NAME
Math::PlanePath::KochSquareflakes -- four-sided Koch snowflakes
SYNOPSIS
use Math::PlanePath::KochSquareflakes;
my $path = Math::PlanePath::KochSquareflakes->new (inward => 0);
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is the Koch curve shape arranged as four-sided concentric snowflakes.
61 10
/ \
63-62 60-59 9
| |
67 64 58 55 8
/ \ / \ / \
69-68 66-65 57-56 54-53 7
| |
70 52 6
/ \
71 51 5
\ /
72 50 4
| |
73 15 49 3
/ / \ \
75-74 17-16 14-13 48-47 2
| | | |
76 18 12 46 1
/ / 4---3 \ \
77 19 . | 11 45 Y=0
\ \ 1---2 / /
78 20 10 44 -1
| | |
79-80 5--6 8--9 42-43 -2
\ \ / /
81 7 41 -3
| |
82 40 -4
/ \
83 39 -5
\ /
84 38 -6
|
21-22 24-25 33-34 36-37 -7
\ / \ / \ /
23 26 32 35 -8
| |
27-28 30-31 -9
\ /
29 -10
^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 10The innermost square N=1 to N=4 is the initial figure. Its sides expand as the Koch curve pattern in subsequent rings. The initial figure is on X=+/-0.5,Y=+/-0.5 fractions. The points after that are integer X,Y.
Inward
The inward=>1 option can direct the sides inward. The shape and lengths etc are the same. The angles and sizes mean there's no overlaps.
69-68 66-65 57-56 54-53 7
| \ / \ / \ / |
70 67 64 58 55 52 6
\ | | /
71 63-62 60-59 51 5
/ \ / \
72 61 50 4
| |
73 49 3
\ /
74-75 17-16 14-13 47-48 2
| | \ / | |
76 18 15 12 46 1
\ \ 4--3 / /
77 19 |11 45 <- Y=0
/ / 1--2 \ \
78 20 7 10 44 -1
| / \ | |
80-79 5--6 8--9 43-42 -2
/ \
81 41 -3
| |
82 29 40 -4
\ / \ /
83 27-28 30-31 39 -5
/ | | \
84 23 26 32 35 38 -6
/ \ / \ / \ |
21-22 24-25 33-34 36-37 -7
^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7Level Ranges
Counting the innermost N=1 to N=4 square as level 0, a given level has
looplen = 4*4^levelmany points. The start of a level is N=1 plus the preceding loop lengths so
Nstart = 1 + 4*[ 1 + 4 + 4^2 + ... + 4^(level-1) ]
= 1 + 4*(4^level - 1)/3
= (4^(level+1) - 1)/3and the end of a level similarly the total loop lengths, or simply one less than the next Nstart,
Nend = 4 * [ 1 + ... + 4^level ]
= (4^(level+2) - 4) / 3
= Nstart(level+1) - 1For example,
level Nstart Nend (A002450,A080674)
0 1 4
1 5 20
2 21 84
3 85 340 The Xstart,Ystart position of the Nstart corner is a Lucas sequence,
Xstart(0) = -0.5
Xstart(1) = -2
Xstart(2) = 4*Xstart(1) - 2*Xstart(0) = -7
Xstart(3) = 4*Xstart(2) - 2*Xstart(1) = -24
...
Xstart(level+1) = 4*Xstart(level) - 2*Xstart(level-1)
0.5, 2, 7, 24, 82, 280, 956, 3264, ... (A003480)This recurrence occurs because the replications are 4 wide when horizontal but 3 wide when diagonal.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::KochSquareflakes->new ()$path = Math::PlanePath::KochSquareflakes->new (inward => $bool)-
Create and return a new path object.
Level Methods
($n_lo, $n_hi) = $path->level_to_n_range($level)-
Return per "Level Ranges" above,
( (4**$level - 1)/3, 4*(4**$level - 1)/3 )
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A003480 (etc)
A003480 -X and -Y coordinate first point of each ring
likewise A020727
A007052 X,Y coordinate of axis crossing,
and also maximum height of a side
A072261 N on Y negative axis (half way along first side)
A206374 N on South-East diagonal (end of first side)
Also:
A332204 X coordinate across one side
A332205 Y coordinate across one sideSEE ALSO
Math::PlanePath, Math::PlanePath::KochSnowflakes
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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/>.