--- name: "Math::PlanePath::MPeaks" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::MPeaks" --- # Math::PlanePath::MPeaks(3pm) ## NAME Math::PlanePath::MPeaks -- points in expanding M shape ## SYNOPSIS ``` use Math::PlanePath::MPeaks; my $path = Math::PlanePath::MPeaks->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This path puts points in layers of an "M" shape ``` 41 49 7 40 42 48 50 6 39 22 43 47 28 51 5 38 21 23 44 46 27 29 52 4 37 20 9 24 45 26 13 30 53 3 36 19 8 10 25 12 14 31 54 2 35 18 7 2 11 4 15 32 55 1 34 17 6 1 3 5 16 33 56 <- Y=0 ^ -4 -3 -2 -1 X=0 1 2 3 4 ``` N=1 to N=5 is the first "M" shape, then N=6 to N=16 on top of that, etc. The centre goes half way down. Reckoning the N=1 to N=5 as layer d=1 then ``` Xleft = -d Xright = d Ypeak = 2*d - 1 Ycentre = d - 1 ``` Each "M" is 6 points longer than the preceding. The verticals are each 2 longer, and the centre diagonals each 1 longer. This step 6 is similar to the `"HexSpiral"`. The octagonal numbers N=1,8,21,40,65,etc k\*(3k-2) are a straight line of slope 2 going up to the left. The octagonal numbers of the second kind N=5,16,33,56,etc k\*(3k+2) are along the X axis to the right. ### N Start The default is to number points starting N=1 as shown above. An optional `"n_start"` can give a different start, in the same pattern. For example to start at 0, ``` n_start => 0 40 48 39 41 47 49 38 21 42 46 27 50 37 20 22 43 45 26 28 51 36 19 8 23 44 25 12 29 52 35 18 7 9 24 11 13 30 53 34 17 6 1 10 3 14 31 54 33 16 5 0 2 4 15 32 55 ``` ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$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. ## OEIS Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include \ (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 ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::PyramidSides ## HOME PAGE \ ## LICENSE Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde This file is part of Math-PlanePath. Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see \.