Math::PlanePath::DiamondSpiral -- integer points around a diamond shaped spiral
use Math::PlanePath::DiamondSpiral; my $path = Math::PlanePath::DiamondSpiral->new; my ($x, $y) = $path->n_to_xy (123);
This path makes a diamond shaped spiral.
19 3 / \ 20 9 18 2 / / \ \ 21 10 3 8 17 1 / / / \ \ \ 22 11 4 1---2 7 16 <- Y=0 \ \ \ / / 23 12 5---6 15 ... -1 \ \ / / 24 13--14 27 -2 \ / 25--26 -3 ^ -3 -2 -1 X=0 1 2 3
This is not simply the
SquareSpiral rotated, it spirals around faster, with side lengths following a pattern 1,1,1,1, 2,2,2,2, 3,3,3,3, etc, if the flat kink at the bottom (like N=13 to N=14) is treated as part of the lower right diagonal.
Going diagonally on the sides as done here is like cutting the corners of the
SquareSpiral, which is how it gets around in fewer steps than the
HeptSpiralSkewed for similar cutting just 3, 2 or 1 of the corners.
N=1,5,13,25,etc on the Y negative axis is the "centred square numbers" 2*k*(k+1)+1.
The default is to number points starting N=1 as shown above. An optional
n_start can give a different start, with the same shape etc. For example to start at 0,
n_start => 0 18 / \ 19 8 17 / / \ \ 20 9 2 7 16 / / / \ \ \ 21 10 3 0-- 1 6 15 \ \ \ / / 22 11 4-- 5 14 ... \ \ / / 23 12--13 26 \ / 24--25
N=0,1,6,15,28,etc on the X axis is the hexagonal numbers k*(2k-1). N=0,3,10,21,36,etc on the negative X axis is the hexagonal numbers of the "second kind" k*(2k-1) for k<0. Combining those two is the triangular numbers 0,1,3,6,10,15,21,etc, k*(k+1)/2, on the X axis alternately positive and negative.
N=0,2,8,18,etc on the Y axis is 2*squares, 2*Y^2. N=0,4,12,24,etc on the negative Y axis is 2*pronic, 2*Y*(Y+1).
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::DiamondSpiral->new ()
$path = Math::PlanePath::DiamondSpiral->new (n_start => $n)
Create and return a new diamond spiral object.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number
$n on the path.
$n < 1 the return is an empty list, it being considered the path starts at 1.
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates
$y are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1, so the entire plane is covered.
($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)
The returned range is exact, meaning
$n_hi are the smallest and biggest in the rectangle.
Within each row N increases as X moves away from the Y axis, and within each column similarly N increases as Y moves away from the X axis. So in a rectangle the maximum N is at one of the four corners.
| x1,y2 M---|----M x2,y2 | | | -------O--------- | | | | | | x1,y1 M---|----M x1,y1 |
For any two columns x1 and x2 with x1<x2, the values in column x2 are all bigger if x2>-x1. This is so even when x1 and x2 are on the same side of the origin, ie. both positive or both negative.
For any two rows y1 and y2, the values in the part of the row with X>0 are bigger if y2>=-y1, and in the part of the row with X<=0 it's y2>-y1, or equivalently y2>=-y1+1. So the biggest corner is at
max_x = (x2 > -x1 ? x2 : x1) max_y = (y2 >= -y1+(max_x<=0) ? y2 : y1)
The minimum is similar but a little simpler. In any column the minimum is at Y=0, and in any row the minimum is at X=0. So at 0,0 if that's in the rectangle, or the edge on the side nearest the origin when not.
min_x = / if x2 < 0 then x2 | if x1 > 0 then x1 \ else 0 min_y = / if y2 < 0 then y2 | if y1 > 0 then y1 \ else 0
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A010751 (etc) n_start=1 A130883 N on X axis, 2*n^2-n+1 A058331 N on Y axis, 2*n^2 + 1 A001105 N on column X=1, 2*n^2 A084849 N on X negative axis, 2*n^2+n+1 A001844 N on Y negative axis, centred squares 2*n^2+2n+1 A215471 N with >=5 primes among its 8 neighbours A215468 sum 8 neighbours N A217015 N permutation points order SquareSpiral rotate -90, value DiamondSpiral N at each A217296 inverse permutation n_start=0 A010751 X coordinate, runs 1 inc, 2 dec, 3 inc, etc A000384 N on X axis, hexagonal numbers A001105 N on Y axis, 2*n^2 (and cf similar A184636) A014105 N on X negative axis, second hexagonals A046092 N on Y negative axis, 2*pronic A003982 1,0 according as N on Y negative axis, being delta abs(X)+abs(Y) which is 1 when move "outward" to next ring n_start=-1 A188551 N positions of turns, from N=1 up A188552 and which are primes
Copyright 2010, 2011, 2012, 2013 Kevin Ryde
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