Ring magnets that repel on both sides

I'm trying to figure out what kind of ring magnets repel on both sides.  That is, if I place one around a pencil and a second around the same pencil, they will always repel regardless of which side is up.

Before you try to tell me they don't or can't exist, let me assure you that I know they exist.  I played with a couple of them as a kid at my grandfater's house.  He passed away some time ago and I don't know what happened to the magnets, so now I'm trying to figure out what they were (and get some, if they are available somewhere).  I have no idea what the magnetic field looks like around one of these, but plan on figuring it out if I ever get my hands on some (assuming there isn't already an explanation online somewhere that I haven't found yet).

The specific magnets that I handled were about 1.5 inches in diameter with a large center hole.  They were shaped somewhat like a doughnut.  That is, they were not flat like regular ring magnets and the cross section was a polygon (octagonal, if I remember correctly).  The only way I recall the magnets attracting one another was diagonally (i.e., imagine placing two on a flat surface near each other and raise the edge of one up and on top of the other so that they are barely overlapping).

My grandfather had a lot of weird stuff that came from industrial sources and I seem to recall that these magnets may have been used in power or telephone applications in some way, but I don't know exactly how.

I've searched online and have been unable to find anything to help me identify these magnets.  Any help is appreciated.
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Carl BohmanAsked:
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if you had a standard bar magnet, it is usually rectangular, with the poles at the thin ends.  But you could presumably magnetize it the other way, so that the 'wide' ends are the north and south poles - so you now have a magnet where the long sides are the poles - if you know wrap it into a circle, one pole is on the outside and the other pole is on the inside, so it will always repel if you put two of them on a pencil
A magnet has N and S poles.  You can never have one without the other.

If you take four identical ring magnets (with N and S poles) and epoxy them together in pairs burying all the S poles (for example) inside you might get the sort of effect you remember.

         NS-SN   would repel   NS-SN   from either side.

These composite magnets would be much weaker than individual components.
The magnet could have a little air-gap (so it is not a perfect ring)
Are you sure the 2 magnets were identical ?
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Imagine a cylindrical tube magnet with N at one end and S at the other. Its topology is equal to a ring/donut. If you started stretching the opening at the S end, you could eventually flatten it out so that N is around the center opening and S is around the outer edge.

Two such magnets would repel unless the center of one approached the edge of the other.

flatten it out so that N is around the center opening and S is around the outer edge
A more practical way to construct such a ring may be in wedges with trapezoidally  shaped sections.
This could also be consistent with the description of the ring as polygonal.
and with the description of how the rings could be made to attract.
It's indeed a more practical construction. My description was intended more for the possible visualization of the magnetic field. But only the reader will know if it helped.

Carl BohmanAuthor Commented:
@d-glitch: Your idea is interesting and probably would work to some degree, but the magnets had the appearance of a single piece.  They were an unpainted, dull black (similar to most ring and disk magnets) with no visible seams.  The magnets were also fairly strong in strength, but I don't know if that matters or not.

@tliotta: There were no gaps or seams and all the magnets (more than two) acted identically and could be interchanged.

@ozo: That could be how they built the ring in the first place, but I have no way of knowing.  Unfortunately, I don't know the exact magnetic field, only a few observable properties.

For visualization purposes, I searched for a good image and found the ones on these pages:

The color is wrong (silver instead of dull black) and the size of the central hole is a little too big relative to the ring, but it may help to visualize.

I'm still searching for more details and will post any I find.
Carl BohmanAuthor Commented:
@deighton: That sounds like it could very well be the answer.  Web searches do show that magnets like that exist.  Now if I could just find a way to get my hands on a couple to see if they work as the ones I remember...
There were no gaps or seams and all the magnets (more than two) acted identically and could be interchanged.

If they were manipulated just as I described (perhaps poorly described), they would indeed have no gaps/seams, act identically and be completely interchangeable. They would almost certainly be relatively weak, though. Any process of reforming from a tubular shape to a ring/donut-like shape would disturb magnetic domains in the material.

The illustrative thought was to show how the field lines could appear. It would be interesting to learn the fabrication process and to see the field lines.

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