--- name: "Math::PlanePath::CubicBase" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::CubicBase" --- # Math::PlanePath::CubicBase(3pm) ## NAME Math::PlanePath::CubicBase -- replications in three directions ## SYNOPSIS ``` use Math::PlanePath::CubicBase; my $path = Math::PlanePath::CubicBase->new (radix => 4); my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This is a pattern of replications in three directions 0, 120 and 240 degrees. ``` 18 19 26 27 5 16 17 24 25 4 22 23 30 31 3 20 21 28 29 2 50 51 58 59 2 3 10 11 1 48 49 56 57 0 1 8 9 <- Y=0 54 55 62 63 6 7 14 15 -1 52 53 60 61 4 5 12 13 -2 34 35 42 43 -3 32 33 40 41 -4 38 39 46 47 -5 36 37 44 45 -6 ^ -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 ``` The points are on a triangular grid by using every second integer X,Y, as per "Triangular Lattice" in Math::PlanePath. All points on that triangular grid are visited. The initial N=0,N=1 is replicated at +120 degrees. Then that trapezoid at +240 degrees ``` +-----------+ +-----------+ \ 2 3 \ \ 2 3 \ +-----------+ \ \ \ 0 1 \ \ 0 1 \ +-----------+ --------- -----------+ \ 6 7 \ replicate +120deg \ \ rep +240deg \ 4 5 \ +----------+ ``` Then that bow-tie N=0to7 is replicated at 0 degrees again. Each replication is 1/3 of the circle around, 0, 120, 240 degrees repeating. The relative layout within a replication is unchanged. ``` ----------------------- \ 18 19 26 27 \ \ \ \ 16 17 24 25 \ ---------- ---------- \ 22 23 30 31 \ \ \ \ 20 21 28 29 \ --------- ------------ +----------- ----------- \ 50 51 58 59 \ 2 3 \ 10 11 \ \ +-----------+ \ \ 48 49 56 57 \ 0 1 \ 8 9 \ ---------- --------- +----------- ---------+ \ 54 55 62 63 \ 6 7 \ 14 15 \ \ \ \ \ \ 52 53 60 61 \ 4 5 \ 12 13 \ -------------- +----------+------------ \ 34 35 42 43 \ \ \ \ 32 33 40 41 \ ---------+ ----------- \ 38 39 46 47 \ \ \ \ 36 37 44 45 \ ----------------------- ``` The radial distance doubles on every second replication, so N=1 and N=2 are at 1 unit from the origin, then N=4 and N=8 at 2 units, then N=16 and N=32 at 4 units. N=64 is not shown but is then at 8 units away (X=8,Y=0). This is similar to the `"ImaginaryBase"`, but where that path repeats in 4 directions based on i=squareroot(-1), here it's 3 directions based on w=**cuberoot**(1) = -1/2+i\*[**sqrt**(3)](/sqrt/3.md)/2. ### Radix The `"radix"` parameter controls the "r" used to break N into X,Y. For example radix 4 gives 4x4 blocks, with r-1 replications of the preceding level at each stage. ``` 3 radix => 4 12 13 14 15 2 8 9 10 11 1 4 5 6 7 Y=0 -> 0 1 2 3 -1 28 29 30 31 -2 24 25 26 27 -3 20 21 22 23 -4 16 17 18 19 -5 44 45 46 47 ... 40 41 42 43 36 37 38 39 32 33 34 35 60 61 62 63 56 57 58 59 52 53 54 55 48 49 50 51 ^ -15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 ``` Notice the parts always replicate away from the origin, so the block N=16 to N=31 is near the origin at X=-4, then N=32,48,64 are further away. In this layout the replications still mesh together perfectly and all points on the triangular grid are visited. ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::CubicBase-\>new ()" - "$path = Math::PlanePath::CubicBase-\>new (radix =\> $r)" Create and return a new path object. - "($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. ### Level Methods - "($n\_lo, $n\_hi) = $path-\>level\_to\_n\_range($level)" Return `"(0, $radix**$level - 1)"`. ## SEE ALSO Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf ## HOME PAGE \ ## LICENSE Copyright 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 \.