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
^
1110 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 bowtie 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.
Radix
The radix
parameter controls the "r" used to break N into X,Y. For example radix 4 gives 4x4 blocks, with r1 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
^
151413121110 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
SEE ALSO
Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.