View Full Version : Math puzzle with magnets

See this site: http://www.dealextreme.com/details.dx/sku.13506

Very cool magnetic balls. If you allow an even number of them to gather and self-align, what is their average magnetic polarity, and how are the poles aligned on average?

To visualize - paint the north pole hemisphere red, the south pole blue. Will the self alignment result in an even amount of blue and red, will some be stood on edge, will all be stood on edge...

Sure it's OT!

aostling

02-01-2010, 12:49 AM

This sort of thing interests me. Let's start with a simple case of four spherical magnets in a tetrahedral stack (three on the bottom with one nested on top). The magnets themselves already know the answer of how they will align, but I don't.

If you have these magnets, I'd like to know what they do.

The stack is unstable. They are almost exactly repelling as much as attracting. ( I have actually tried it)

This sort of thing interests me. Let's start with a simple case of four spherical magnets in a tetrahedral stack (three on the bottom with one nested on top). The magnets themselves already know the answer of how they will align, but I don't.

If you have these magnets, I'd like to know what they do.

I'm tempted to buy a lot of them and see what they do in their spare time. As a magnetic geometry they might be full of surprises. I'm really curious as to what natural 2 and 3 dimensional geometries they can naturally form.

I need a government grant!

The stack is unstable. They are almost exactly repelling as much as attracting. ( I have actually tried it)

But what if they form natural but sparse arrays?

Edit: I just did the math and for all the possibilities they present at first blush, the damn things form simple diagonal rank and file alignments like stars on the flag (US), and what ever becomes of the perimeter is left to chance. They can be coerced to form triangles and all extrapolations of triangles as any set of balls can do.

In other words, it's boring :(

http://upload.wikimedia.org/wikipedia/commons/e/e5/US_Naval_Jack.svg

Natural arrays such as packed spheres always represent the lowest energy state. For magnets that means S-N-S-N etc comes first. Chances are pretty good that they will form up enough magnets in a group that the overall length of each group will vary statistically and make it at least very difficult to make any regular array. That is just a guess on my part but it "feels" right. I also think that solving the problem is likely not possible. It took plenty long enough to demonstrate the the hexagonal 3D packing of spheres was the minimal energy configuration.

edit: How do you account for possible defects? How sensitive is the math to variances in the field strength?

I'm certain that if you can arrange them in a way to keep them from packing they will form interesting geometric strings. As soon as they're allowed to pack that's where they'll stay. The hexagon is the only stable arrangement I can predict. Even in cube form.

edit: How do you account for possible defects? How sensitive is the math to variances in the field strength?

Well, in science the ideal form is the principle form considered as non-ideal forms are chaotic to a degree - the actual form not being defined and all.

For instance, how might a single layer of magnetic snow flakes (no two alike) align on a plane? Could they even do so? Entropy suggests yes but I don't have the math skill to prove it :)

bborr01

02-01-2010, 01:31 AM

I spent the day today at the Tucson gem and mineral show. They had piles of magnets, the black shiny ones that are made of some kind of stone. (my tired brain can't remember what kind of stone)

I grabbed a handful and formed them into a circle and sure enough, the grouped up full circle, just like Evan had said a few weeks ago in another thread.

Pretty cool.

Brian

I spent the day today at the Tucson gem and mineral show. They had piles of magnets, the black shiny ones that are made of some kind of stone. (my tired brain can't remember what kind of stone)

I grabbed a handful and formed them into a circle and sure enough, the grouped up full circle, just like Evan had said a few weeks ago in another thread.

Pretty cool.

Brian

Yep - that would be a happy form as it is sparse and stable. It should be easily possible to place a pair of 90º crossing lines (1 by 1 grid) in the circle as well. As the grid becomes more dense it will destabilize and collapse, then fill all gaps to the extent possible.

In a packed array of irregular sizes there will be holes that cannot be filled, but that still represents the densest possible packing for that arrangement of elements.

beanbag

02-01-2010, 01:46 AM

Edit: removed commentary coz I think I said something very wrong

I did this mental exercise. Spherical magnets, all perfectly identical. Hell, this is science where the impossible is possible :)

Two such will stabilize in a line. There will be perfect polar aligment.

Three will stabilize as a triangle. The polar alignments will not be perfect.

Four will form a stable diamond shape which is two triangles. Polar alignments will be irregular.

And so on. As the 2 dimensional array grows, the polar alignment of the innermost orbs will approach perfection but this will stray as you approach the perimeter.

This pattern continues if you grow the array in 3 dimensions.

I considered what becomes of magnetic hexagon solids - these provide the opportunity for unhappy alignments because the compressive force of the array can force an individual element into a non-perfect alignment. Spherical magnets in a 3-D array will self-align to the maximum degree of happy - ignoring surface friction.

beanbag

02-01-2010, 02:26 AM

In my earlier commentary, I made an analogy to the physics of magnetic spin systems, but the main difference here is that the interactions of two magnets also depends on their relative position to each other. For example, if you have one sphere fixed, then if you put another magnet on top the poles will be aligned, but if you instead put it to the side it will be anti-aligned. Some of these features may still carry over, but I no longer want to use physics terms to describe it. (even if I gave a link, you wouldn't bother to look it up anyway.) I will guess that this problem has been studied and solved. I just don't know what the answer is.

One dumb guess is that the spheres pack as close as possible (fcc or hcp) and all the poles point in the same direction, just like one big magnet with holes cut out. Another guess is that in 2D (one layer only), there are alternating rows with poles anti-aligned. Of course there could be more intricate patterns as well with the poles tilted at some weird angle.

To solve this requires you to write out the interaction energy between a pair of spheres, i.e. some kind of dipole-dipole interaction. Then sum all the energies over all configurations. So yes, finding the lowest energy config depends a lot of the shape of the magnetic field of each sphere, coz it will tell you whether a particular configuration is "sorta" unhappy, or "kinda" unhappy, or "somewhat" unhappy.

In practice, it's not as hard as it sounds to solve since there are various symmetries and self-consistency checks you can do to cut out much of the work.

Edit: what I wrote was for infinite arrays

This sounds a lot like a tiling problem to me.

Black_Moons

02-01-2010, 06:54 AM

On a related note, I happen to have seriously about a hundred 1/4" square and cylinder magnets, They *can* form larger 3d cubes if coaxed, but like to sorta explode too if you poke at the cube too much, because each row is opposite of the row next to it, if you shift a row one magnet distance it now trys to explode insted of stay togethor.

beanbag

02-01-2010, 08:02 AM

I did this mental exercise. Spherical magnets, all perfectly identical. Hell, this is science where the impossible is possible :)

Three will stabilize as a triangle. The polar alignments will not be perfect.

If so, what is the polar alignment? And why not 3 in a row?

Four will form a stable diamond shape which is two triangles. Polar alignments will be irregular.

If so, what is the polar alignment? And why not a tetrahedron or any of the Tetris shapes?

And so on. As the 2 dimensional array grows, the polar alignment of the innermost orbs will approach perfection but this will stray as you approach the perimeter.

So in plain english, there is a tradeoff between surface energy and bulk energy as N increases? ;)

beanbag

08-22-2011, 03:47 AM

On my recent trip to Alaska I went to a gift shop and bought some non-native magnetic spheres. It looks the same as the kind used in magnetic rocks you find in a gift shop. (Hematine, an artificial form of hematite)

For 3 spheres, the most stable arrangement is either a line or a triangle with the poles of the top piece of the triangle parallel to the base. In retrospect this makes sense because it is the closest to forming a ring.

For 4 spheres, I think it is a square with the poles pointed at 45 degrees in the plane of the square.

I can slightly get a tetrahedron to stick together.

I think in general the poles like to lie in a plane, so the most stable shapes are rings and not-fully-packed planes.

For empirical experimentation in this domain you really need to use super magnets. Because the field strength is so high they swamp out all other confounding factors such as friction and gravity.

Thruthefence

08-22-2011, 07:55 PM

Here's some magnet puzzles:

http://unitednuclear.com/index.php?main_page=product_info&cPath=70_71&products_id=307

And I wonder what you could do with this stuff ?:

http://unitednuclear.com/index.php?main_page=product_info&cPath=70_71&products_id=157

By ordering from United Nuclear you are guaranteed to get on a list with Homeland Security. They have been investigated more than once and "They" keep a very close eye on his dealings.

Thruthefence

08-22-2011, 10:23 PM

I'm not surprised.

This is the world we have chosen, I suppose.

lazlo

08-23-2011, 12:28 AM

By ordering from United Nuclear you are guaranteed to get on a list with Homeland Security. They have been investigated more than once and "They" keep a very close eye on his dealings.

LOL, Bob Lazar. Quite the character :)

I'm not sure about "guaranteed". Bob's other company, Firefox, has sold components and chemicals to the high-end (as in, BIG) rocketry community for 30 years, and United Nuclear sells legitimate products including calibrated radiation sources, to many companies and government agencies.

Then again, Bob almost single-handedly put Area 51 on the map, and in pop culture, and he makes quite a few *cough* interesting claims about what goes on there...

"Hi, I'm Bob Lazar. I worked on propulsion systems for extra-terrestrial vehicles for the United States Government" :D

http://www.youtube.com/v/N3LbkAbmJQw

He was harassed pretty heavy by the government some years ago. They confiscated some of his inventory and were threatening charges for various "offenses". They really didn't like that he was selling isotopes to anybody.

beanbag

08-23-2011, 06:27 AM

I am actually glad that I did not get the rare earth magnets. They are too powerful and the coating will eventually flake off. These hematine spheres are just strong enough to be fun to play around with and they can also do that neat trick where you throw two in the air together and they make a rattling/buzzing sound when the bounce off each other.