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)/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)".
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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 <http://www.gnu.org/licenses/>.