Math::PlanePath::CellularRule54 -- cellular automaton points
use Math::PlanePath::CellularRule54; my $path = Math::PlanePath::CellularRule54->new; my ($x, $y) = $path->n_to_xy (123);
This is the pattern of Stephen Wolfram's "rule 54" cellular automaton
arranged as rows,
29 30 31 . 32 33 34 . 35 36 37 . 38 39 40 7 25 . . . 26 . . . 27 . . . 28 6 16 17 18 . 19 20 21 . 22 23 24 5 13 . . . 14 . . . 15 4 7 8 9 . 10 11 12 3 5 . . . 6 2 2 3 4 1 1 <- Y=0 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The initial figure N=1,2,3,4 repeats in two-row groups with 1 cell gap between figures. Each two-row group has one extra figure, for a step of 4 more points than the previous two-row.
The rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers N=1,6,15,28, etc k*(2k-1). The hexagonal numbers of the "second kind" 1, 3, 10, 21, 36, etc j*(2j+1) are a steep sloping line upwards in the middle too. Those two taken together are the triangular numbers 1,3,6,10,15 etc, k*(k+1)/2.
The 18-gonal numbers 18,51,100,etc are the vertical line at X=-3 on every fourth row Y=5,9,13,etc.
The left end of each row is
Nleft = Y*(Y+2)/2 + 1 if Y even Y*(Y+1)/2 + 1 if Y odd
The right end is
Nright = (Y+1)*(Y+2)/2 if Y even (Y+1)*(Y+3)/2 if Y odd = Nleft(Y+1) - 1 ie. 1 before next Nleft
The row width Xmax-Xmin is 2*Y but with the gaps the number of visited points in a row is less than that, being either about 1/4 or 3/4 of the width on even or odd rows.
rowpoints = Y/2 + 1 if Y even 3*(Y+1)/2 if Y odd
For any Y of course the Nleft to Nright difference is the number of points in the row too
rowpoints = Nright - Nleft + 1
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 15 16 17 18 19 20 21 22 23 5 12 13 14 4 6 7 8 9 10 11 3 4 5 2 1 2 3 1 0 <- Y=0 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::CellularRule54->new ()
$path = Math::PlanePath::CellularRule54->new (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
.
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a couple of forms,
http://oeis.org/A118108 (etc)
A118108 whole-row used cells as bits of a bignum A118109 1/0 used and unused cells across rows
Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows
http://mathworld.wolfram.com/Rule54.html
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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