Search This Blog

Loading...

Wednesday, April 30, 2014

Planes in Spaaace!


;; Into how many regions can six planes divide space?

;; Too hard!

;; So make it harder:

;; Into how many regions can m hyperplanes divide n dimensional space?

;; Now look at the easiest case.

;; Into how many lines can m points divide a line?

;; We start off with one line and no points

;; n = 1 
;; m  points lines
;; 0  0      1

;; Every time we add a point, it splits a line

;; n = 1 
;; m  points lines
;; 0  0      1
;; 1  1      2
;; 1  1      2

(defn regions [n m]
  (cond (= n 1) (list m (inc m))
        :else 'chsaispas))

(regions 1 10) ;-> (10 11)
;; Adding ten points turns a line into 11 lines and 10 points.

;; Ok, what about lines dividing a plane?

;; We start off with one area, no lines and no points.

;; n = 2 
;; m  points lines areas
;; 0  0      0     1

;; When we add a line, it splits the area into two areas, so we have
;; no points, one line and two areas

;; n = 2 
;; m  points lines areas
;; 0  0      0     1
;; 1  0      1     2

;; When we add a second line, unless we've accidentally put it
;; parallel to our first line, in which case we aren't doing as well
;; as we could do, it will cross the first line.

;; So our new line will be actually be one point and two lines, 
;; and the one point will divide the first line in two, 
;; and the two lines will cut the two areas into four.

;; 0      1     2  ;; State with one line
;; 1      2        ;; Added one point and two lines
;;        1     2  ;; Which have split one line and two areas
;; 1      4     4  ;; So two lines divide a plane into one point, four lines and four areas

;; n = 2 
;; m  points lines areas
;; 0  0      0     1
;; 1  0      1     2
;; 2  1      4     4

;; When we add a third line, it will cross both previous lines, and so it will be 
;; a line divided by two points 

(regions 1 2) ;-> (2 3)

(map + '(1 4 4) 
     (concat '(0) (regions 1 2)) 
     (concat (regions 1 2) '(0))) ;-> (3 9 7)

;; If you draw the figure, you'll see that it indeed has a triangle with three vertices, 
;; which split the three lines into 9, and there are seven distinct regions.

;; So, taking a blind leap in the dark...

(defn regions [n m]
  (cond (= n 1) (list m (inc m))
        (= m 0) (concat (repeat n 0) '(1))
        :else (map + 
                   (regions n (dec m))
                   (concat '(0) (regions (dec n) (dec m))) 
                   (concat (regions (dec n) (dec m)) '(0)))))

(regions 1 0) ;-> (0 1)
(regions 1 1) ;-> (1 2)
(regions 1 2) ;-> (2 3)
(regions 1 3) ;-> (3 4)

(regions 2 0) ;-> (0 0 1)
(regions 2 1) ;-> (0 1 2)
(regions 2 2) ;-> (1 4 4)
(regions 2 3) ;-> (3 9 7)
(regions 2 4) ;-> (6 16 11)
(regions 2 5) ;-> (10 25 16)

(regions 3 1) ;-> (0 0 1 2)
(regions 3 2) ;-> (0 1 4 4)
(regions 3 3) ;-> (1 6 12 8)
(regions 3 4) ;-> (4 18 28 15)
(regions 3 5) ;-> (10 40 55 26)
(regions 3 6) ;-> (20 75 96 42)

;; Into how many regions can six planes divide space?

(last (regions 3 6)) ;-> 42



Into How Many Regions Can Six Planes Divide Space?

I recently misread an easy recreational puzzle as 'Into how many regions can six planes divide space?'.

It took me about half an hour to get an answer. (Which was the correct answer to the question, but not the answer to the real puzzle). But it turned out to be great fun to think about.

I'm pretty sure that my answer's correct (and wikipedia agrees), but although the combination of mathematical proof and worldwide consensus should probably be enough, I'm never really convinced by a mathematical idea until I've seen it confirmed by lots of examples.

So I've got programs planned, and if they turn out well, then I'll post them here. But they won't be any fun to read unless you've tried to work it out yourself.

If it's not clear, the idea is that you've got a big block of sponge cake, and a very large knife, and you want to cut it into as many pieces as possible with six cuts.

So if the problem said 'one' rather than 'six', the answer would be two, and if it said 'two', then the answer would be 'four'.

1,2,4,8,..... how does this sequence continue?

Followers