--- name: "Math::PlanePath::QuintetCentres" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::QuintetCentres" --- # Math::PlanePath::QuintetCentres(3pm) ## NAME Math::PlanePath::QuintetCentres -- self-similar "plus" shape centres ## SYNOPSIS ``` use Math::PlanePath::QuintetCentres; my $path = Math::PlanePath::QuintetCentres->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This a self-similar curve tracing out a "+" shape like the `"QuintetCurve"` but taking the centre of each square visited by that curve. ``` 92 12 / | 124-... 93 91--90 88 11 | \ \ / \ 122-123 120 102 94 82 89 86--87 10 \ / | / | / / | | 121 119 103 101-100 95 81 83--84--85 9 \ \ \ \ \ 114-115-116 118 104 32 99--98 96 80 78 8 | |/ / / | |/ |/ \ 112-113 110 117 105 31 33--34 97 36 79 76--77 7 \ / \ \ \ \ / \ | 111 109-108 106 30 42 35 38--37 75 6 |/ / / | | / 107 29 43 41--40--39 74 5 \ \ | 24--25--26 28 44 46 72--73 70 68 4 | |/ |/ \ \ / \ / \ 22--23 20 27 18 45 48--47 71 56 69 66--67 3 \ / \ / \ | / \ | 21 6 19 16--17 49 54--55 58--57 65 2 / \ | | \ | / 4-- 5 8-- 7 15 50--51 53 59 64 1 \ | / |/ | \ 0-- 1 3 9 14 52 60--61 63 <- Y=0 |/ | \ |/ 2 10--11 13 62 -1 |/ 12 -2 ^ -1 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 ``` The base figure is the initial the initial N=0 to N=4. It fills a "+" shape as ``` ..... . . . 4 . . \. ........\.... . . .\ . . 0---1 . 3 . . . | ./ . ......|./.... . |/. . 2 . . . ..... ``` ### Arms The optional `"arms"` parameter can give up to four copies of the curve, each advancing successively. For example `"arms=>4"` is as follows. Notice the N=4\*k points are the plain curve, and N=4\*k+1, N=4\*k+2 and N=4\*k+3 are rotated copies of it. ``` 69 ... 7 / | \ 121 113 73 65--61 53 120 6 / \ / \ \ \ / \ / ... 117 105-109 77 29 57 45--49 116 5 | / / | | \ 101 81 25 33--37--41 96-100-104 112 4 | \ \ | |/ 50 97--93 85 21 13 88--92 80 108 72 3 / | |/ |/ \ \ / \ / \ 54 46--42 89 10 17 5-- 9 84 24 76 64--68 2 \ | / | | / \ | 58 38 14 6-- 2 1 16--20 32--28 60 1 / | \ \ | / 62 30--34 22--18 3 0-- 4 12 36 56 <- Y=0 | \ / | |/ | \ 70--66 78 26 86 11-- 7 19 8 91 40--44 52 -1 \ / \ / \ \ / | / | |/ 74 110 82 94--90 15 23 87 95--99 48 -2 / | | \ \ | 114 106-102--98 43--39--35 27 83 103 -3 \ | |/ / | 118 51--47 59 31 79 111-107 119 ... -4 / \ / \ \ \ / \ / 122 55 63--67 75 115 123 -5 \ |/ ... 71 -6 ^ -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 ``` The pattern an ever expanding "+" shape with first cell N=0 at the origin. The further parts are effectively as follows, ``` +---+ | | +---+--- +---+ | | | +---+ +---+ +---+ | 2 | 1 | | +---+ +---+---+ +---+ | | 3 | 0 | +---+ +---+ +---+ | | | +---+ +---+---+ | | +---+ ``` At higher replication levels the sides become wiggly and spiralling, but they're symmetric and mesh to fill the plane. ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::QuintetCentres-\>new ()" - "$path = Math::PlanePath::QuintetCentres-\>new (arms =\> $a)" Create and return a new path object. - "($x,$y) = $path-\>n\_to\_xy ($n)" Return the X,Y coordinates of point number `$n` on the path. Points begin at 0 and if `"$n < 0"` then the return is an empty list. Fractional positions give an X,Y position along a straight line between the integer positions. - "$n = $path-\>n\_start()" Return 0, the first N in the path. - "($n\_lo, $n\_hi) = $path-\>rect\_to\_n\_range ($x1,$y1, $x2,$y2)" In the current code the returned range is exact, meaning `$n_lo` and `$n_hi` are the smallest and biggest in the rectangle, but don't rely on that yet since finding the exact range is a touch on the slow side. (The advantage of which though is that it helps avoid very big ranges from a simple over-estimate.) ### Level Methods - "($n\_lo, $n\_hi) = $path-\>level\_to\_n\_range($level)" Return `"(0, 5**$level - 1)"`, or for multiple arms return `"(0, $arms *` `5**$level - 1)"`. There are 5^level points in a level, or arms\*5^level for multiple arms, numbered starting from 0. ## FORMULAS ### X,Y to N The `"xy_to_n()"` calculation is similar to the `"FlowsnakeCentres"`. For a given X,Y a modulo 5 remainder is formed ``` m = (2*X + Y) mod 5 ``` This distinguishes the five squares making up the base figure. For example in the base N=0 to N=4 part the m values are ``` +-----+ | m=3 | 1 +-----+-----+-----+ | m=0 | m=2 | m=4 | <- Y=0 +-----+-----+-----+ | m=1 | -1 +-----+ X=0 1 2 ``` From this remainder X,Y can be shifted down to the 0 position. That position corresponds to a vector multiple of X=2,Y=1 and 90-degree rotated forms of that vector. That vector can be divided out and X,Y shrunk with ``` Xshrunk = (Y + 2*X) / 5 Yshrunk = (2*Y - X) / 5 ``` If X,Y are considered a complex integer X+iY the effect is a remainder modulo 2+i, subtract that to give a multiple of 2+i, then divide by 2+i. The vector X=2,Y=1 or 2+i is because that's the N=5 position after the base shape. The remainders can then be mapped to base 5 digits of N going from high to low and making suitable rotations for the sub-part orientation of the curve. The remainders alone give a traversal in the style of `"QuintetReplicate"`. Applying suitable rotations produces the connected path of `"QuintetCentres"`. ## OEIS Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include \ (etc) ``` A099456 level Y end, being Im((2+i)^k) arms=2 A139011 level Y end, being Re((2+i)^k) ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::QuintetCurve, Math::PlanePath::QuintetReplicate, Math::PlanePath::FlowsnakeCentres ## 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 \.