You may be aware of mathematical objects called 'sets'. If not, they're basically just collections of things: for example, {3,4,5} is a set of three numbers, and {0,1,2,3,...} is an infinite set of numbers. But, some sets have very weird properties . And the above video illustrates one of these, by means of a little experiment:
- Pick any subset of the (x,y)-plane. It could be a circle; a square; a collection of random points; or anything else: it really doesn't matter (as long as it contains at least two points ). Call that set S.
- Split S into two non-empty parts (this is why you needed at least two points ). Call one A, and the other B.
- Take A, and shift it one unit to the left. Does A=S? Probably not.
- Now, take B, and rotate it clockwise by 1 radian (that is, 180/π degrees). Does B = S? Again, probably not.
But amazingly, there is a set that it's possible to break into two; then translate one bit, and rotate the other (neither of which should change the size of the set); and end up with both A=S and B=S! It's not something that it's possible to draw on a graph - but, it is possible to describe it mathematically, as follows:
- Imagine that, instead of the (x,y)-plane, we have the complex plane. Now, let 'p' denote the point ei (so, by Euler's formula, p = cos(1) + i sin(1).
- Now, let S be the set of all polynomials in p (expressions like "p2 + 1", or "5p3 + 4p + 7"), with non-negative integer coefficients. This isn't a nice visual shape: it's a cloud of scattered points .
- Now, how do we split S into subsets A and B? Well, A will be all the polynomials that have a non-zero constant term (for example: "p2 + 1", "2p4 + p + 5"), and B will be all the ones that have a zero constant term (for example: "p2", "2p4 + p").
- Now, let's imagine we add 1 to anything in S. No matter which element of S we pick, adding 1 will get us something with a non-zero constant term, i.e. something in A. Or, working backwards: if we start with the elements of A, and subtract 1 from each element, we can get every element of S. And "subtracting 1 from every element of A" is the same thing as "shifting A to the left by one unit". So, after shifting A to the left by one unit, we will get S!
- Next, let's imagine we multiply anything in S by p. No matter which element of S we pick, this will give us a new polynomial with a zero constant term (for example, "p2 + 1" becomes "p3 + p", which has a zero constant term, and thus is in B). Or, working backwards: if we start with the elements of B, and divide each element by p, we can get every element of S. And "dividing each element of B by p" is the same thing as "rotating B clockwise by one radian". So, after rotating B, we will get S!
A bit of a shame we can't see this in action - but, I guess this set is pretty much impossible to visualise . Nevertheless, I hope it made at least some sense to you!
If you have any questions, then by all means ask away, and I'll do my best to answer them. Otherwise, I hope you found this interesting!
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Moonface (in 'Woman runs 49 red lights in ex's car')' Wrote: If only she had ran another 20 lights.
(Thanks to Nilla for the avatar, and Detective Osprey for the sig!)
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