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 Post subject: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 2:17 am 
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Hi Twisty Puzzlers fans,

Paso Doble ("two-step") is based on a suggestion by Bram Cohen and several other people. Whereas Gear Cube has a gearing ratio of 1:1, Paso Doble has a gearing ratio of 2:1. The gearing mechanism is similar to Slice Gear Cube.

Watch the YouTube video.
Buy the puzzle at my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar
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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 6:21 am 
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Very interesting, this must be very hard to solve!
Why have you decided to make open cubies?

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 6:44 am 
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Mattias wrote:
Why have you decided to make open cubies?
To show the mechanism.

Oskar

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 7:31 am 
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This implements one of the asymmetric axis systems of the 3x3x3.
Bram discussed the symmetric ones here:
viewtopic.php?f=1&t=23191
For the theorists out there: This variant has the same number of permutations as the normal rubiks Cube.

I assume the edge length is 57mm, right?
What does it weigh?


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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 10:40 am 
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Andreas Nortmann wrote:
For the theorists out there: This variant has the same number of permutations as the normal rubiks Cube.
Really!? I'm amazed. I wouldn't have thought that possible for what I'm considering an order=1 puzzle. Surely you must not be counting the permutations which differ solely due to face center orientation as 3 of the face centers on this puzzle can only be rotated by 180 degrees. So now I'm interested in finishing out this table.
Image

Oskar,

Congrats!!! That turns very well. May I ask if there is more apparent friction when the opposite face needs to turn 180 degrees as opposed to when the opposite face only turns 45 degrees? If so its not apparent in the video. And considering the success of this puzzle, do you feel 1:3 and 1:4 gear ratios are possible?

Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 10:45 am 
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There must be a really tricky mechanism inside! Does it have more than one geared layer?

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 11:08 am 
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Timur wrote:
There must be a really tricky mechanism inside! Does it have more than one geared layer?
This picture from Oskar's Shapeways shop should answer that question.

Image

Another question which popped into my head... What is God's number for this puzzle? With the same number of permutations (not counting face center orientation) and faces gear-bandaged to only allow 1 independent type of layer turn per axis (instead of 2 as on a normal Rubik's cube) I feel sure the God's number MUST be greater then 20. And with the greatly reduced number of turn options I'm hoping the God's number would be much easier to calculate. Not sure if that is actually the case but I'd be very interested to know what it is.

And not that the "hardness" of a puzzle is very well defined, but I'd be willing to go out on a limb here and state this must be the hardest doctrinaire subset of the 3x3x3 possible.

Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 11:20 am 
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wwwmwww wrote:
May I ask if there is more apparent friction when the opposite face needs to turn 180 degrees as opposed to when the opposite face only turns 45 degrees?
I do not notice significant difference.
wwwmwww wrote:
do you feel 1:3 and 1:4 gear ratios are possible?
1:4 is easy, just reuse the 1:2's from this puzzle. 1:3 is more tricky.
Andreas Nortmann wrote:
I assume the edge length is 57mm, right?
What does it weigh?
60x60x60 mm, 90 gram.

Oskar

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 4:02 pm 
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Can the Geared Mixup concept be applied to this puzzle? >_>


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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 12, 2012 7:52 pm 
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Oskar wrote:
I do not notice significant difference.
NICE!!!
Oskar wrote:
1:4 is easy, just reuse the 1:2's from this puzzle. 1:3 is more tricky.
Not certain I follow. 1:4 would still require a redesign wouldn't it? You'd have to apply 1:2 twice on each axis to get 1:4 so wouldn't you need twice as many gears if you tried to reuse the 1:2 gearing mech from the puzzle?
Jared wrote:
Can the Geared Mixup concept be applied to this puzzle? >_>
I'm going to say "no". To enable mixup turns the opposite faces need to align after either face has turned 45 degrees, and that doesn't happen here. However a puzzle with a gear ratio of 1:3 on all axes I think could be used to allow mixup turns. There when one face has been turned by 45 degrees the opposite face has turned 135 degrees and they are again in alignment. I have NO idea what such a puzzle would look like... especially if the gears were exposed in the surface as with the Geared Mixup. Sounds like a great project for Oskar.

Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 12:18 am 
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wwwmwww wrote:
Oskar wrote:
1:4 is easy, just reuse the 1:2's from this puzzle. 1:3 is more tricky.
Not certain I follow. 1:4 would still require a redesign wouldn't it? You'd have to apply 1:2 twice on each axis to get 1:4 so wouldn't you need twice as many gears if you tried to reuse the 1:2 gearing mech from the puzzle?


1:4 just has the 4 side blocked from rotation. 1:1 is antislice gearing. 1:3 is trickier in that you can do it trivially for one pair of opposite faces by having the axle go through the center of the puzzle, but doing it for a second axis causes problems.


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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 2:24 am 
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Bram wrote:
1:4 just has the 4 side blocked from rotation. 1:1 is antislice gearing. 1:3 is trickier in that you can do it trivially for one pair of opposite faces by having the axle go through the center of the puzzle, but doing it for a second axis causes problems.

1:3 is already implemented for all three axis:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1564
wwwmwww wrote:
Andreas Nortmann wrote:
For the theorists out there: This variant has the same number of permutations as the normal rubiks Cube.
Really!? I'm amazed. I wouldn't have thought that possible for what I'm considering an order=1 puzzle. Surely you must not be counting the permutations which differ solely due to face center orientation as 3 of the face centers on this puzzle can only be rotated by 180 degrees. So now I'm interested in finishing out this table.
You never ask easy question? :shock: Anyway: When I dig out my old notes and the GAP-code your answer should be answerable.

Andreas


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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 9:40 am 
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Andreas Nortmann wrote:
Bram wrote:
1:4 just has the 4 side blocked from rotation. 1:1 is antislice gearing. 1:3 is trickier in that you can do it trivially for one pair of opposite faces by having the axle go through the center of the puzzle, but doing it for a second axis causes problems.

1:3 is already implemented for all three axis:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1564
Should be able to make it with this mech too. http://www.youtube.com/watch?v=L3vhZ98ezvE

Although it would have the annoying problem of being limited to 270 degree turns so you'd have to do a lot of back and forth. It'd be like the 2nd center pictured here from TomZ's Constrained Cube.
Image


EDIT: nvmd, this won't work on 3 orthogonal axes.

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 10:04 am 
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wwwmwww wrote:
1:4 would still require a redesign wouldn't it? You'd have to apply 1:2 twice on each axis to get 1:4 so wouldn't you need twice as many gears if you tried to reuse the 1:2 gearing mech from the puzzle?
That is correct. The reason why 1:3 is difficult, is that the smallest gear (most inside, lowest number of teeth) would be smaller than the axes that it is moving.

Oskar

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 11:30 am 
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Oskar wrote:
That is correct.
Is there enough room to be able to double up on the gears?
Oskar wrote:
The reason why 1:3 is difficult, is that the smallest gear (most inside, lowest number of teeth) would be smaller than the axes that it is moving.
In that case couldn't you then apply 1:1.5 twice to keep the smallest gear from being too small?

Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 3:08 pm 
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Andreas Nortmann wrote:
Bram wrote:
1:3 is trickier in that you can do it trivially for one pair of opposite faces by having the axle go through the center of the puzzle, but doing it for a second axis causes problems.

1:3 is already implemented for all three axis:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1564


The slice cube is easy to implement, but mixing and matching two slice axes with something else for the third can be tricky.


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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 4:44 pm 
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wwwmwww wrote:
couldn't you then apply 1:1.5 twice
What good would a 1:2.25 gearing ratio do? 1:2 followed by 2:3 would do the trick.
Bram wrote:
The slice cube is easy to implement, but mixing and matching two slice axes with something else for the third can be tricky.
It could be implemented as a cubic 1x1x2 with Gear-Cube-like gears.

Oskar

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sun May 13, 2012 8:22 pm 
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Oskar wrote:
What good would a 1:2.25 gearing ratio do? 1:2 followed by 2:3 would do the trick.
Arg... I was thinking 1.5+1.5 was 3. But gears multiply... I guess that is why you are the one making gear puzzles and I haven't even attempted to design one yet.

I'm still thinking you could design 4 cubes, one would be 1:1 on each axis, one would be 1:2 on each axis (this very puzzle), one would be 1:3 on each axis, and the last would be 1:4 on each axis. And from these I think you'd have the parts to make any of the 20 puzzles I listed above.

For example your Gear Slice puzzle is ALMOST the first puzzle. Could I take one of the 1:1 axes from that puzzle and put it in the Paso Doble and have a puzzle which was 1:2 on two axes and 1:1 on the third?

Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Mon May 14, 2012 11:14 am 
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wwwmwww wrote:
So now I'm interested in finishing out this table.
Here you go:

SuperCube
49152
61152952320
98304
61152952320
1384064104783675392000
1384064104783675392000
1384064104783675392000
21233664
1384064104783675392000
57669337699319808000
1384064104783675392000
1384064104783675392000
1384064104783675392000
1384064104783675392000
1384064104783675392000
57669337699319808000
24576
1384064104783675392000
57669337699319808000
5461111524556800

3x3x3
6144
3822059520
12288
3822059520
43252003274489856000
43252003274489856000
43252003274489856000
1327104
43252003274489856000
1802166803103744000
43252003274489856000
43252003274489856000
43252003274489856000
43252003274489856000
43252003274489856000
1802166803103744000
768
43252003274489856000
1802166803103744000
170659735142400


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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Mon May 14, 2012 9:43 pm 
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Andreas Nortmann wrote:
Here you go:
Thanks. Here is the finished table.

Image

Actually, I'd love to add a column in for God's number for each of these as well but I don't think that will be filled in any time soon. The figure of 20 I believe only applies to the normal 3x3x3 and not the Super Cube which I believe has been proven to be at least 24 if I recall correctly.

Also I wanted to mention the names in this table which are followed by an "*" haven't yet been built with the mech I have pictured in my head. Oskar and I seem to be in the same page. I can tell a few of you aren't.
Bram wrote:
1:4 just has the 4 side blocked from rotation.
Andreas Nortmann wrote:
1:3 is already implemented for all three axis:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1564
True the 1:4 solving wise is the same puzzle as the Fused Cube and the 1:3 solves as a Slice-turn-only Cube. But the mech inside the puzzle would need to be very different to turn the way I'd want them too. Actually I'd consider the Fused Cube to have a 1:0 gearing ratio on each axis and the Slice-turn-only Cube would have a gearing ratio of 1:-1.

This is how a 1:3 axis should turn:
Image

And this is how a 1:4 axis should turn:
Image

Why would one want to make a 1:4 over a 1:0 puzzle? Or a 1:3 puzzle over a 1:-1 puzzle?

(1) I believe if one were to make the 4 pure (all axes the same) puzzles on the above list I believe it would enable one to make ANY of the 20 puzzles on the list with the parts. Oskar is already almost half way there.

(2) Visually a Fused Cube made this way would look much cooler then simply gluing cubies together. The side which is "blocked from rotation" does rotate... it just can't be scrambled.

(3) A puzzle which has 1:3 on all 3 axes and which was built into a Mixup Type puzzle could actually reach states that a Slice-turn-only Mixup Cube (aka my Doctor Cube could not as when one later was turned by 45 degrees the opposite face would turn 135 degrees and be back into a position to allow Mixup Moves.

(4) And just because I now think we can... or at least Oskar could if he wanted to. I still have many non-geared puzzles I want to design before I start playing with gears.

Carl

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Last edited by wwwmwww on Thu May 17, 2012 3:19 pm, edited 1 time in total.

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Thu May 17, 2012 3:03 pm 
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wwwmwww wrote:
Here is the finished table.
Image
Carl,

Here is my challenge for you: design and prototype the Multi Gear Cube kit with which one can build all 20 puzzles as you describe.

To help you, I made this parts kit for you. The kit contains
    Core, center, hollow edge and hollow corner
    Matching sets of 8- and 16-teeth gears for the 1:2 ratio
    Matching sets of 10- and 15-teeth gears for the 2:3 ratio
    A self-matching set of 14-teeth gears for the 1:1 ratio
Together, they can make the 1:1, 1:2,1:3 and 1:4 ratios that you need, as we discussed before. Note the chirality of the gears

Enjoy!

Oskar
Attachment:
Multi Gear Cube 14-14 gear set.jpg
Multi Gear Cube 14-14 gear set.jpg [ 103.47 KiB | Viewed 3814 times ]

Attachment:
Multi Gear Cube 10-15 gear set.jpg
Multi Gear Cube 10-15 gear set.jpg [ 101.05 KiB | Viewed 3814 times ]

Attachment:
Multi Gear Cube 8-16 gear set.jpg
Multi Gear Cube 8-16 gear set.jpg [ 86.71 KiB | Viewed 3814 times ]

Attachment:
Multi Gear Cube regular parts.jpg
Multi Gear Cube regular parts.jpg [ 148.04 KiB | Viewed 3814 times ]

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Thu May 17, 2012 6:23 pm 
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Oskar wrote:
Carl,

Here is my challenge for you: design and prototype the Multi Gear Cube kit with which one can build all 20 puzzles as you describe.

To help you, I made this parts kit for you.
Nice! I will give this a shot. I'm already rushing to get one puzzle finished in time for IPP so I may not be able to do much with this before then. And also keep in mind I was the guy saying two 1:1.5 gear ratios would give you 1:3 and that I have yet to make any puzzles which make use of gears. So if there is an obvious flaw in my logic (not saying there isn't one) but I'm guessing this may be your way of showing me that flaw as if there is a trap I'll surely fall into it. Not anything wrong with that... the best way to learn is by trying. Its just that I'm sort of expecting something because if it were as easy as I'm thinking you'd probably have done it by now and not be asking me to do it.

Two questions:

You never answered this:
wwwmwww wrote:
Could I take one of the 1:1 axes from that puzzle [Gear Slice] and put it in the Paso Doble and have a puzzle which was 1:2 on two axes and 1:1 on the third?
I'm thinking yes. If not can you tell me why not?

Also you've said
Oskar wrote:
1:4 is easy, just reuse the 1:2's from this puzzle.
However I know its not that simple. I can't buy a Paso Doble and using the same gears re-assemble it as a puzzle with 1:4 on each axis. I haven't tried to open your zip file yet (I'll down load it tonight... are these SolidWorks files?) but the images in your post show 6 gear levels. Looking at your Paso Doble model it appears to make use of 6 gear levels all by itself. If that is correct then wouldn't the 1:4 puzzle require 12 gear levels with concentric axis under each face? If so this kit which covers all cases may need to be a cube with a 100mm edge length... not that I'm opposed to large puzzles but it could be rather expensive to print.

I guess if I'm going to get very serious I'll likely need to buy a Paso Doble first so I can make sure I understand it. If I haven't bought one by then could you bring one to IPP or even better bring the pieces of one to IPP so I can study how everything goes together.

Thanks,
Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Fri May 18, 2012 3:37 am 
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wwwmwww wrote:
Two questions:
Carl,

The approach for Paso Doble will work for all of your 20 puzzles. Paso Doble has three gear trains, connecting the three pairs of opposite faces. Each gear train uses two gear layers, hence the six gear layers in total. Per gear layer there is one gearing ratio.

Here is an example for your 1:3-ratio gear train, see sketch. It has a 8:16 followed by a 10:15. The top axle is connected to the 8 gear, the 16 gear is connected to the 10 gear, and the 15 gear is connected to the bottom axle.

Oskar
Attachment:
Multi Gear Cube 1-3 example.jpg
Multi Gear Cube 1-3 example.jpg [ 133.84 KiB | Viewed 3711 times ]

P.S.
wwwmwww wrote:
Its just that I'm sort of expecting something because if it were as easy as I'm thinking you'd probably have done it by now and not be asking me to do it.
It is as easy as you are thinking. The reason that I do not want to develop this 20-puzzle kit is that I do not see a buyer willing to pay $600 for it. But if you find me a sponsor for either the kit or the set of 20 puzzles, I would likely change my mind.

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Fri May 18, 2012 6:51 pm 
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Oskar wrote:
Here is my challenge for you: design and prototype the Multi Gear Cube kit with which one can build all 20 puzzles as you describe.
Formally accepted. It was the picture in your last post which pushed me over the edge. I believe I finally see exactly what is going on here and that only 6 gear shells are needed in all cases. Plus I already have a few changes I want to make... If I don't have to work this weekend I'll try to start on this then. My IPP puzzle is now off to Shapeways and I'll be waiting anyways. Oh this will be so fun... thanks Oskar!!!
Oskar wrote:
Here is an example for your 1:3-ratio gear train, see sketch. It has a 8:16 followed by a 10:15. The top axle is connected to the 8 gear, the 16 gear is connected to the 10 gear, and the 15 gear is connected to the bottom axle.
So every time the bottom layer is turned 90 degrees the top layer turns 270 degrees and vice versa. Perfect! The 16 gear / 10 gear pair on the left and right are identical... correct?
Oskar wrote:
It is as easy as you are thinking. The reason that I do not want to develop this 20-puzzle kit is that I do not see a buyer willing to pay $600 for it. But if you find me a sponsor for either the kit or the set of 20 puzzles, I would likely change my mind.
Oh... I would have been willing to sponsor this after IPP but to be honest with you I like this challenge too much to pass it up. In the end, I'm sure it will cost me less then $600. I will learn alot more. And I already have the form I want this kit to take in my head and when I have this much 3D geometry in my head the only way to get it out is to model it.

Some questions...
(1) I was about to ask the dimensions of the of the puzzle but I opened a few files and found its 60x60x60. Then I noticed Andreas has already asked and you've already answered this very question. So skip this one

(2) What software are you using? These aren't Soildworks files but I am able to open and play with them so I think I can do everything I want to with them.

(3) Aside from the obvious openings in the edges and corners is everything solid?

(4) I may be able to pull this out of the files, not sure yet, but could you give me the radii of the spherical shells you used in case I need to re-create a part or two from scratch in Solidworks.
a. Core radius
b. inner and outer radius of all 6 gear shells
c. inner and outer radius of the spherical shell extending from the face center piece

(5) Are all the face centers on the Paso Doble identical? I'm pretty sure they aren't the same as the face center in your zip file as they don't have a system for locking into the gears as is seen in a few of the pictures above (see the one I copied from your Shapeways shop). No problem I can re-make something similiar. I was almost going to ask why not make the face center and the gear it attaches to physically one piece but the reason why you can't do that 2 out of 3 times is clear.

Attachment:
PasoDoble.png
PasoDoble.png [ 376.52 KiB | Viewed 3604 times ]


(5b) After typing question 5 I decided to go ahead and repost that picture here as I had a question about it. What is the recess I have circled in red for? It looks like the recess for a screw head, but the screw head recess is actually in the face center which is above this part. I don't think its critical but just wanted to make sure.

(6) The only thing I don't have a clue how to make from scratch are the gears themselves but as you were kind enough to provide them I don't need to know. However I'd love to learn at some point. Would you consider making a video showing this at some point... something like Tom's sticker template video? As these aren't Solidworks files I'm guessing these are easier to make in the software you are using but I would think it should be possible in Solidworks.

By the way, even at $600 if you consider that you can actually make 40 puzzles (20 normal and 20 super cubes) that comes to $15 a puzzle, which I'd consider a steal of a deal for a Shapeways puzzle.

Carl

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 Post subject: Re: Paso Doble by OSKAR
PostPosted: Sat May 19, 2012 3:32 am 
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wwwmwww wrote:
Formally accepted.
Great!
wwwmwww wrote:
The 16 gear / 10 gear pair on the left and right are identical... correct?
Correct!
wwwmwww wrote:
What software are you using?
These are IGES files, exported from Rhino.
wwwmwww wrote:
Aside from the obvious openings in the edges and corners is everything solid?
Indeed. You can do all the hollowing that you need.
wwwmwww wrote:
could you give me the radii of the spherical shells
11.2, 11.6, 13.6, 14.0, 16.0, 16.4, 18.4, 18.8, 20.8, 21.2, 23.2, 23.6, 25.6, 26.0 millimeter.
wwwmwww wrote:
Are all the face centers on the Paso Doble identical?
Yes. However I leave it to you to design a good connection with the axles.
wwwmwww wrote:
What is the recess I have circled in red for?
For the screw head.
wwwmwww wrote:
how to make from scratch are the gears
I use Excell to draw the curves that are the sharp edges of the gears. Then I use the "Sweep2" command to get a surface between four connecting curves. Solidworks has a similar function.
wwwmwww wrote:
that comes to $15 a puzzle
OK, note me down for $15 for the 1:2+1:3+1:4 :-)

Oskar

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