61–70 of 87 posts
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I believe this result has important applications in cylindrical stack encryption; it's very difficult to produce a solution, but very easy to verify a given solution.
Brb, I'm gonna create CylinderCoin.
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You could dedicate a cylinder to utilities, and it will connect directly to all the others.
That was more or less what I had in mind. On the other hand, it might be more cost-efficient in general to design the cylinders to connect endwise and form a hexagon; it'd increase travel time for the station's inhabitants, but reduce the need for unique construction tooling -- instead of seven types of cylinders, each of whose interconnections are placed differently from those of all the others, you build one type o…
Earlier quoted context omitted.
That was more or less what I had in mind. On the other hand, it might be more cost-efficient in general to design the cylinders to connect endwise and form a hexagon; it'd increase travel time for the station's inhabitants, but reduce the need for unique construction tooling -- instead of seven types of cylinders, each of whose interconnections are placed differently from those of all the others, you build one type o…
Or you could forego connectors and have them all intersect slightly. Once in space, pop out the panels where the intersections will happen.
Suppose, for example, you want to be able to assemble, from the same components, a flat-hexagon station, and an "asterisk" station like what you'd get if you drew radii from each vertex of a hexagon and then took the hexagon away -- a configuration which might be very useful, for example, as a "transfer point"; if the inter-segment connectors are adaptable to whatever standard governs spacecraft airlocks, then you can dock at least six spacecraft at once and interchange cargo among any or all of them. If you add a seventh "socket" to your central connector, normal to the plane of the other six, then you can tie your "transfer point" to a larger station, too, and assemble a larger structure, as for example might be very useful at a nearby Lagrange point, as a way station for ships inbound to and outbound from the Earth-Moon system.
(On the other hand, perhaps I simply spent too much time playing with Tinkertoy as a child.)
It has been known for a long time that one can arrange 7 cylinders to be mutually touching. That was written about by Martin Gardner decades ago, and was set as a puzzle. The result had the cylinders touching at the end of one with the length of the other, so the question arose, can one arrange to have seven cylinders all mutually touching, without using the ends. The easiest way to say this is to ask for seven infin…
I linked to another related puzzle elsewhere in this discussion: there appears to be a solution for 9 cylinders of infinite length but different radii . It's not clear what's known about the case of 10 cylinders. So there are known solutions for 5 coins, 7 identical infinite cylinders, 7 identical finite cylinders (maybe more), and 9 different infinite cylinders.
This is going to sound like trolling, but it's not - I'm honestly curious. Why is this important? Is it just cool, or is there some real world application? Was someone paying for this research for some reason, or was it just a mathematician's hobby? EDIT: For the record, I don't have any problem with "just cool" research. I do that kind of research often (albeit, not as smart), and totally understand the value in it.…
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Interesting. I would have thought that one could solve such things exactly by representing each unique known irrational that arises (root 2, pi, etc) by its own rational multiplier, and then overloading the relevant equality checks. Of course, you'd need to anticipate/implement each irrational type that might arise (roots, the geometric transcendental pi, and so on.) [Leaving the next sentence in, for comedy value. I…
How would you test whether two generic irrational numbers were equal? Obviously you can numerically approximate them and if you see any difference in the numerical approximation then they must be different - but if they seem the same up to e.g. 10 decimal places, what do you do next?
[[ Approximate algorithm, in case I'm not being clear: you need sqrt(35), represent it as 1×sqrt(5) × 1×sqrt(7). You simplify each expression evaluated down to roughly what you'd write on paper in RAM, and then you do exact comparisons - is that the same coefficients of the same number of the same prime sqrts? For greater than/less than you cast them to a hundred-sig-fig float, and if those are still equal, keep going down the rabbit hole. Obviously the RAM requirements for complicated numbers would be large, but that's the same with rationals - this is just taking it to the extreme. I'd be very surprised if some function of sqrts was exactly the same as some other function of different sqrts. ]]
This is going to sound like trolling, but it's not - I'm honestly curious. Why is this important? Is it just cool, or is there some real world application? Was someone paying for this research for some reason, or was it just a mathematician's hobby? EDIT: For the record, I don't have any problem with "just cool" research. I do that kind of research often (albeit, not as smart), and totally understand the value in it.…
I'm not sure what you mean by "real world problems". Most mathematicians do mathematics because they find it beautiful, rewarding, fun, and for various other reasons that have little to do with creating things in the physical world. Mathematics is extremely interconnected and some minor discovery in a certain field can later become extremely important in some seemingly unrelated field, similarly - real world applicat…
Of course, the same could be said of basic research, but it is further removed and so requires more forward thinking to appreciate.
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I linked to another related puzzle elsewhere in this discussion: there appears to be a solution for 9 cylinders of infinite length but different radii . It's not clear what's known about the case of 10 cylinders. So there are known solutions for 5 coins, 7 identical infinite cylinders, 7 identical finite cylinders (maybe more), and 9 different infinite cylinders.
Is there an example of a similiar non-trivial problem for which there is an inexistance proof? I'm not a mathematician so I have a hard time picturing a way of proving this sort of thing when the number of objects is "tricky" (not too high, not too low), apart from simply showing a counterexample.
Great design for a no-gravity space station. Easy to go everywhere from everywhere.
I'd love to see a jungle gym in this shape. I would definitely climb it!
It was great: whether by design or luck, the poles you had to climb got more and more horizontal as you got higher from the ground, then they became inverted in a way that made the topmost part easy and safer. The bottom then was a test: if you failed you had not far to fall, if you succeeded you would likely be ok to the top.