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 is "undef".
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/math-planepath/index.html>
LICENSE¶
Copyright 2011, 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.
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/>.