle with this shape anyway, unintentionally fudging it by making all the mixup pieces identical, just how non-planar would each "face" be?
To be exact you'd need to calculate Eithan's angle C. I suspect that its within a few degrees of 135 so I suspect you might not even be able to tell the puzzle was fudged. As I suspect you'd also need small gaps of the surface to allow turning and this would also help hind any fudging.
Jared wrote:
I hope Eitan will post in this topic with his FTDI experience too.
Please PM and invite him... the more the merrier.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:52 am

Joined: Fri Nov 05, 2010 2:20 am
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Oskar wrote:

Note that Mixup Cube has many "filler pieces" at the inside, which we could make visible at the outside. All holes would be filled, and no fudging is needed.

Then it would be a mixup plus deltoidal icositetrahedron. Right?

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:54 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Looks non-shape-shifting to me. Or do I misunderstand the question?
True... but you can also put faces in the place of the edges of a normal Mixup Cube and get a non-shape shifting puzzle. My math tells me the faces on your puzzle can't be planer as the red and green angles here aren't the same angle on a Deltaoidal Icositetrahedron.

I'm going to guess you just designed face centers and put them in the place of the edges and maybe tweeked the corners so they best matched the face centers when they were in the edge positions. Correct?
Attachment:

MixUpDI.png [ 157.7 KiB | Viewed 2480 times ]

Either that or something is very far off in my understanding of geometry.... which may very well be the case. If I am off my rocker can you tell me why my Octagonal Bipyramid approach fails?

Carl

P.S. It also occurred to me that maybe this looks so nice because you did actually cut up a Deltaoidal Icositetrahedron. If so can you remove one edge and rotate it 90 degrees and put it back in?

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Last edited by wwwmwww on Sun Aug 11, 2013 2:08 am, edited 4 times in total.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 1:55 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
rubikcollector123 wrote:
Then it would be a mixup plus deltoidal icositetrahedron. Right?

That's right.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 2:11 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
My math tells me the faces on your puzzle can't be planer as the red and green angles here aren't the same angle on a Deltaoidal Icositetrahedron. I'm going to guess you just designed face centers and put them in the place of the edges and maybe tweeked the corners so they best matched the face centers when they were in the edge positions. Correct?
My CAD drawing has eight identical "corners" and eighteen identical "centers/edges". Each of the "centers/edges" has fourfold rotational symmetry. There was no tweeking or fudging.

Oskar

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 2:23 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
My CAD drawing has eight identical "corners" and eighteen identical "centers/edges". Each of the "centers/edges" has fourfold rotational symmetry. There was no tweeking or fudging.
Are the faces planar? Fudging typically applies to mechanism and I've used it in this thread where maybe I should have just stated tweeking the geometry of the surface. If your puzzle has those properties then I don't believe its still a Deltaoidal Icositetrahedron. If it is then my understanding of the Deltaoidal Icositetrahedron is in error and I'm not seeing my mistake at the moment. Then again its after midnight so maybe I just need some sleep.

Goodnight,
Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 2:26 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Are the faces planar?
Yes. Attached is the geometry that I used in IGES format. Remove the ".txt" extension that I added so Sandy would not block the upload.

Oskar

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 2:41 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
wwwmwww wrote:
Are the faces planar?
Yes.
Ok... I've found an error in my Octagonal Bipyramid approach.
Attachment:

MixUpDI2.png [ 158.04 KiB | Viewed 2462 times ]

I made the angle between the planes A and B equal to 135 degrees (I called this 45 degrees above as I was considering one plane extended through the other). I should have made the angle between the two edges equal to 135 degrees. So I'm sure Oskar is correct even without looking at his file. Tomorrow I'll try to calculate what the angle between those two planes needs to be if the edges meet at 135 degrees and I'll re-run my POV-Ray code.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 3:32 am

Joined: Mon Aug 18, 2008 10:16 pm
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!!

I can't wait to see what you come up with!

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 5:00 am

Joined: Mon Nov 30, 2009 1:03 pm
One can close up those curvy holes with beveled caps, as shown below. This is the same solution that I used for the original Mixup Cube design.

Oskar
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Mixup Deltaoidal Icositetrahedron - view 2.jpg [ 81.18 KiB | Viewed 2434 times ]

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Mixup Deltaoidal Icositetrahedron - view 3.jpg [ 60.53 KiB | Viewed 2434 times ]

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 7:30 am

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
Wouldn't the caps do some funny overlapping motion as they go? The lines just don't look right.

By the way... One of your new designs can be made without shapeways.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 8:24 am

Joined: Wed Jan 07, 2009 6:46 pm
Location: Evanston, IL
Hey everyone! Thanks Jared for alerting me to the existence of this thread. The short answer is that I'm pretty sure the DI version of the Mixup Cube wouldn't shape-shift. I'm a little confused why the yellow one in this thread has all those empty spots. Can someone explain that, please?

Btw, with the overhaul of my Shapeways shop happening this month (thanks to Tom's packing program) the improved FTDI should be available soon.

Eitan

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 8:28 am

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
It leaves gaps because in cubic form, the mixup plus pieces are hidden. Kind of like how a dino cube is 'perfect' for cubes and it doesn't work for many other geometries unless weird cuts are used. Helicopter cubes are like this as well.

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 12:28 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else

Oskar, is it possible to make pieces that could be attached to a WitEden Mixup or Mixup Plus cube, or would you have to trim those puzzles down? (For the Mixup Cube you could just make replacement caps, but I'm not sure if you'd have to cut the corners...)

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 2:40 pm

Joined: Mon Nov 30, 2009 1:03 pm
pirsquared wrote:
I'm a little confused why the yellow one in this thread has all those empty spots. Can someone explain that, please?
The cuts are conical. That explains the curvy pieces and the gap between them.
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Mixup Deltaoidal Icositetrahedron - view 5.jpg [ 78.81 KiB | Viewed 2330 times ]
Jared wrote:
Oskar, is it possible to make pieces that could be attached to a WitEden Mixup or Mixup Plus cube, or would you have to trim those puzzles down? (For the Mixup Cube you could just make replacement caps, but I'm not sure if you'd have to cut the corners...)
Of course that can be done, but the result would be rather bulky.
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Mixup Deltaoidal Icositetrahedron - view 4.jpg [ 114.83 KiB | Viewed 2330 times ]
Oskar

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 4:47 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Yeah, I guess with a full print you could make it smaller and cheaper. Or maybe you could see if WitEden could just make these too.

Also... can we still call it a Mixup puzzle when it does not change shape?

(I hope you'll render a Mixup Plus version also, just to see it at least... the extra wedge pieces would be nicely symmetrical. With narrow center layers like the WitEden Mixup Plus it would look quite nice.)

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 6:07 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
Ok... I've found an error in my Octagonal Bipyramid approach.
Ok... redid the math.
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MixupDI_Math.png [ 167.7 KiB | Viewed 2294 times ]

And it turns out the angle I needed was 20.9410204722438 degrees. I had been careless and used 22.5 degrees in my code. With that fixed here are the 3 dice.
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Deltoid3B.png [ 38.49 KiB | Viewed 2294 times ]

And the intersection now looks like this:
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Deltoid4.png [ 95.72 KiB | Viewed 2294 times ]

And the issue you see with the colors on the surface tells me POV-Ray is finding coincident surfaces as it should. So YES the edges have 4 fold symmetry and are identical to the faces and I was 100% wrong yesterday. Sorry about that.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 11, 2013 6:22 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
Yeah, I guess with a full print you could make it smaller and cheaper. Or maybe you could see if WitEden could just make these too.
I'm generally not a collector of shape mods... but I have to say if WitEden offered this I'd have to pick one up even though I already have their Mixup Cube. I think all they'd have to design is new center caps and new corners so I think its certainly a possibility.
Jared wrote:
Also... can we still call it a Mixup puzzle when it does not change shape?
Speaking of naming issues I just realized this is now the PERFECT shape for the redesign of my Thorny Cube which has been on the back burner. However it too would no longer be Thorny or even a Cube. Hmmm... I guess I'll cross that bridge when get there.
Jared wrote:
(I hope you'll render a Mixup Plus version also, just to see it at least... the extra wedge pieces would be nicely symmetrical. With narrow center layers like the WitEden Mixup Plus it would look quite nice.)
I agree. And as I haven't purchased the Mixup Plus puzzles yet, it just felt backwards getting them before the true Mixup Cube, I'd prefer them in this shape it that were an option WitEden offered.

Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Thu Aug 15, 2013 8:29 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Well, Oskar went and made it available on his Shapeways shop.

http://www.shapeways.com/model/1274297/mixup-kites.html

Wish I could afford one...

(Edited to change link - the name of the puzzle changed.)

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 Post subject: Re: Mixup Cube QuestionPosted: Sat Aug 17, 2013 7:01 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
Well, Oskar went and made it available on his Shapeways shop.
NICE!!!
Oskar wrote:
One can close up those curvy holes with beveled caps, as shown below.
Oskar,

Did you add fillets after you added the beveled caps? It looks like the holes have been opened up a bit again on the version you have posted on Shapeways. Just curious as to why.
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MUKites.jpg [ 116.42 KiB | Viewed 2090 times ]

Thanks,
Carl

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 Post subject: Re: Mixup Cube QuestionPosted: Sun Aug 18, 2013 3:40 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Did you add fillets after you added the beveled caps? It looks like the holes have been opened up a bit again on the version you have posted on Shapeways.
Yes, those are 0.3 mm fillets. Otherwise, the points would be razor sharp, which is both dangerous and locking/damage-prone. Your question makes a good end-of-video question.

Oskar

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