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[math] here's a cool problem that proves something that feels like it would be really hard to prove, or dubious that it's even true at all
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it's called the Borsuk Ulam Theorem
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it's more general, but the specific example I'm using states this:
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there is guaranteed to always be two antipodes on the Earth's surface - two points on exact opposite sides of the world - that have the exact same temperature and air pressure
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now we're making some small assumptions here for simplicity
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1 is that the temperature and air pressure on earth are Continuous - in other words, there's no point where the temperature immediately jumps from hot to cold without some degree of gradient between the two
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2 is that the earth is a sphere. it isn't quite but it's close enough
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so. let's prove that this is the case
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we'll do this by mapping the sphere into something else! we're gonna squish it
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at each point on the sphere, there are two measured numbers - one for temp, one for pressure
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we can use those two values as coordinates
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https://images.plurk.com/6DGuAXpaG9y446UY82N811.png so, each point on the sphere plots onto a point in 2D, based on its temp/pressure
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of course, some places will have the same temp/pressure as others, but we're fine with that. it just means multiple points on the sphere map to the same point on the 2D grid
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we can take this mapping and call it f(p)
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we say, "f(p) = (x,y)"
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which means, "for point p, its coordinates on the mapping plane are x, y"
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p is not a number, it's a 3D vector
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pointing to a place on the sphere, with 0 as the center of the earth
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so, since it's a vector, we can do vector math to it
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specifically if p is one end of an antipode, -p is the other end
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so, we can reframe our original claim in more concrete terms:
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there exists some p such that f(p) = f(-p)
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or, if you subtract f(-p) from both sides:
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f(p) - f(-p) = (0,0)
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so. time to prove that!
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because we have this mapping, we aren't just restricted to mapping points! we can also map lines
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https://images.plurk.com/hYNi7iX98puw8yepLLQUq.png like this.
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we've taken the path around the equator, and mapped it to some squiggly line in the mapped space
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two things are guaranteed to be true about this specific line
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1. it will form a closed loop.
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because the start and end are in the same place.
BattroidBattery
oh okay I missed the part about assuming continuity and was getting bothered about nothing
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ok hang on I need to stop and brain at this to make sure I understand this next part
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oh okay I see! I explained this mapping wrong
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it's not a mapping of f(p) that makes the squiggle here
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it's a mapping of g(p), defined as:
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g(p) = f(p) - f(-p)
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so based on that, when you make it halfway around the loop
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you'll end up at the antipode of p, or -p
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so, on the equator, the starting point and the halfway point are opposites
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and accordingly, the second half of the loop is the first half rotated 180 degrees
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and mathswise, the opposite point is g(-p)
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g(-p) = f(-p) - f(p)
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in other words, g(-p) = -g(p)
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anyway
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to prove that we have a p where f(p) = f(-p) is to say that there exists a point p such that g(p) = (0,0)
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(0,0) = the origin
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I'm repeating a lot of stuff because this fucks with my head
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anyway if you play with the restrictions of a loop made by two squiggles rotated 180 degrees from each other, you'll quickly find that you can't have the origin be outside of the loop
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the second half will always wrap around and contain the origin. it is always inside the loop, unless the loop passes through the origin directly - in which point, hey, we found the point where g(p) = (0,0) already, so we're done
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but most of the time, origin is inside the loop
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https://images.plurk.com/3Z3IIHP63XVkR9HRz2h7bb.png what we do next is this
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start pulling the loop up the sphere, towards the north pole
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as you do, it loses the convenient symmetry of the original loop
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but it contracts
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and when it reaches the north pole, the loop will collapse into a point Somewhere
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but because the origin is known to be inside the loop
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https://images.plurk.com/3D9xSMDisVhcu9hc55wKti.png at some point, the loop has to cross over the origin to collapse around the north pole
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and when it does, that's a point where g(p) = (0,0)
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in other words, a point where f(p) - f(-p) = (0,0)
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in other words, a point where f(p) = f(-p)
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because the loop has to start containing the origin and has to end as a single point somewhere
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and the translation between those two shapes will always cross over the origin at some point
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OKAY TIME TO MAKE THIS EVEN MORE MATH
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first, a thing you're going to have to Just Trust Me On, about the above proof
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the general statement of the above is
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"in any continuous mapping of a sphere to a 2D plane, there will exist antipodal points on the sphere that map to the same point on the plane"
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and the Just Trust Me part is:
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this proof extends into higher dimensions
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in any continuous mapping of a 4D hypersphere into a 3D space, there will exist antipodal points on the hypersphere that map to the same point on the plane
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still true in 5D, 6D, etc
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we will hold it in our hearts that this is the case
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anyway throw all this bullshit about spheres out
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let's talk about a completely different problem
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the Stolen Necklace Problem
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the problem suggests that Alice and Bob have stolen a necklace containing a variety of jewels
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and that each type of jewel occurs an even number of times
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they want to cut up the necklace such that Alice and Bob can both take segments of it in such a way that each of them has the same number of each type of jewel
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so, if there are 8 sapphires, 4 emeralds, and 6 diamonds, each thief must end up with 4 sapphires, 2 emeralds, and 3 diamonds
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the claim is that, if there are N types of jewels on the necklace
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then it can be divided into a satisfactory set of segments in N cuts.
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this is hard to verify experimentally, let alone prove rigorously
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but let's get weird with it
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https://images.plurk.com/4tfjrbZA0rvMNR9Dhclx0x.png start with a simplest case - only two jewel colors
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there are 8 sapphires and 10 emeralds
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so 2 cuts have to divide it so that each person can take home 4 sapphires and 5 emeralds
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https://images.plurk.com/2GjtDiK4buh78KZouCSdGy.png in this case, you can do it like this
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but this is too discrete for what we're doing with it
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so let's generalize
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https://images.plurk.com/4ZkDCHsrA5lTwlCMJwHckB.png a line of length 1
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divided into 18 segments, of length 1/18
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each segment colored blue or green
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now the problem is to find two cuts anywhere on the line that result in alice and bob getting the same total length of blue and green
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there were 18 segments initially, so 8/18ths of the line is blue and 10/18ths of the line is green
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and each side must have 4/18ths blue and 5/18ths green, adding up to their 1/2 of the original line
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before advancing further: despite technically being allowed to place the cuts in mid-jewel now, this is still the same problem.
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https://images.plurk.com/3w54bRayu7GA4SADEnxPsS.png if you did something like this
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then you gave, say, 30% of an emerald to alice and 70% of it to bob
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then the thing is that alice and bob both need a total of 5 emeralds worth of green line
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so somewhere else, you need to give alice 70% of an emerald and give bob 30% of an emerald
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so, the cuts that are in mid-jewel will always be in pairs of the same color, so you can just slide them over and get the same result without cutting a jewel
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so if we can solve this problem, we can solve the discrete version
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now
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let's rethink how we're defining 'cutting'
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we've been thinking of it as picking two points on the line
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https://images.plurk.com/5nkBqcTkeKCyAnzYB9GIPZ.png but let's think of it like this instead
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the division is defined as three numbers, a, b, and c, such that a+b+c = 1
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then, each piece must be given to alice or bob, which will call negative and positive
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so.
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what's a sphere.
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mathematically the unit sphere is all points (x,y,z) such that x^2 + y^2 + z^2 = 1
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so if you say x^2 = a, y^2 = b, z^2 = c
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then a+b+c = 1
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in other words, our line division definition above can be mapped to the surface of a sphere.
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for each, you take the square root, so
x = root(1/6)
y = root(1/3)
z = root(1/2)
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and square it to get to your sum of one
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and just as each segment can be given to alice or bob
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x, y, and z can each be negative or positive
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because a^2 and (-a)^2 are equivalent
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every possible division of the necklace into three segments is represented by a point on the sphere
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so, what would it mean for a division to be even?
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it would mean that if you swap all three pieces, you get the same value
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so, (a,b,c) = (-a,-b,-c)
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two antipodal points with the same value
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does such a pair of points exist on the sphere?
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yeah.
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we proved it back in the first half.
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by the same reasoning, there must be a division of the necklace into three segments such that each side of the division is equal in value
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and because the borsuk-ulam theorem applies to higher dimensions, this is still true as we add more colors of jewels to the necklace - each new color is just another dimension of the sphere
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these two problems, one about sphere topology and one about necklace grifting,
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are the same fucking problem
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