noble (3) Math::PlanePath::AlternatePaperMidpoint.3pm.gz
NAME
Math::PlanePath::AlternatePaperMidpoint -- alternate paper folding midpoints
SYNOPSIS
use Math::PlanePath::AlternatePaperMidpoint;
my $path = Math::PlanePath::AlternatePaperMidpoint->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is the midpoints of each alternate paper folding curve (Math::PlanePath::AlternatePaper).
8 | 64-65-...
| |
7 | 63
| |
6 | 20-21 62
| | | |
5 | 19 22 61-60-59
| | | |
4 | 16-17-18 23 56-57-58
| | | |
3 | 15 26-25-24 55 50-49-48-47
| | | | | |
2 | 4--5 14 27-28-29 54 51 36-37 46
| | | | | | | | | |
1 | 3 6 13-12-11 30 53-52 35 38 45-44-43
| | | | | | | |
Y=0 | 0--1--2 7--8--9-10 31-32-33-34 39-40-41-42
+----------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
The "AlternatePaper" curve begins as follows and the midpoints are numbered from 0,
|
9
|
--8--
| |
7 |
| |
--2-- --6--
| | |
1 3 5
| | |
*--0-- --4--
These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc. For this "AlternatePaperMidpoint" they're
turned 45 degrees and mirrored so the 0,1,2 upward diagonal becomes horizontal along the X axis, and the
2,3,4 downward diagonal becomes a vertical at X=2, extending to X=2,Y=2 at N=4.
The midpoints are distinct X,Y positions because the alternate paper curve traverses each edge only once.
The curve is self-similar in 2^level sections due to its unfolding. This can be seen in the midpoints as
for example N=0 to N=16 above is the same shape as N=16 to N=32, but the latter rotated +90 degrees and
numbered in reverse.
Arms
The midpoints fill an eighth of the plane and eight copies can mesh together perfectly when mirrored and
rotated by 90, 180 and 270 degrees. The "arms" parameter can choose 1 to 8 curve arms successively
advancing.
For example "arms => 8" begins as follows. N=0,8,16,24,etc is the first arm, the same as the plain curve
above. N=1,9,17,25 is the second, N=2,10,18,26 the third, etc.
90-82 81-89 7
arms => 8 | | | |
... 74 73 ... 6
| |
66 65 5
| |
43-35 42-50-58 57-49-41 4
| | | |
91-.. 51 27 34-26-18 17-25-33 3
| | | | |
83-75-67-59 19-11--3 10 9 32-40 2
| | | |
84-76-68-60 20-12--4 2 1 24 48 ..-88 1
| | | | | |
92-.. 52 28 5 6 0--8-16 56-64-72-80 <- Y=0
| | | |
44-36 13 14 7-15-23 63-71-79-87 -1
| | | | |
37-29-21 22-30-38 31 55 ..-95 -2
| | | |
45-53-61 62-54-46 39-47 -3
| |
69 70 -4
| |
... 77 78 ... -5
| | | |
93-85 86-94 -6
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
With eight arms like this every X,Y point is visited exactly once, because the 8-arm "AlternatePaper"
traverses every edge exactly once ("Arms" in Math::PlanePath::AlternatePaper).
The arm numbering doesn't correspond to the "AlternatePaper", due to the rotate and reflect of the first
arm. It ends up arms 0 and 1 of the "AlternatePaper" corresponding to arms 7 and 0 of the midpoints
here, those two being a pair going horizontally corresponding to a pair in the "AlternatePaper" going
diagonally into a quadrant.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::AlternatePaperMidpoint->new ()"
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 = $path->n_start()"
Return 0, the first N in the path.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 2**$level - 1)", or for multiple arms return "(0, $arms * (2**$level - 1)*$arms)". This
is the same as the "DragonMidpoint".
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A016116> (etc)
A334576 X coordinate
A334577 Y coordinate
A016116 X/2 at N=2^k, being X/2=2^floor(k/2)
SEE ALSO
Math::PlanePath, Math::PlanePath::AlternatePaper
Math::PlanePath::DragonMidpoint, Math::PlanePath::R5DragonMidpoint, Math::PlanePath::TerdragonMidpoint
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 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/>.