Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::SierpinskiArrowheadCentres -- self-similar triangular path traversal
SYNOPSIS
use Math::PlanePath::SierpinskiArrowheadCentres;
my $path = Math::PlanePath::SierpinskiArrowheadCentres->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path is variation on Sierpinski's curve from
Waclaw Sierpinski, "Sur une Courbe Dont Tout Point est un Point de Ramification", Comptes Rendus
Hebdomadaires des Séances de l'Académie des Sciences, volume 160, January-June 1915, pages 302-305.
<http://gallica.bnf.fr/ark:/12148/bpt6k31131/f302.image.langEN>
The path here takes the centres of each triangle represented by the arrowhead segments. The points
visited are the same as the "SierpinskiTriangle" path, but traversing in a connected sequence (rather
than across rows).
... ...
/ /
. 30 . . . . . 65 . ...
/ \ /
28----29 . . . . . . 66 68 9
\ \ /
27 . . . . . . . 67 8
\
26----25 19----18----17 15----14----13 7
/ \ \ / /
24 . 20 . 16 . 12 6
\ / /
23 21 . . 10----11 5
\ / \
22 . . . 9 4
/
4---- 5---- 6 8 3
\ \ /
3 . 7 2
\
2---- 1 1
/
0 <- Y=0
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The base figure is the N=0 to N=2 shape. It's repeated up in mirror image as N=3 to N=6 then rotated
across as N=6 to N=9. At the next level the same is done with N=0 to N=8 as the base, then N=9 to N=17
up mirrored, and N=18 to N=26 across, etc.
The X,Y coordinates are on a triangular lattice using every second integer X, per "Triangular Lattice" in
Math::PlanePath.
The base pattern is a triangle like
.-------.-------.
\ / \ /
\ 2 / \ 1 /
\ / \ /
.- - - -.
\ /
\ 0 /
\ /
.
Higher levels replicate this within the triangles 0,1,2 but the middle is not traversed. The result is
the familiar Sierpinski triangle by connected steps of either 2 across or 1 diagonal.
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * *
* * * *
* *
* * * * * * * *
* * * *
* * * *
* *
* * * *
* *
* *
*
See the "SierpinskiTriangle" path to traverse by rows instead.
Level Ranges
Counting the N=0,1,2 part as level 1, each replication level goes from
Nstart = 0
Nlevel = 3^level - 1 inclusive
For example level 2 from N=0 to N=3^2-1=9. Each level doubles in size,
0 <= Y <= 2^level - 1
- (2^level - 1) <= X <= 2^level - 1
The Nlevel position is alternately on the right or left,
Xlevel = / 2^level - 1 if level even
\ - 2^level + 1 if level odd
The Y axis ie. X=0, is crossed just after N=1,5,17,etc which is is 2/3 through the level, which is after
two replications of the previous level,
Ncross = 2/3 * 3^level - 1
= 2 * 3^(level-1) - 1
Align Parameter
An optional "align" parameter controls how the points are arranged relative to the Y axis. The default
shown above is "triangular". The choices are the same as for the "SierpinskiTriangle" path.
"right" means points to the right of the axis, packed next to each other and so using an eighth of the
plane.
align => "right"
| |
7 | 26-25 19-18-17 15-14-13
| / | |/ /
6 | 24 20 16 12
| | / /
5 | 23 21 10-11
| |/ |
4 | 22 9
| /
3 | 4--5--6 8
| | |/
2 | 3 7
| |
1 | 2--1
| /
Y=0 | 0
+--------------------------
X=0 1 2 3 4 5 6 7
"left" is similar but skewed to the left of the Y axis, ie. into negative X.
align => "left"
\ |
26-25 19-18-17 15-14-13 | 7
| \ \ | | |
24 20 16 12 | 6
\ | | |
23 21 10-11 | 5
\ | \ |
22 9 | 4
| |
4--5--6 8 | 3
\ \ | |
3 7 | 2
\ |
2--1 | 1
| |
0 | Y=0
--------------------------+
-7 -6 -5 -4 -3 -2 -1 X=0
"diagonal" puts rows on diagonals down from the Y axis to the X axis. This uses the whole of the first
quadrant, with gaps.
align => "diagonal"
| |
7 | 26
| \
6 | 24-25
| |
5 | 23 19
| | |\
4 | 22-21-20 18
| \
3 | 4 17
| |\ |
2 | 3 5 16-15
| | \ \
1 | 2 6 10 14
| \ | |\ \
Y=0 | 0--1 7--8--9 11-12-13
+--------------------------
X=0 1 2 3 4 5 6 7
These diagonals visit all points X,Y where X and Y written in binary have no 1-bits in the same places,
ie. where X bitand Y = 0. This is the same as the "SierpinskiTriangle" with align=diagonal.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::SierpinskiArrowheadCentres->new ()"
"$path = Math::PlanePath::SierpinskiArrowheadCentres->new (align => $str)"
Create and return a new arrowhead path object. "align" is a string, one of the following as
described above.
"triangular" the default
"right"
"left"
"diagonal"
"($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.
If $n is not an integer then the return is on a straight line between the integer points.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 3**$level - 1)".
FORMULAS
N to X,Y
The align="diagonal" style is the most convenient to calculate. Each ternary digit of N becomes a bit of
X and Y.
ternary digits of N, high to low
xbit = state_to_xbit[state+digit]
ybit = state_to_ybit[state+digit]
state = next_state[state+digit]
There's a total of 6 states which are the permutations of 0,1,2 in the three triangular parts. The
states are in pairs as forward and reverse, but that has no particular significance. Numbering the
states by "3"s allows the digit 0,1,2 to be added to make an index into tables for X,Y bit and next
state.
state=0 state=3
+---------+ +---------+
|^ 2 | | |\ 0 | |
| \ | | | \ | |
| \ | | | v | |
|----+----| |----+----|
| |^ | | || |
| 0 || 1 | | 0 || 1 |
|--->|| | |<---|v |
+---------+ +---------+
state=6 state=9
+---------+ +---------+
| | | | | |
| 1 | | | 1 | |
|--->| | |<---| |
|----+----| |----+----|
|^ |\ 2 | || |^ |
||0 | \ | || 2 | \0 |
|| | v | |v | \ |
+---------+ +---------+
state=12 state=15
+---------+ +---------+
|| 0 | | |^ | |
|| | | || 2 | |
|v | | || | |
|----+----| |----+----|
|\ 1 | | |^ 1 | |
| \ | 2 | | \ | 0 |
| v |--->| | \ |<---|
+---------+ +---------+
The initial state is 0 if an even number of ternary digits, or 6 if odd. In the samples above it can be
seen for example that N=0 to N=8 goes upwards as per state 0, whereas N=0 to N=2 goes across as per state
6.
Having calculated an X,Y in align="diagonal" style it can be mapped to the other alignments by
align coordinates from diagonal X,Y
----- -----------------------------
triangular X-Y, X+Y
right X, X+Y
left -Y, X+Y
N to dX,dY
For fractional N the direction of the curve towards the N+1 point can be found from the least significant
digit 0 or 1 (ie. a non-2 digit) and the state at that point.
This works because if the least significant ternary digit of N is a 2 then the direction of the curve is
determined by the next level up, and so on for all trailing 2s until reaching a non-2 digit.
If N is all 2s then the direction should be reckoned from an initial 0 digit above them, which means the
opposite 6 or 0 of the initial state.
Rectangle to N Range
An easy over-estimate of the range can be had from inverting the Nlevel formulas in "Level Ranges" above.
level = floor(log2(Ymax)) + 1
Nmax = 3^level - 1
For example Y=5, level=floor(log2(11))+1=3, so Nmax=3^3-1=26, which is the left end of the Y=7 row, ie.
rounded up to the end of the Y=4 to Y=7 replication.
SEE ALSO
Math::PlanePath, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::SierpinskiTriangle
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/>.