Garrett Lisi and E8: “The universe is maximally beautiful.”

Don Knuth: importunate permutation

The wonderful Cliff Stoll presents a Steiger Millionaire challenge; always offering me Acme Klein Bottles and his peculiar brand of crystal ball.

Consider the factored integers N=[(0), (1), (2), (3), (2,2), 5, (2,3), 7, (2,2,2)...], with each factor representing a point in Euclidean space. Each of these points is at a distance d from the origin, and can be ordered by d.

-- Ieso.hs -- Integers in Euclidean space order -- Uses the Haskell Number Theory functions, from the "Haskell for Maths" -- site: http://www.polyomino.f2s.com/david/haskell/numbertheory.html import Data.List import Primes -- Return distance between two points in Euclidean space. Points are lists -- of possibly unequal lengths, padded with zeros as needed. dist_points x y = sqrt (realToFrac ((foldl (+) 0 (map sq (zipWith (-) x1 y1))))) where x1 = pad (lengthLongest [x,y]) x y1 = pad (lengthLongest [x,y]) y sq x = x * x lengthLongest l = maximum (map length l) pad len x = (x ++ zeroPadding) where zeroPadding = genericTake (len - genericLength x) (repeat 0) -- Return list of all prime factors of n primeFactors 1 = [1] primeFactors n = flatten (map expandedFactors (primePowerFactors n)) where expandedFactors x = genericTake (snd x) (repeat (fst x)) flatten :: [[a]] -> [a] flatten = foldl (++) [] -- Return distance between two factored integers distance n m = dist_points (primeFactors n) (primeFactors m) -- Return list of distances between origin and integers up to n, -- sorted by distance d. Where ties occur, the lowest n comes first, i.e., -- where d=5 for both n=48 and n=5, the order is ... 5, 48 .... -- Neil Sloane has suggested a different way of breaking ties is use of -- lexicographic order, ie for n=48 [2,2,2,3] comes before n=5 [5] - see -- below. distanceOrigin n = map snd (sort d) where d = [(distance 0 i, i) | i <- [0..n]] -- In order to list the integers within a certain distance d from the origin, -- we have to calculate distances up to 2^(d^2/4), e.g. to list ALL -- integers within a distance of 10, use range 2...2^25. -- I believe a list is "complete" up to where a power of 2 appears, -- because numbers greater than that power of 2 can't be closer -- to the origin (except ties, depending on they are handled). However, -- I am curious about how to determine the minimum number of -- integers you have to calculate in order to make a list complete, -- e.g., how high do we have to look to be sure we've found the -- 20 integers closest to the origin? main = do -- Calculate integers within ~7 units of origin print "n ordering for ties" let d = [(distance 0 i, i) | i <- [0..8192]] print (map snd (sort d)) -- Recalculate using lexicographic order to break ties print "lexicographical ordering for ties" let l = [(distance 0 i, primeFactors i, i) | i <- [0..8192]] print (map lastx (sort l)) where lastx (x, y, z) = z

For example, d=8.66 for N=125 at the point (5,5,5), while d=5.29 for N=128 at (2,2,2,2,2,2,2) :

*Ieso> distance 0 125 8.660254037844387 *Ieso> distance 0 128 5.291502622129181

Here is a simple view of several of these integers that lie in 3-space, with the distance between the origin and N=125 highlighted.

The sequence N=[0..8192] in Euclidean space ordered by d begins

*Ieso> distanceOrigin 8192 0,1,2,4,3,8,6,16,12,9,32,24,18,64,5,48,36,27,128,10,96,72,54,256,20,192,15,144,108,81,512,40,384,30,288,216,162,1024

(One way to think about this may be that integers with higher **Ω(n)** should be relatively clustered around the origin).

(The sequence is listed as OEIS A168521).