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October 8th, 2012, 11:42 AM   #1
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Number dimensions

1, [color=#BF0040]2[/color], 4, [color=#BF0040]3[/color], 8, 6, 16, 12, 32, 9, 24, 64, 18, 48, 128, 36, 96, 256,
27, [color=#BF0040]5[/color], 72, 192, 512, 54, 10, 144, 384, 1024, 108, 20, 288, 768, 81,
15, 2048, 216, 40, 576, 1536, 162, 30, 4096, 432, 80, 1152, 3072,
324, 60, 8192, 864, 160, 2304, 243, 45, 6144, 648, 120, 16384, 1728,
320, 4608, 486, 90, 12288, 1296, 240, 32768, 3456, 640, 9216, 972,
180, 24576, 2592, 480, 65536, 6912, 1280, 729, 135, 25, [color=#BF0040]7[/color], 18432,
1944, 360, 49152, 5184, 960, 131072, 13824, 2560, 1458, 270, 50, 14,
36864, 3888, 720, 98304, 10368, 1920, 262144

This is the order of integers when ordered according to their distance from the origin in hyperspace when expressed as points in 20 dimensional space corresponding to the primes having to be multiplied together to form the number (padded with 1 so that all numbers are expressed in 20 dimensions). For example, in this space the integer 12 = 1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*2*2*3 and would be represented by the point (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3) and would sit at a distance of SquareRoot 34 from the origin. Interesting to see how powers of two are common but primes are spread out over large intervals. To get an accurate ordered list up to that point (given that there are an infinite number of dimensions corresponding to primes), it is necessary to factorize a huge quantity of numbers (up to 2ąd, where d is the dimension where the numbers are expressed)
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October 8th, 2012, 12:58 PM   #2
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Re: Number dimensions

It seems to make more sense to express them in infinite-dimensional space with unused coordinates set to 0 instead of 1. Then you're looking at A067666 and the order goes 1, 2, 4, 3, 8, 6, 16, 12, 9, 32, 24, 18, 64, 5, 48, 36, 27, 128, 10, ...
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October 8th, 2012, 03:00 PM   #3
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Re: Number dimensions

It certainly simplifies things to use 0 instead of 1. It's just that it seems odd to me to use 0 when the coordinates are defined as being factors of the number they represent.

I feel that by using 0's, I can represent lower-dimensional objects in higher dimensions (0 just means that they don't exist in that dimension), but by using 1's, I can really create higher-dimensional objects, which seems to me a more satisfactory way of expressing a bunch of numbers in the same dimensions. Unfortunately, I have no idea why this approach would be useful at all! But it's always fun to play with numbers and try different things.
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October 8th, 2012, 06:17 PM   #4
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Re: Number dimensions

The problem with 1s is that as you change the dimension, you change the numbers. The problems with finitely many dimensions is that you can't represent all numbers. Using infinitely-many dimensions also solves another problem: the arbitrariness of which numbers go in which dimension. With infinitely many you can have a dimension set aside for 2, 3, 4, 5, 7, 8, 9, 11, ... so that all numbers divisible by 3 have the same coordinate in that dimension.
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