--- name: "Math::PlanePath::HeptSpiralSkewed" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::HeptSpiralSkewed" --- # Math::PlanePath::HeptSpiralSkewed(3pm) ## NAME Math::PlanePath::HeptSpiralSkewed -- integer points around a skewed seven sided spiral ## SYNOPSIS ``` use Math::PlanePath::HeptSpiralSkewed; my $path = Math::PlanePath::HeptSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This path makes a seven-sided spiral by cutting one corner of a square ``` 31-30-29-28 3 | \ 32 14-13-12 27 2 | | \ \ 33 15 4--3 11 26 1 | | | \ \ \ 34 16 5 1--2 10 25 <- Y=0 | | | | | 35 17 6--7--8--9 24 -1 | | | 36 18-19-20-21-22-23 -2 | 37-38-39-40-41-... -3 ^ -3 -2 -1 X=0 1 2 3 ``` The path is as if around a heptagon, with the left and bottom here as two sides of the heptagon straightened out, and the flat top here skewed across to fit a square grid. ### N Start The default is to number points starting N=1 as shown above. An optional `"n_start"` can give a different start, in the same pattern. For example to start at 0, ``` 30 29 28 27 n_start => 0 31 13 12 11 26 32 14 3 2 10 25 33 15 4 0 1 9 24 34 16 5 6 7 8 23 35 17 18 19 20 21 22 36 37 38 39 40 ... ``` ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::HeptSpiralSkewed-\>new ()" - "$path = Math::PlanePath::HeptSpiralSkewed-\>new (n\_start =\> $n)" Create and return a new path object. - "$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 N in the path as centred in a square of side 1, so the entire plane is covered. ## 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, ``` top length = d 30-29-28-27 | \ 31 26 diagonal length = d left | \ length 32 25 = 2*d | \ 33 0 24 | | right 34 . 23 length = d-1 | | 35 17-18-19-20-21-22 | . bottom length = 2*d-1 ``` The SW diagonal is N=0,5,17,36,etc which is ``` N = (7d-11)*d/2 + 2 # starting d=1 first loop ``` This can be inverted to get d from N ``` d = floor( (sqrt(56*N+9)+11)/14 ) ``` The side lengths are as shown above. The first loop is d=1 and for it the "right" vertical length is zero, so no such side on that first loop 0 \<= N \< 5. ## OEIS Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include \ (etc) ``` n_start=1 A140065 N on Y axis n_start=0 A001106 N on X axis, 9-gonal numbers A218471 N on Y axis A022265 N on X negative axis A179986 N on Y negative axis, second 9-gonals A195023 N on X=Y diagonal A022264 N on North-West diagonal A186029 N on South-West diagonal A024966 N on South-East diagonal ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiralSkewed ## 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 \.