Math::PlanePath::CubicBase -- replications in three directions
use Math::PlanePath::CubicBase; my $path = Math::PlanePath::CubicBase->new (radix => 4); my ($x, $y) = $path->n_to_xy (123);
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)/2.
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.
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 $n < 0
then the return is an empty list.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, $radix**$level - 1)
.
Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012, 2013, 2014 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.
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