--- name: "Math::PlanePath::PentSpiral" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::PentSpiral" --- # Math::PlanePath::PentSpiral(3pm) ## NAME Math::PlanePath::PentSpiral -- integer points in a pentagonal shape ## SYNOPSIS ``` use Math::PlanePath::PentSpiral; my $path = Math::PlanePath::PentSpiral->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This path makes a pentagonal (five-sided) spiral with points spread out to fit on a square grid. ``` 22 3 23 10 21 2 24 11 3 9 20 1 25 12 4 1 2 8 19 <- Y=0 26 13 5 6 7 18 ... -1 27 14 15 16 17 33 -2 28 29 30 31 32 -2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 ``` Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The lower diagonals are 1 across and 1 down, so n=17 is at x=4,y=-2 and n=18 is x=5,y=-1. But the upper angles go 2 across and 1 up, so n=20 is x=4,y=1 then n=21 is x=2,y=2. The effect is to make the sides equal length, except for a kink at the lower right corner. Only every second square in the plane is used. In the top half (y\>=0) those points line up, in the lower half (y\<0) they're offset on alternate rows. ### N Start The default is to number points starting N=1 as shown above. An optional `"n_start"` can give a different start, in the same pattern. For example to start at 0, ``` n_start => 0 38 39 21 37 ... 40 22 9 20 36 57 41 23 10 2 8 19 35 56 42 24 11 3 0 1 7 18 34 55 43 25 12 4 5 6 17 33 54 44 26 13 14 15 16 32 53 45 27 28 29 30 31 52 46 47 48 49 50 51 ``` ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::PentSpiral-\>new ()" - "$path = Math::PlanePath::PentSpiral-\>new (n\_start =\> $n)" Create and return a new pentagon spiral object. - "$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 each point in the path as a square of side 1. ## FORMULAS ### N to X,Y It's convenient to work in terms of Nstart=0 and to take each loop as beginning on the South-West diagonal, ``` 21 loop d=3 -- -- 22 20 -- -- 23 19 -- -- 24 0 18 \ / 25 . 17 \ / 26 13----14----15----16 \ . ``` The SW diagonal is N=0,4,13,27,46,etc which is ``` N = (5d-7)*d/2 + 1 # starting d=1 first loop ``` This can be inverted to get d from N ``` d = floor( (sqrt(40*N + 9) + 7) / 10 ) ``` Each side is length d, except the lower right diagonal slope which is d-1. For the very first loop that lower right is length 0. ## OEIS Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include \ (etc) ``` n_start=1 (the default) A192136 N on X axis, (5*n^2 - 3*n + 2)/2 A140066 N on Y axis A116668 N on X negative axis A005891 N on South-East diagonal, centred pentagonals A134238 N on South-West diagonal n_start=0 A000566 N on X axis, heptagonal numbers A005476 N on Y axis A028895 N on South-East diagonal ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiral ## HOME PAGE \ ## LICENSE Copyright 2010, 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 \.