NAME
Math::PlanePath::QuintetCentres -- self-similar "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 self-similar curve tracing out a "+" shape like the QuintetCurve but taking the centre of each square visited by that curve.
92 12
/ |
124-... 93 91--90 88 11
| \ \ / \
122-123 120 102 94 82 89 86--87 10
\ / | / | / / | |
121 119 103 101-100 95 81 83--84--85 9
\ \ \ \ \
114-115-116 118 104 32 99--98 96 80 78 8
| |/ / / | |/ |/ \
112-113 110 117 105 31 33--34 97 36 79 76--77 7
\ / \ \ \ \ / \ |
111 109-108 106 30 42 35 38--37 75 6
|/ / / | | /
107 29 43 41--40--39 74 5
\ \ |
24--25--26 28 44 46 72--73 70 68 4
| |/ |/ \ \ / \ / \
22--23 20 27 18 45 48--47 71 56 69 66--67 3
\ / \ / \ | / \ |
21 6 19 16--17 49 54--55 58--57 65 2
/ \ | | \ | /
4-- 5 8-- 7 15 50--51 53 59 64 1
\ | / |/ | \
0-- 1 3 9 14 52 60--61 63 <- Y=0
|/ | \ |/
2 10--11 13 62 -1
|/
12 -2
^
-1 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13The base figure is the initial the initial N=0 to N=4. It fills a "+" shape as
.....
. .
. 4 .
. \.
........\....
. . .\ .
. 0---1 . 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 65--61 53 120 6
/ \ / \ \ \ / \ /
... 117 105-109 77 29 57 45--49 116 5
| / / | | \
101 81 25 33--37--41 96-100-104 112 4
| \ \ | |/
50 97--93 85 21 13 88--92 80 108 72 3
/ | |/ |/ \ \ / \ / \
54 46--42 89 10 17 5-- 9 84 24 76 64--68 2
\ | / | | / \ |
58 38 14 6-- 2 1 16--20 32--28 60 1
/ | \ \ | /
62 30--34 22--18 3 0-- 4 12 36 56 <- Y=0
| \ / | |/ | \
70--66 78 26 86 11-- 7 19 8 91 40--44 52 -1
\ / \ / \ \ / | / | |/
74 110 82 94--90 15 23 87 95--99 48 -2
/ | | \ \ |
114 106-102--98 43--39--35 27 83 103 -3
\ | |/ / |
118 51--47 59 31 79 111-107 119 ... -4
/ \ / \ \ \ / \ /
122 55 63--67 75 115 123 -5
\ |/
... 71 -6
^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6The 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
$non the path. Points begin at 0 and if$n < 0then 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
$n_loand$n_hiare 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 over-estimate.)
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 5This 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 2From 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 90-degree 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) / 5If 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 sub-part 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/math-planepath/index.html
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/>.