NAME
Math::PlanePath::QuintetCurve  selfsimilar "plus" shaped curve
SYNOPSIS
use Math::PlanePath::QuintetCurve;
my $path = Math::PlanePath::QuintetCurve>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path is Mandelbrot's "quartet" trace of spiralling selfsimilar "+" shape.
Benoit B. Mandelbrot, "The Fractal Geometry of Nature", W. H. Freeman and Co., 1983, ISBN 0716711869, section 7, "Harnessing the Peano Monster Curves", pages 7273.
125... 9392 11
  
123124 94 91908988 10
  
122121120 103102 95 8283 8687 9
      
115116 119 104 10110099 96 81 8485 8
      
113114 117118 105 3233 9897 807978 7
    
112111110109 106 31 34353637 7677 6
    
108107 30 4342 3938 75 5
    
2526 29 44 4140 7374 4
    
2324 2728 454647 7271706968 3
  
2221201918 4948 555657 6667 2
    
567 1617 5051 54 5958 65 1
      
01 4 98 15 5253 6061 64 < Y=0
     
23 1011 14 6263 1
 
1213 2
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
Mandelbrot calls this a "quartet", taken as 4 parts around a further middle part (like 4 players around a table). The module name "quintet" here is a mistake, though it does suggest the base5 nature of the curve.
The base figure is the initial N=0 to N=5.
5


01 4 base figure
 
 
23
It corresponds to a traversal of the following "+" shape,
.... 5
. ^
. <

0>1 .. 4 ....
v ^  .
. > > .
.  v .
.... 2>3 ....
. v .
. .
. .
. .. .
The "v" and ">" notches are the side the figure is directed at the higher replications. The 0, 2 and 3 subcurves are the right hand side of the line and are a plain repetition of the base figure. The 1 and 4 parts are to the left and are a reversal (rotate the base figure 180 degrees). The first such reversal is seen in the sample above as N=5 to N=10.
.....
. .
567 ...
. .  .
.  . reversed figure
... 98 ...
 .
 .
10 ...
Mandelbrot gives the expansion without designating start and end. The start is chosen here so the expansion has subcurve 0 forward (not reverse). This ensures the next expansion has the curve the same up to the preceding level, and extending from there.
In the base figure it can be seen the N=5 endpoint is rotated up around from the N=0 to N=1 direction. This makes successive higher levels slowly spiral around.
base b = 2 + i
N = 5^level
angle = level * arg(b) = level*atan(1/2)
= level * 26.56 degrees
In the sample shown above N=125 is level=3 and has spiralled around to angle 3*26.56=79.7 degrees. The next level goes to X negative in the second quadrant. A full circle around the plane is approximately level 14.
Arms
The optional arms => $a
parameter can give 1 to 4 copies of the curve, each advancing successively. For example arms=>4
is as follows. N=4*k points are the plain curve, and N=4*k+1, N=4*k+2 and N=4*k+3 are rotated copies of it.
6965 ...
  
..117113109 73 61575349 120
   
101105 77 2529 4145 100104 116
       
9793 81 21 3337 9296 108112
   
5046 8985 1713 9 8884807672
   
54 4238 10 6 1 5 202428 6468
      
58 3034 14 2 0 4 16 3632 60
      
6662 262218 7 3 812 4044 56
   
7074788286 111519 8791 4852
   
110106 9490 3935 23 83 9599
       
114 10298 4743 3127 79 107103
   
118 51555963 75 111115119..
  
... 6771
The curve is essentially an ever expanding "+" shape with one corner at the origin. Four such shapes pack as follows. O is the origin and each * is the end of the part on its right.
++
 
+* ++
  B 
++ ++ +*
 C   
++ +O+ ++
   A 
*+ ++ ++
 D  
++ +*+
 
++
At higher replication levels the sides are wiggly and spiralling and the centres of each rotate around, but their sides are symmetric and mesh together perfectly to fill the plane.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::QuintetCurve>new ()
$path = Math::PlanePath::QuintetCurve>new (arms => $a)

Create and return a new path object.
($x,$y) = $path>n_to_xy ($n)

Return the X,Y coordinates of point number
$n
on the path. Points begin at 0 and if$n < 0
then the return is an empty list.Fractional positions give an X,Y position along a straight line between the integer positions.
$n = $path>n_start()

Return 0, the first N in the path.
($n_lo, $n_hi) = $path>rect_to_n_range ($x1,$y1, $x2,$y2)

In the current code the returned range is exact, meaning
$n_lo
and$n_hi
are the smallest and biggest in the rectangle, but don't rely on that yet since finding the exact range is a touch on the slow side. (The advantage of which though is that it helps avoid very big ranges from a simple overestimate.)
Level Methods
($n_lo, $n_hi) = $path>level_to_n_range($level)

Return
(0, 5**$level)
, or for multiple arms return(0, $arms * 5**$level)
.There are 5^level + 1 points in a level, numbered starting from 0. On the second and subsequent arms the origin is omitted (so as not to repeat that point) and so just 5^level for them, giving 5^level+1 + (arms1)*5^level = arms*5^level + 1 many points starting from 0.
FORMULAS
Various properties are in my R5 Dragon mathematical writeup in section "Quartet Curve",
X,Y to N
The current approach uses the QuintetCentres
xy_to_n()
. Because the tiling in QuintetCurve
and QuintetCentres
is the same, the X,Y coordinates for a given N are no more than 1 away in the grid.
The way the two lowest shapes are arranged in fact means that for a QuintetCurve
N at X,Y then the same N on the QuintetCentres
is at one of three locations
X, Y same
X, Y+1 up
X1, Y+1 up and left
X1, Y left
This is so even when the "arms" multiple paths are in use (the same arms in both coordinates).
Is there an easy way to know which of the four offsets is right? The current approach is to give each to QuintetCentres
to make an N, put that N back through n_to_xy()
to see if it's the target $n
.
SEE ALSO
Math::PlanePath, Math::PlanePath::QuintetCentres, Math::PlanePath::QuintetReplicate, Math::PlanePath::Flowsnake
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.