Math::PlanePath::MPeaks -- points in expanding M shape
use Math::PlanePath::MPeaks; my $path = Math::PlanePath::MPeaks->new; my ($x, $y) = $path->n_to_xy (123);
This path puts points in layers of an "M" shape
41 49 7 40 42 48 50 6 39 22 43 47 28 51 5 38 21 23 44 46 27 29 52 4 37 20 9 24 45 26 13 30 53 3 36 19 8 10 25 12 14 31 54 2 35 18 7 2 11 4 15 32 55 1 34 17 6 1 3 5 16 33 56 <- Y=0 ^ -4 -3 -2 -1 X=0 1 2 3 4
N=1 to N=5 is the first "M" shape, then N=6 to N=16 on top of that, etc. The centre goes half way down. Reckoning the N=1 to N=5 as layer d=1 then
Xleft = -d Xright = d Ypeak = 2*d - 1 Ycentre = d - 1
Each "M" is 6 points longer than the preceding. The verticals are each 2 longer, and the centre diagonals each 1 longer. This step 6 is similar to the HexSpiral
.
The octagonal numbers N=1,8,21,40,65,etc k*(3k-2) are a straight line of slope 2 going up to the left. The octagonal numbers of the second kind N=5,16,33,56,etc k*(3k+2) are along the X axis to the right.
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start, in the same pattern. For example to start at 0,
n_start => 0 40 48 39 41 47 49 38 21 42 46 27 50 37 20 22 43 45 26 28 51 36 19 8 23 44 25 12 29 52 35 18 7 9 24 11 13 30 53 34 17 6 1 10 3 14 31 54 33 16 5 0 2 4 15 32 55
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::MPeaks->new ()
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n
on the path.
For $n < 0.5
the return is an empty list, it being considered there are no negative points.
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y
. $x
and $y
are each rounded to the nearest integer which has the effect of treating points as a squares of side 1, so the half-plane y>=-0.5 is entirely covered.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A045944 (etc)
n_start=1 (the default) A045944 N on X axis >= 1, extra initial 0 being octagonal numbers second kind A056106 N on Y axis, extra initial 1 A056109 N on X negative axis <= -1 n_start=0 A049450 N on Y axis, extra initial 0, 2*pentagonal n_start=2 A027599 N on Y axis, extra initial 6,2
Math::PlanePath, Math::PlanePath::PyramidSides
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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