;; My recursion tells me that n planes can divide space into ;; 2, 4, 8, 15, 26, and finally 42 regions. ;; But I'd like some sort of empirical confirmation of this result. ;; Forms in 3-space can be represented as ordered sets of three numbers (def a-form [1 2 3]) ;; As can vectors (def a-point [2 3 5]) ;; And we evaluate the form on the vector (defn contract [form vector] (reduce + (map * form vector))) ;; (or if you prefer, take the dot product, or contract the tensors) (contract a-form a-point) ;-> 23 ;; Any plane can be defined in terms of a 1-form and a number. (def a-plane {:form [1 2 3] :val 4}) ;; Now if we have a plane or a vector, we can evaluate the form on the ;; vector, and compare the result with the value. This tells us which ;; side of the plane the vector is on. (defn side [plane point] (- (contract (plane :form) point) (plane :val))) (side a-plane [2 3 5]) ;-> 19 (this point is on the positive side) (side a-plane [2 3 -5]) ;-> -11 (on the negative side) (side a-plane [2 3 -4/3]) ;-> 0N (in the plane itself) ;; Ok, now we need a way of taking vectors and forms at random. ;; The cauchy distribution is easy to sample from and has nice fat tails (defn cauchy[] (Math/tan (* Math/PI (- (rand) 0.5)))) (repeatedly 20 cauchy) ;-> (-0.43989542100474244 -0.6517139433588255 1.58518947555566 0.001268073580101198 3.6164981498788262 0.44928717656825584 0.3365831420089349 0.4646894211443393 0.8802485518044282 1.8146747880005754 0.1608864471756546 -0.23538854021056904 8.836583912257565 3.8174659241864703 0.5387819323291936 -0.18830386363467239 -1.0430272980416788 0.3310448308016341 -0.10735190850334911 0.3426157380908667) (defn make-point [] (repeatedly 3 cauchy)) (defn make-plane [] {:form (repeatedly 3 cauchy) :val (cauchy)}) (make-point) ;-> (33.032354006369815 -29.428219536044043 -37.796430533111334) (make-plane) ;-> {:form (-45.36910184399889 -1.6741101969009575 9.952054197916382), :val 0.9505471630252558} (def points (repeatedly #(make-point))) (def planes (repeatedly #(make-plane))) ;; And we'll need a function to tell us the sign of a number (defn sign[x] (if (< x 0) '- '+)) (map sign [ -1 -2 -3 0 -0.5 1.3]) ;-> (- - - + - +) ;; Now if we take a set of planes and a point, (defn sig [point planes] (for [p planes] (sign (side p point)))) ;; We can check which side of each plane the point is on (sig (first points) (take 3 planes)) ;-> (+ - +) ;; Every different region gives a different signature. ;; The more planes, the more signatures. (count (frequencies (take 10 (map #(sig % (take 1 planes)) points)))) ;-> 2 (count (frequencies (take 10 (map #(sig % (take 2 planes)) points)))) ;-> 4 (count (frequencies (take 10 (map #(sig % (take 3 planes)) points)))) ;-> 6 (count (frequencies (take 10 (map #(sig % (take 4 planes)) points)))) ;-> 7 (count (frequencies (take 10 (map #(sig % (take 5 planes)) points)))) ;-> 7 (count (frequencies (take 10 (map #(sig % (take 6 planes)) points)))) ;-> 7 ;; But the more planes we have, the smaller the smallest regions are ;; and thus the chance of a point falling in every one goes down. ;; The more points we take, the more likely we are to get one in every region (count (frequencies (take 100 (map #(sig % (take 1 planes)) points)))) ;-> 2 (count (frequencies (take 100 (map #(sig % (take 2 planes)) points)))) ;-> 4 (count (frequencies (take 100 (map #(sig % (take 3 planes)) points)))) ;-> 7 (count (frequencies (take 100 (map #(sig % (take 4 planes)) points)))) ;-> 11 (count (frequencies (take 100 (map #(sig % (take 5 planes)) points)))) ;-> 18 (count (frequencies (take 100 (map #(sig % (take 6 planes)) points)))) ;-> 21 (count (frequencies (take 1000 (map #(sig % (take 1 planes)) points)))) ;-> 2 (count (frequencies (take 1000 (map #(sig % (take 2 planes)) points)))) ;-> 4 (count (frequencies (take 1000 (map #(sig % (take 3 planes)) points)))) ;-> 8 (count (frequencies (take 1000 (map #(sig % (take 4 planes)) points)))) ;-> 15 (count (frequencies (take 1000 (map #(sig % (take 5 planes)) points)))) ;-> 26 (count (frequencies (take 1000 (map #(sig % (take 6 planes)) points)))) ;-> 38 (count (frequencies (take 10000 (map #(sig % (take 1 planes)) points)))) ; 2 (count (frequencies (take 10000 (map #(sig % (take 2 planes)) points)))) ; 4 (count (frequencies (take 10000 (map #(sig % (take 3 planes)) points)))) ; 8 (count (frequencies (take 10000 (map #(sig % (take 4 planes)) points)))) ; 15 (count (frequencies (take 10000 (map #(sig % (take 5 planes)) points)))) ; 26 (count (frequencies (take 10000 (map #(sig % (take 6 planes)) points)))) ; 40 (count (frequencies (take 100000 (map #(sig % (take 1 planes)) points)))) ; 2 (count (frequencies (take 100000 (map #(sig % (take 2 planes)) points)))) ; 4 (count (frequencies (take 100000 (map #(sig % (take 3 planes)) points)))) ; 8 (count (frequencies (take 100000 (map #(sig % (take 4 planes)) points)))) ; 15 (count (frequencies (take 100000 (map #(sig % (take 5 planes)) points)))) ; 26 (count (frequencies (take 100000 (map #(sig % (take 6 planes)) points)))) ; 41 (count (frequencies (take 1000000 (map #(sig % (take 1 planes)) points)))) ; 2 (count (frequencies (take 1000000 (map #(sig % (take 2 planes)) points)))) ; 4 (count (frequencies (take 1000000 (map #(sig % (take 3 planes)) points)))) ; 8 (count (frequencies (take 1000000 (map #(sig % (take 4 planes)) points)))) ; 15 (count (frequencies (take 1000000 (map #(sig % (take 5 planes)) points)))) ; 26 (count (frequencies (take 1000000 (map #(sig % (take 6 planes)) points)))) ; 42 ;; I'm painfully conscious of having stopped the experiment at the ;; exact point where I got the answer I expected. But my poor little ;; computer is not going to be up to running this for 10000000 points. ;; But this can't just be coincidence, surely?

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## Tuesday, May 20, 2014

### Planes In Space : Random Sampling

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