# Why is 7x7 round?



## GalPro (Jun 2, 2009)

Hi,
I once saw a video on youtube showing the math behind the round shape of the 7x7,

does anyone have a link to it?

ty


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## Ethan Rosen (Jun 2, 2009)




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## rjohnson_8ball (Jun 2, 2009)

The way I see it a V7 could be made with flat sides. Just imagine adding plastic to the cubies on a pillowed V7 until you get flat sides. It could be done. However there would be more stress and a bit too much flexibility at the extremities if it were not pillowed. I would worry about its stability.


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## Swoncen (Jun 2, 2009)

I didn't know that or thought of that. Thanks for that information!


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## masterofthebass (Jun 2, 2009)

rjohnson_8ball said:


> The way I see it a V7 could be made with flat sides. Just imagine adding plastic to the cubies on a pillowed V7 until you get flat sides. It could be done. However there would be more stress and a bit too much flexibility at the extremities if it were not pillowed. I would worry about its stability.



It would be just as stable as the 6x6. I imagine that I would almost prefer the longer edge length, rather than the pillowedness. Past 7x7 though, I think it would get too ridiculous. I wonder if someone who has some experience with putty would be willing to do this?


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## rjohnson_8ball (Jun 3, 2009)

I wish the first V6 was pillowed. I would even like the V5 pillowed. (Imagine adding/removing material on a V5 until it looked pillowed.)


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## Logan (Jun 3, 2009)

masterofthebass said:


> ...I wonder if someone who has some experience with putty would be willing to do this?


 We should ask Tony Fisher.


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## 4Chan (Jun 3, 2009)

I do recall seeing a thread similar to making a cubic 7x7 by casting parts for extra wide outer layers.

It was somewhere on twistypuzzles a few months back, however, i was unsure if anyone did it.


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## Lord Voldemort (Jun 3, 2009)

Ethan Rosen said:


>



This is a good video.
He explains it well.


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## rahulkadukar (Jun 3, 2009)

I think there is a mathematical explanation of this on Verdes site


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## rjohnson_8ball (Jun 3, 2009)

It looks like the explanation at http://www.v-cubes.com/inventor.php resembles the Youtube video above. Although it indicates flat sides for big cubes is impractical (due to stress and torque on pieces), it is not impossible.


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## jcuber (Jun 4, 2009)

Personally, when cubes get as big as 7x7, I find the pillowing helpful and comfortable.


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## mcciff2112 (Jun 4, 2009)

jcuber said:


> Personally, when cubes get as big as 7x7, I find the pillowing helpful and comfortable.



Fingertricks are much more difficult though.


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## peedu (Jun 5, 2009)

I've been using these two mathematical expressions to explain why a corner of a cube would fall off if cube is bigger than 6x6x6.
6x6x6: 2 x sqr2 < 3
7x7x7: 2,5 x sqr2 > 3,5

Peedu


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## GalPro (Jun 5, 2009)

I didn't really understand your calculation


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## Stefan (Jun 5, 2009)

Here's a picture I just drew to illustrate the problem (click it to enlarge). Wanted to do this for a while already.


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## Ellis (Jun 5, 2009)

I still don't fully understand why a corner going slightly over the edge is such a huge problem. Can someone elaborate? 

I can hold my V6 from just a corner that has an exposed stalk and there's pretty much no chance of it coming out, unless the whole cube goes with it. 

I know I'm wrong here. I just need someone to point out the error I've made that I've never been able to see.


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## TomZ (Jun 5, 2009)

Ellis said:


> I still don't fully understand why a corner going slightly over the edge is such a huge problem. Can someone elaborate?
> I can hold my V6 from just a corner that has an exposed stalk and there's pretty much no chance of it coming out, unless the whole cube goes with it.
> I know I'm wrong here. I just need someone to point out the error I've made that I've never been able to see.



The problem is very, very simple. As you can see on your V6, while there is little chance of the corner coming off, you can see that the stalk or stem holding it in is quite thin. The problem is that for 7x7x7 and higher, the thickness of this stem goes below zero. Now there's your problem!

Now you might say, why can't I make the stem any thicker? The answer to that is a bit more complicated. The problem is that for the cube to function, there need to be certain symmetries. The simplest way to view it is that the internal geometry of the cube needs to be circular (or conical, spherical...). Making the "circle" larger makes the stem thicker, but for a 7x7x7 to be made you need the circle to go outside the cube. V-Cubes solve this problem by changing the proportions slightly and adding extra material to the cube to have it approach a sphere. You can obviously make the circle slightly "stretched". This is done on the floppy cube where the faces flex slightly to allow for the pieces to pass.


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## Ellis (Jun 5, 2009)

TomZ said:


> Ellis said:
> 
> 
> > I still don't fully understand why a corner going slightly over the edge is such a huge problem. Can someone elaborate?
> ...



ok thanks, I could see that being a pretty big problem 

I've never heard it explained like that. I've only seen stuff like Stephan's diagram and the video which only talk about the corner being fully exposed over the edge. Maybe I was just completely misunderstanding the argument all along.


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## Stefan (Jun 5, 2009)

Ellis said:


> I can hold my V6 from just a corner that has an exposed stalk and there's pretty much no chance of it coming out


Yeah. Because the V6 works.

Look at the circle under your thumbnail in your picture. That's below the blue surface. The bottom of that corner cubie is still further below. But if you look at my drawing, with a 7x7x7 made of cubes, it would be outside, *above* the blue surface. How are you going to turn your imagined stalk *through* the blue surface?


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## Stefan (Jun 5, 2009)

Ellis said:


> I've only seen stuff like Stephan's diagram and the video which only talk about *the corner being fully exposed over the edge*.


Yes. On all three inner sides (the diagram only shows it for one side). How do you connect the stalk if the corner cube is cut off on all three inner sides?


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## Kickflip1993 (Jun 5, 2009)

a 7x7x7 has been made, but the layer below flexes so the cubies can rotate
a pillowed 5x5x5 has been made too

maybe when i have time in summer i will make a non pillowed 7x7x7


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## Ellis (Jun 5, 2009)

StefanPochmann said:


> Ellis said:
> 
> 
> > I've only seen stuff like Stephan's diagram and the video which only talk about *the corner being fully exposed over the edge*.
> ...



Yea, it makes sense. When I see a diagram like what you posted I don't really imagine what the stalk would look like on the inside. I see a line going over the edge, but that's as far as I imagine. I do see the problem now.


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## qqwref (Jun 5, 2009)

As Stefan and others point out, it is impossible to make a cubical 7x7 with every layer the same size (without using some kind of tricks, like Etienne de Foras's 7x7 which bulged out when you did an outer layer turn). However, it is possible to make a cubical 7x7 with thicker outer layers - in fact, the thicker the outer layer (in proportion to the rest of the cube), the more stable the corners can be. That is because a thicker outer layer allows you to use a thicker corner stalk.

The problem is that having such a thick outer layer will significantly increase the size of the puzzle (or else significantly decrease the size of the already-tiny center pieces). For 7x7x7 this is not a huge problem, but the larger the cube gets, the larger the outer layers will have to be. On an 11x11x11 you would have to make them about twice as wide for the mechanism to even function! That begins to not only look kind of ugly, but also put a lot of stress on the pieces (because the spindly stalks are now carrying these huge boxes).

Verdes's answer to this problem was pillowing. Although it might look like the centers are being pushed out - and this is typically how pillowing is done on the small cubes (which don't need to be pillowed at all) - what is actually happening is that the corners, edges, and most of the centers have effectively been shaved down. Thus, instead of having an extra-wide outer layer, it seems to be about the same size as the other layers, and ends up rounded. The height of the center-centers show the true width of the outer layers. Pillowing also has the nice effect of making the puzzle more comfortable to hold when it becomes large.


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## V-cube7_101 (Jun 6, 2009)

Memyself andpi on youtube has a good explaination on why the V7 is pillowed.


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## Ethan Rosen (Jun 6, 2009)

V-cube71235678987654 said:


> Memyself andpi on youtube has a good explaination on why the V7 is pillowed.



There are also some really good explanations here: 
http://www.speedsolving.com/forum/showthread.php?t=12506


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## isaacthecuber (Jun 6, 2009)

Ethan Rosen said:


> V-cube71235678987654 said:
> 
> 
> > Memyself andpi on youtube has a good explaination on why the V7 is pillowed.
> ...



Haha, good call.


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