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 arising from complex numbers expressed in a base w*cbrt(2) or other w*sqrt(r) base, where w=-1/2+i*sqrt(3)/2, the cube root of unity at +120 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 6The 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 pattern can be seen by dividing into blocks of 2^k points,
-----------------------
\ 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 \
-----------------------N=0 is the origin, then N=1 to the right. Those two are repeated at +120 degrees up as N=2 and N=3. Then that skew 2x2 repeated at 240 degrees around as N=4 to N=7. The bow-tie shaped block of 8 is repeated around again to 0 degrees as N=8 to N=16. Then the skewed 4x4 block at +120 as N=16 to N=31, and the resulting 32 point block repeated as N=32 to N=64 at +240 degrees. Each replication is 1/3 of the circle around at 0, 120 and 240 degrees. The relative layout within a replication is unchanged.
This is similar to the ImaginaryBase, but where it repeats in 4 directions based on i=squareroot(-1), here it's in 3 directions based on w=cuberoot(1).
Radix
The radix parameter controls the "r" used to break N into X,Y. For example radix => 4 gives 4x4 blocks, with r-1 copies of the preceding level at each stage.
3 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 6Notice the parts always replicate successively away from the origin, so the block N=16 to N=31 is near the origin at X=-4, then N=32 at X=-8, N=48 at X=-12, and N=64 at X=-16 (not shown).
In this layout the replications still mesh together perfectly and all points on that triangular grid are visited.
Axis Values
In the default radix=2, the N=0,1,8,9,etc on the positive X axis are those integers with zeros at two of every three bits starting from zeros in the second and third least significant bit.
X axis Ns = binary ..._00_00_00_ with _ either 0 or 1
in octal, digits 0,1 onlyFor a radix other than binary the pattern is the same. Each "_" is any digit of the given radix, and each 0 must be 0.
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
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.
SEE ALSO
Math::PlanePath, Math::PlanePath::ImaginaryBase
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2012 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 <http://www.gnu.org/licenses/>.