NAME
Math::PlanePath::QuintetCentres  selfsimilar "plus" shape centres
SYNOPSIS
use Math::PlanePath::QuintetCentres;
my $path = Math::PlanePath::QuintetCentres>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This a selfsimilar curve tracing out a "+" shape like the QuintetCurve
but taking the centre of each square visited by that curve.
92 12
/ 
124... 93 9190 88 11
 \ \ / \
122123 120 102 94 82 89 8687 10
\ /  /  / /  
121 119 103 101100 95 81 838485 9
\ \ \ \ \
114115116 118 104 32 9998 96 80 78 8
 / / /  / / \
112113 110 117 105 31 3334 97 36 79 7677 7
\ / \ \ \ \ / \ 
111 109108 106 30 42 35 3837 75 6
/ / /   /
107 29 43 414039 74 5
\ \ 
242526 28 44 46 7273 70 68 4
 / / \ \ / \ / \
2223 20 27 18 45 4847 71 56 69 6667 3
\ / \ / \  / \ 
21 6 19 1617 49 5455 5857 65 2
/ \   \  /
4 5 8 7 15 5051 53 59 64 1
\  / /  \
0 1 3 9 14 52 6061 63 < Y=0
/  \ /
2 1011 13 62 1
/
12 2
^
1 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13
The base figure is the initial the initial N=0 to N=4. It fills a "+" shape as
.....
. .
. 4 .
. \.
........\....
. . .\ .
. 01 . 3 .
. .  ./ .
......./....
. /.
. 2 .
. .
.....
Arms
The optional arms
parameter can give up to four copies of the curve, each advancing successively. For example arms=>4
is as follows. Notice the 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.
69 ... 7
/  \
121 113 73 6561 53 120 6
/ \ / \ \ \ / \ /
... 117 105109 77 29 57 4549 116 5
 / /   \
101 81 25 333741 96100104 112 4
 \ \  /
50 9793 85 21 13 8892 80 108 72 3
/  / / \ \ / \ / \
54 4642 89 10 17 5 9 84 24 76 6468 2
\  /   / \ 
58 38 14 6 2 1 1620 3228 60 1
/  \ \  /
62 3034 2218 3 0 4 12 36 56 < Y=0
 \ /  /  \
7066 78 26 86 11 7 19 8 91 4044 52 1
\ / \ / \ \ /  /  /
74 110 82 9490 15 23 87 9599 48 2
/   \ \ 
114 10610298 433935 27 83 103 3
\  / / 
118 5147 59 31 79 111107 119 ... 4
/ \ / \ \ \ / \ /
122 55 6367 75 115 123 5
\ /
... 71 6
^
7 6 5 4 3 2 1 X=0 1 2 3 4 5 6
The pattern an ever expanding "+" shape with first cell N=0 at the origin. The further parts are effectively as follows,
++
 
++ ++
  
++ ++ ++
 2  1  
++ +++ ++
  3  0 
++ ++ ++
  
++ +++
 
++
At higher replication levels the sides become wiggly and spiralling, but they're symmetric and mesh to fill the plane.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::QuintetCentres>new ()
$path = Math::PlanePath::QuintetCentres>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  1)
, or for multiple arms return(0, $arms * 5**$level  1)
.There are 5^level points in a level, or arms*5^level for multiple arms, numbered starting from 0.
FORMULAS
X,Y to N
The xy_to_n()
calculation is similar to the FlowsnakeCentres
. For a given X,Y a modulo 5 remainder is formed
m = (2*X + Y) mod 5
This distinguishes the five squares making up the base figure. For example in the base N=0 to N=4 part the m values are
++
 m=3  1
++++
 m=0  m=2  m=4  < Y=0
++++
 m=1  1
++
X=0 1 2
From this remainder X,Y can be shifted down to the 0 position. That position corresponds to a vector multiple of X=2,Y=1 and 90degree rotated forms of that vector. That vector can be divided out and X,Y shrunk with
Xshrunk = (Y + 2*X) / 5
Yshrunk = (2*Y  X) / 5
If X,Y are considered a complex integer X+iY the effect is a remainder modulo 2+i, subtract that to give a multiple of 2+i, then divide by 2+i. The vector X=2,Y=1 or 2+i is because that's the N=5 position after the base shape.
The remainders can then be mapped to base 5 digits of N going from high to low and making suitable rotations for the subpart orientation of the curve. The remainders alone give a traversal in the style of QuintetReplicate
. Applying suitable rotations produces the connected path of QuintetCentres
.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A106665 (etc)
A099456 level Y end, being Im((2+i)^k)
arms=2
A139011 level Y end, being Re((2+i)^k)
SEE ALSO
Math::PlanePath, Math::PlanePath::QuintetCurve, Math::PlanePath::QuintetReplicate, Math::PlanePath::FlowsnakeCentres
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/>.