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Math::PlanePath::CellularRule57 -- cellular automaton 57 and 99 points
Contents
Description
This is the pattern of Stephen Wolfram's "rule 57" cellular automaton
<http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
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
NStart
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".
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License
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.
perl v5.32.0 2021-01-23 Math::PlanePath::CellularRule57(3pm)
Name
Math::PlanePath::CellularRule57 -- cellular automaton 57 and 99 points
See Also
Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule54,
Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows
<http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
Synopsis
use Math::PlanePath::CellularRule57;
my $path = Math::PlanePath::CellularRule57->new;
my ($x, $y) = $path->n_to_xy (123);
PNG Icons
Emojis