The outer cylinder is stacked in a cubic pattern, the inner cylinder stacked with a hexagonal pattern, which is denser around the circumference. This difference in stacking density is what causes the curvature.

a series of end views

Note the different stacking: cubic for the outer cylinder, hexagonal for the inner cylinder

unfinished base, but made from larger balls to show the stacking clearly

subunits of the inner cylinder. Detailed instructions for the tube here: http://www.dotpedia.com/creations/view/1897

Detailed instructions for the tube here: http://www.dotpedia.com/creations/view/1897

the snake made from the smaller version of the inner tube
more detail here: http://www.dotpedia.com/creations/view/1897

http://www.dotpedia.com/creations/view/1897 doesn't show how to build the 800-dot cylinder, with the hexagonally stacked inner cylinder and the square lattice for the outer. How do I build this amazing shape?

Boyd, the video above does show how (it appeared to be missing, so I reattached it to this page and it now appears to be back). Basically, you make the inner cylinder by fusing five of the 8ring snakes (see my torus / snake entry here or on YouTube) side by side, and the outer cylinder is constructed by rows laid up and down along the length of the cylinder

Damian, what do you mean by "800 dots will make a 20 row cylinder"? If you build a perfect cylinder with 800 dots, it will have only 10 rows, not 20. Each row requires 40 magnets for the outer layer and 40 magnets for the inner, for a total of 80 magnets. And ten such rows will require 800 magnets.

Perhaps you are thinking of the 20 rows to mean half the number of magnets in each circle around the cylinder?

Boyd, Yes, you are right !
I've corrected the comment above to say 10 rows.
sorry about that :-) (although it probably doesn't matter to you, I've seen how many magnets you have, it could well matter to others)

Thanks for the clarification. I just wanted to make sure that I wasn't missing something. I'm messing around with your flexible tubes in various sizes. I built the perfect cylinder and showed it to my wife and sons. It's so cool!!!

I worked out the math to predict that the ideal ring size for the perfect cylinder is 40 for the two layers to fit together the best, and I'm playing with different ways of making your flexible tubes. I might upload a YouTube video, with credit to you and with links to your dotpedia and Youtube work on the subject. Bravo, bravo, bravo!

I wonder whether the perfect cylinder had been discovered or invented or studied (by mathematicians or crystallographers) prior to your discovering it. Have you seen it anywhere else, and how did you come up with it? Being fascinated with various lattice packings (I'm a physicist), I think the perfect cylinder is really, really neat. I hunted around to see if there's a nanotube molecular structure that uses this packing (hexagonal packing inner layer, square packing outer layer), but no luck. Do you know of one?

Boyd, that would really be fascinating (if a crystal structure could be found). I'm not sure what atom could replicate the bond requirements, it would need to be a molecular sub-structure of some kind I guess, to have a sufficient number of outward facing bonds.
one interesting thought is that, considering the axially oriented rows, you could remove every second ball from every second row.the inner layer would become the same shape as a cabin nano tube, although obviously a quite different bond arrangement. The outer layer would become a square grid that is most likely unstable for magnets, unless the inner layer would stabilise it. That configuration could simplify the bond geometry for crystal structures. I mean that it would reduce the number of neighbors that each sphere(or repeating subset of spheres) has. The smallest repeating subset would be a group of the spheres.
I'm just thinking about geometry here, since I know very little (in a practical sense) about crystallography.
Also, surely the size variation of the different molecular sub units in a crystal complicates the packing problem?
I never understood why you can't make a tetrahedral diamond-like lattice from silicon, or graphene style nanotubes from silicon.

Thanks, Damian. It sounds like both of us don't know whether the perfect cylinder has been observed in nature.

1. To your knowledge, have any mathematicians studied the "perfect cylinder," a 40-ring double-walled cylinder with the inner wall having hexagonal packing and the outer wall having square packing?

2. Are you the first magnet enthusiast to build the shape, as far as you know?

3. How did you come up with the shape? If I had come up with your flexible tube design, I doubt whether I would have tried to attach straight axial chains of magnets to the outside of the tube, with alternating magnetic direction (N, S, N, S, …). Did you stumble upon the design by chance or were you actually setting out to try several different ways to add a second layer?

ha! I apologise for my lack of specificity :-)
1. No, I don't know of any other work, mathematical or otherwise, that uses this packing.

I would point out, that as a solution to the packing problem, it is of limited utility because it is limited to two layers.

2. It is completely my own idea, and is an elaboration of the tube in this cross http://dotpedia.com/creations/view/2326 , which technique is also unprecedented as far as I know, as was the flexible tube at its core, and the method of joining tubes side-wise. I keep an eye on the various forms (Dotpedia, YouTube and Flickr), and nothing like it has ever appeared there, or (almost certainly) I would have seen it or somebody would have mentioned it.

3. I came up with it after making the cross ( http://dotpedia.com/creations/view/2326 ), reasoning that because the packing of the square grid which is formed in that way was less dense, a two layer curved surface would result if I decreased the curvature of the inner layer to the point where where the outer rows were touching. This was achieved by estimate and trial and error. Although the maths is relatively straight forward once you can see the structure, I would not have been certain enough of the final resting places of the magnets to optimise the structure before building it.
As you can perhaps see on the photo of the snake, the cross, the (working) gears, and perhaps other photos on my Flickr page , the issue of curved layering is something I find quite fascinating. There are various other original flat layering methods and curved strut techniques which I have showcased on the forums as well.
You might also be interested in the lattices and complex surfaces I have made.

Thanks so much for the detailed answers. The packing is limited to two layers, but that doesn't diminish its interest in my mind. It's so neat to be able to match a hexagonal packing with a square packing, two layers or not. And it certainly appears that you've invented it, at least as far as making it with magnets goes. I looked at your cross, which is really cool as well. Bravo!

Damian, I found a fast way of building double-walled tubes that are geometrically identical to but magnetically different from yours. See http://youtu.be/TuDL1TvAEY4. In the video, I build a 25-ring tube in 6 minutes. In the written description of this video, I offer proof that rings of 40 magnets best match the natural curvature of the double wall.

Try #2 for the correct link: Damian, I found a fast way of building double-walled tubes that are geometrically identical to but magnetically different from yours. In my tutorial video (http://youtu.be/TuDL1TvAEY4), I build a 25-ring tube in 6 minutes. In the written description of this video, I offer proof that rings of 40 magnets best match the natural curvature of the double wall.

Try #3 for the correct link (dotpedia doesn't like characters after the URL): Damian, I found a fast way of building double-walled tubes that are geometrically identical to but magnetically different from yours. In my tutorial video ( http://youtu.be/TuDL1TvAEY4 ), I build a 25-ring tube in 6 minutes. In the written description of this video, I offer proof that rings of 40 magnets best match the natural curvature of the double wall.

- 3 years ago- 3 years ago- 3 years ago- 3 years ago- 3 years ago- 3 years agoPerhaps you are thinking of the 20 rows to mean half the number of magnets in each circle around the cylinder?

Boyd

- 3 years agoI've corrected the comment above to say 10 rows.

sorry about that :-) (although it probably doesn't matter to you, I've seen how many magnets you have, it could well matter to others)

- 3 years agoThanks for the clarification. I just wanted to make sure that I wasn't missing something. I'm messing around with your flexible tubes in various sizes. I built the perfect cylinder and showed it to my wife and sons. It's so cool!!!

I worked out the math to predict that the ideal ring size for the perfect cylinder is 40 for the two layers to fit together the best, and I'm playing with different ways of making your flexible tubes. I might upload a YouTube video, with credit to you and with links to your dotpedia and Youtube work on the subject. Bravo, bravo, bravo!

I wonder whether the perfect cylinder had been discovered or invented or studied (by mathematicians or crystallographers) prior to your discovering it. Have you seen it anywhere else, and how did you come up with it? Being fascinated with various lattice packings (I'm a physicist), I think the perfect cylinder is really, really neat. I hunted around to see if there's a nanotube molecular structure that uses this packing (hexagonal packing inner layer, square packing outer layer), but no luck. Do you know of one?

Cheers!

Boyd

- 3 years agoone interesting thought is that, considering the axially oriented rows, you could remove every second ball from every second row.the inner layer would become the same shape as a cabin nano tube, although obviously a quite different bond arrangement. The outer layer would become a square grid that is most likely unstable for magnets, unless the inner layer would stabilise it. That configuration could simplify the bond geometry for crystal structures. I mean that it would reduce the number of neighbors that each sphere(or repeating subset of spheres) has. The smallest repeating subset would be a group of the spheres.

I'm just thinking about geometry here, since I know very little (in a practical sense) about crystallography.

Also, surely the size variation of the different molecular sub units in a crystal complicates the packing problem?

I never understood why you can't make a tetrahedral diamond-like lattice from silicon, or graphene style nanotubes from silicon.

- 3 years ago- 3 years ago1. To your knowledge, have any mathematicians studied the "perfect cylinder," a 40-ring double-walled cylinder with the inner wall having hexagonal packing and the outer wall having square packing?

2. Are you the first magnet enthusiast to build the shape, as far as you know?

3. How did you come up with the shape? If I had come up with your flexible tube design, I doubt whether I would have tried to attach straight axial chains of magnets to the outside of the tube, with alternating magnetic direction (N, S, N, S, …). Did you stumble upon the design by chance or were you actually setting out to try several different ways to add a second layer?

- 3 years ago1. No, I don't know of any other work, mathematical or otherwise, that uses this packing.

I would point out, that as a solution to the packing problem, it is of limited utility because it is limited to two layers.

2. It is completely my own idea, and is an elaboration of the tube in this cross http://dotpedia.com/creations/view/2326 , which technique is also unprecedented as far as I know, as was the flexible tube at its core, and the method of joining tubes side-wise. I keep an eye on the various forms (Dotpedia, YouTube and Flickr), and nothing like it has ever appeared there, or (almost certainly) I would have seen it or somebody would have mentioned it.

3. I came up with it after making the cross ( http://dotpedia.com/creations/view/2326 ), reasoning that because the packing of the square grid which is formed in that way was less dense, a two layer curved surface would result if I decreased the curvature of the inner layer to the point where where the outer rows were touching. This was achieved by estimate and trial and error. Although the maths is relatively straight forward once you can see the structure, I would not have been certain enough of the final resting places of the magnets to optimise the structure before building it.

As you can perhaps see on the photo of the snake, the cross, the (working) gears, and perhaps other photos on my Flickr page , the issue of curved layering is something I find quite fascinating. There are various other original flat layering methods and curved strut techniques which I have showcased on the forums as well.

You might also be interested in the lattices and complex surfaces I have made.

- 3 years agoThanks so much for the detailed answers. The packing is limited to two layers, but that doesn't diminish its interest in my mind. It's so neat to be able to match a hexagonal packing with a square packing, two layers or not. And it certainly appears that you've invented it, at least as far as making it with magnets goes. I looked at your cross, which is really cool as well. Bravo!

Boyd

- 3 years ago- 3 years ago- 3 years ago