NAME
Math::PlanePath::CellularRule57  cellular automaton 57 and 99 points
SYNOPSIS
use Math::PlanePath::CellularRule57;
my $path = Math::PlanePath::CellularRule57>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This is the pattern of Stephen Wolfram's "rule 57" cellular automaton
arranged as rows
51 52 53 54 55 56 10
38 39 40 41 42 43 44 45 46 47 48 49 50 9
33 34 35 36 37 8
23 24 25 26 27 28 29 30 31 32 7
19 20 21 22 6
12 13 14 15 16 17 18 5
9 10 11 4
5 6 7 8 3
3 4 2
2 1
1 < Y=0
9 8 7 6 5 4 3 2 1 X=0 1 2 3 4 5 6 7 8 9
The triangular numbers N=10,15,21,28,etc, k*(k+1)/2, make a 1/2 sloping diagonal upwards.
On rows with odd Y there's a solid block at either end then 1 of 3 cells to the left and 2 of 3 to the right of the centre. On even Y rows there's similar 1 of 3 and 2 of 3 middle parts, but without the solid ends. Those 1 of 3 and 2 of 3 are successively offset so as to make lines going up towards the centre as can be seen in the following plot.
*********** * * * * * ** ** ** ************
* * * * ** ** ** **
********** * * * * ** ** ** ***********
* * * * * ** ** **
********* * * * ** ** ** **********
* * * * ** ** **
******** * * * * ** ** *********
* * * ** ** **
******* * * * ** ** ********
* * * * ** **
****** * * ** ** *******
* * * ** **
***** * * * ** ******
* * ** **
**** * * ** *****
* * * **
*** * ** ****
* * **
** * * ***
* **
* * **
* *
*
*
Mirror
The mirror => 1
option gives the mirror image pattern which is "rule 99". The point numbering shifts but the total points on each row is the same.
51 52 53 54 55 56 10
38 39 40 41 42 43 44 45 46 47 48 49 50 9
33 34 35 36 37 8
23 24 25 26 27 28 29 30 31 32 7
19 20 21 22 6
12 13 14 15 16 17 18 5
9 10 11 4
5 6 7 8 3
3 4 2
2 1
1 < Y=0
9 8 7 6 5 4 3 2 1 X=0 1 2 3 4 5 6 7 8 9
N Start
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start, in the same pattern. For example to start at 0,
n_start => 0
22 23 24 25 26 27 28 29 30 31
18 19 20 21
11 12 13 14 15 16 17
8 9 10
4 5 6 7
2 3
1
0
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::CellularRule57>new ()
$path = Math::PlanePath::CellularRule57>new (mirror => $bool, n_start => $n)

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. $n = $path>xy_to_n ($x,$y)

Return the point number for coordinates
$x,$y
.$x
and$y
are each rounded to the nearest integer, which has the effect of treating each cell as a square of side 1. If$x,$y
is outside the pyramid or on a skipped cell the return isundef
.
SEE ALSO
Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows
http://mathworld.wolfram.com/ElementaryCellularAutomaton.html
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 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/>.