Latrinsorm
07-17-2013, 06:16 PM
Let us imagine a world where people can be classified by a discrete range of attributes that are themselves discretely defined.
Let us imagine that there is no one looking out for us, that we are alone, and there's just chaos (and presumably violence, and random unpredictable evil that comes out of nowhere and rips us to shreds). Every attribute is assigned randomly, on a convenient integer scale from 1 to 6.
If we look at only one attribute, let's call it Strength, then over sufficiently large populations we get a flat graph, like so:
http://img.photobucket.com/albums/v456/johnnyoldschool/Deviance1D_zpsf10f2c34.jpg
-No one is average, because the average value is 3.5. This is a consequence of having an even number of bins, and is uninteresting.
-Even though everyone has the same number of binmates, not everyone's nearest neighbor is equally populous. People with Strengths of 2 to 5 have 50% of people within one bin of themselves (3/6 = .5), but people with Strengths of 1 or 6 have only 33% of people within (3/6 = .3.
-Because we're going to many attributes, it is more convenient to talk about the average distance than the nearest bin. People with 1 Str are 0 away from the 1/6 of the population in 1, 1 from 2, 2 from 3, and so on, so they are 0 * 1/6 + 1 * 1/6 + ... + 5 * 1/6 = 15/6 away from a random person on average. Meanwhile the norms with 3 Str are 2 away from 1, 1 away from 2, and so on, so they are 2 * 1/6 + 1 * 1/6 + ... + 3 * 1/6 = 9/6 away from a random person on average.
If we added another attribute with the same conditions, let's call it Dexterity, again no one would be average, because our average Str and Dex are 3.5, and that is still uninteresting. It gets a little more computationally intense to do average distances, but for any Cartesian points (x,y,z,w,&,...) and (xb,yb,zb,wb,&b,...) we can compute distances by sqrt((x-xb)^2+(y-yb)^2+(z-zb)^2+(w-wb)^2+(&-&b)^2+...), which strictly speaking we already did for one dimension (which is what makes this world beautiful) as sqrt(x^2) = x. I can't graph it, but consider again how every bin has the same population: dice have no memory, so 1 and 6 is just as likely as 6 and 1 as 3 and 3 as 4 and 2 as 2 and 2. It also turns out we have symmetries not just for points like (2,3) to (3,2), which is arithmetically obvious, but points like (3,3) to (4,4), which is not so much. What matters for distance from the random is thus distance from the average. There could be as many as 36 values for distance, but it turns out there are only 6(!). Plus, our ratio between most and least deviant has increased from (15/6)/(9/6) = 1.67 to 3.940222/2.338141 = 1.69.
With this established, we can now understand indefinite dimensions by computing merely the (1,1,1...,1) and (3,3,3...,3) points, whose ratios turn out to look like this...
http://img.photobucket.com/albums/v456/johnnyoldschool/ItDoesTrustMe_zps2fb79295.jpg
...which looks like it wants to converge to something, but my guesses (root 3, ln 6, the rectangular hyperbola) don't seem likely to fit, and unfortunately all humans can be described by exactly 6 attributes so we can't go any further numerically.
Let us imagine that there is no one looking out for us, that we are alone, and there's just chaos (and presumably violence, and random unpredictable evil that comes out of nowhere and rips us to shreds). Every attribute is assigned randomly, on a convenient integer scale from 1 to 6.
If we look at only one attribute, let's call it Strength, then over sufficiently large populations we get a flat graph, like so:
http://img.photobucket.com/albums/v456/johnnyoldschool/Deviance1D_zpsf10f2c34.jpg
-No one is average, because the average value is 3.5. This is a consequence of having an even number of bins, and is uninteresting.
-Even though everyone has the same number of binmates, not everyone's nearest neighbor is equally populous. People with Strengths of 2 to 5 have 50% of people within one bin of themselves (3/6 = .5), but people with Strengths of 1 or 6 have only 33% of people within (3/6 = .3.
-Because we're going to many attributes, it is more convenient to talk about the average distance than the nearest bin. People with 1 Str are 0 away from the 1/6 of the population in 1, 1 from 2, 2 from 3, and so on, so they are 0 * 1/6 + 1 * 1/6 + ... + 5 * 1/6 = 15/6 away from a random person on average. Meanwhile the norms with 3 Str are 2 away from 1, 1 away from 2, and so on, so they are 2 * 1/6 + 1 * 1/6 + ... + 3 * 1/6 = 9/6 away from a random person on average.
If we added another attribute with the same conditions, let's call it Dexterity, again no one would be average, because our average Str and Dex are 3.5, and that is still uninteresting. It gets a little more computationally intense to do average distances, but for any Cartesian points (x,y,z,w,&,...) and (xb,yb,zb,wb,&b,...) we can compute distances by sqrt((x-xb)^2+(y-yb)^2+(z-zb)^2+(w-wb)^2+(&-&b)^2+...), which strictly speaking we already did for one dimension (which is what makes this world beautiful) as sqrt(x^2) = x. I can't graph it, but consider again how every bin has the same population: dice have no memory, so 1 and 6 is just as likely as 6 and 1 as 3 and 3 as 4 and 2 as 2 and 2. It also turns out we have symmetries not just for points like (2,3) to (3,2), which is arithmetically obvious, but points like (3,3) to (4,4), which is not so much. What matters for distance from the random is thus distance from the average. There could be as many as 36 values for distance, but it turns out there are only 6(!). Plus, our ratio between most and least deviant has increased from (15/6)/(9/6) = 1.67 to 3.940222/2.338141 = 1.69.
With this established, we can now understand indefinite dimensions by computing merely the (1,1,1...,1) and (3,3,3...,3) points, whose ratios turn out to look like this...
http://img.photobucket.com/albums/v456/johnnyoldschool/ItDoesTrustMe_zps2fb79295.jpg
...which looks like it wants to converge to something, but my guesses (root 3, ln 6, the rectangular hyperbola) don't seem likely to fit, and unfortunately all humans can be described by exactly 6 attributes so we can't go any further numerically.