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It's possible to put 24 lines on a cubic surface. 27 lines is possible, but I don't have a great picture for that surface. It turns out that the 24 lines can be built with Zome. I'm thinking that might be the basis of a great puzzle, if those lines could be transformed into identical spiraled rods, so that all the intersections were handled by grooves. Can anyone put all those pieces of math together to make a physical realization?

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    What do you mean by spiralled rods and grooves? Do you have a picture of a similar construction?2012-12-31
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    http://www.youtube.com/watch?v=7qCEleiBwcg -- the spiral burr, but the spirals for this new object wouldn't be packed so tightly.2012-12-31
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    Put a random helix around each line. With probability 1, none of the helices intersect, and there is a nonzero minimum distance between all of them. Thicken the helices by less than half the minimum distance and you're done. Is this what you wanted?2012-12-31
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    Rahul, that's pretty much right, but it needs to be optimized, the helices need to be able to twist in and out of position, and the crossings need to make the structure stable. If all that can be done with identical rods, then the object would be rather desirable to have, I think.2012-12-31
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    Well, any helix can be screwed in and out of position without disturbing anything else. And I think my construction should work for identical rods too; the random variables will just be the rotation of each helix about its axis. The structural integrity of the arrangement is a more interesting issue though.2012-12-31

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