--- name: "Math::PlanePath::TriangleSpiralSkewed" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::TriangleSpiralSkewed" --- # Math::PlanePath::TriangleSpiralSkewed(3pm) ## NAME Math::PlanePath::TriangleSpiralSkewed -- integer points drawn around a skewed equilateral triangle ## SYNOPSIS ``` use Math::PlanePath::TriangleSpiralSkewed; my $path = Math::PlanePath::TriangleSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This path makes an spiral shaped as an equilateral triangle (each side the same length), but skewed to the left to fit on a square grid, ``` 16 4 |\ 17 15 3 | \ 18 4 14 2 | |\ \ 19 5 3 13 1 | | \ \ 20 6 1--2 12 ... <- Y=0 | | \ \ 21 7--8--9-10-11 30 -1 | \ 22-23-24-25-26-27-28-29 -2 ^ -2 -1 X=0 1 2 3 4 5 ``` The properties are the same as the spread-out `"TriangleSpiral"`. The triangle numbers fall on straight lines as they do in the `"TriangleSpiral"` but the skew means the top corner goes up at an angle to the vertical and the left and right downwards are different angles plotted (but are symmetric by N count). ### Skew Right Option `"skew => 'right'"` directs the skew towards the right, giving ``` 4 16 skew="right" / | 3 17 15 / | 2 18 4 14 / / | | 1 ... 5 3 13 / | | Y=0 -> 6 1--2 12 / | -1 7--8--9-10-11 ^ -2 -1 X=0 1 2 ``` This is a shear "X -\> X+Y" of the default skew="left" shown above. The coordinates are related by ``` Xright = Xleft + Yleft Xleft = Xright - Yright Yright = Yleft Yleft = Yright ``` ### Skew Up ``` 2 16-15-14-13-12-11 skew="up" | / 1 17 4--3--2 10 | | / / Y=0 -> 18 5 1 9 | | / -1 ... 6 8 |/ -2 7 ^ -2 -1 X=0 1 2 ``` This is a shear "Y -\> X+Y" of the default skew="left" shown above. The coordinates are related by ``` Xup = Xleft Xleft = Xup Yup = Yleft + Xleft Yleft = Yup - Xup ``` ### Skew Down ``` 2 ..-18-17-16 skew="down" | 1 7--6--5--4 15 \ | | Y=0 -> 8 1 3 14 \ \ | | -1 9 2 13 \ | -2 10 12 \ | 11 ^ -2 -1 X=0 1 2 ``` This is a rotate by -90 degrees of the skew="up" above. The coordinates are related ``` Xdown = Yup Xup = - Ydown Ydown = - Xup Yup = Xdown ``` Or related to the default skew="left" by ``` Xdown = Yleft + Xleft Xleft = - Ydown Ydown = - Xleft Yleft = Xdown + Ydown ``` ### N Start 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, ``` 15 n_start => 0 |\ 16 14 | \ 17 3 13 ... | |\ \ \ 18 4 2 12 31 | | \ \ \ 19 5 0--1 11 30 | | \ \ 20 6--7--8--9-10 29 | \ 21-22-23-24-25-26-27-28 ``` With this adjustment for example the X axis N=0,1,11,30,etc is (9X-7)\*X/2, the hendecagonal numbers (11-gonals). And South-East N=0,8,25,etc is the hendecagonals of the second kind, (9Y-7)\*Y/2 with Y negative. ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::TriangleSpiralSkewed-\>new ()" - "$path = Math::PlanePath::TriangleSpiralSkewed-\>new (skew =\> $str, n\_start =\> $n)" Create and return a new skewed triangle spiral object. The `"skew"` parameter can be ``` "left" (the default) "right" "up" "down" ``` - "$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 ### Rectangle to N Range Within each row there's a minimum N and the N values then increase monotonically away from that minimum point. Likewise in each column. This means in a rectangle the maximum N is at one of the four corners of the rectangle. ``` | x1,y2 M---|----M x2,y2 maximum N at one of | | | the four corners -------O--------- of the rectangle | | | | | | x1,y1 M---|----M x1,y1 | ``` ## OEIS Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include \ (etc) ``` n_start=1, skew="left" (the defaults) A204439 abs(dX) A204437 abs(dY) A010054 turn 1=left,0=straight, extra initial 1 A117625 N on X axis A064226 N on Y axis, but without initial value=1 A006137 N on X negative A064225 N on Y negative A081589 N on X=Y leading diagonal A038764 N on X=Y negative South-West diagonal A081267 N on X=-Y negative South-East diagonal A060544 N on ESE slope dX=+2,dY=-1 A081272 N on SSE slope dX=+1,dY=-2 A217010 permutation N values of points in SquareSpiral order A217291 inverse A214230 sum of 8 surrounding N A214231 sum of 4 surrounding N n_start=0 A051682 N on X axis (11-gonal numbers) A081268 N on X=1 vertical (next to Y axis) A062708 N on Y axis A062725 N on Y negative axis A081275 N on X=Y+1 North-East diagonal A062728 N on South-East diagonal (11-gonal second kind) A081266 N on X=Y negative South-West diagonal A081270 N on X=1-Y North-West diagonal, starting N=3 A081271 N on dX=-1,dY=2 NNW slope up from N=1 at X=1,Y=0 n_start=-1 A023531 turn 1=left,0=straight, being 1 at N=k*(k+3)/2 A023532 turn 1=straight,0=left n_start=1, skew="right" A204435 abs(dX) A204437 abs(dY) A217011 permutation N values of points in SquareSpiral order but with 90-degree rotation A217292 inverse A214251 sum of 8 surrounding N n_start=1, skew="up" A204439 abs(dX) A204435 abs(dY) A217012 permutation N values of points in SquareSpiral order but with 90-degree rotation A217293 inverse A214252 sum of 8 surrounding N n_start=1, skew="down" A204435 abs(dX) A204439 abs(dY) ``` The square spiral order in A217011,A217012 and their inverses has first step at 90-degrees to the first step of the triangle spiral, hence the rotation by 90 degrees when relating to the `"SquareSpiral"` path. A217010 on the other hand has no such rotation since it reckons the square and triangle spirals starting in the same direction. ## SEE ALSO Math::PlanePath, Math::PlanePath::TriangleSpiral, Math::PlanePath::PyramidSpiral, Math::PlanePath::SquareSpiral ## 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 \.