Math::PlanePath::AlternatePaperMidpoint -- alternate paper folding midpoints
use Math::PlanePath::AlternatePaperMidpoint; my $path = Math::PlanePath::AlternatePaperMidpoint->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
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 = $path->n_start()
Return 0, the first N in the path.
($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
.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A016116 (etc)
A016116 X/2 at N=2^k, being X/2=2^floor(k/2)
Math::PlanePath, Math::PlanePath::AlternatePaper
Math::PlanePath::DragonMidpoint, Math::PlanePath::R5DragonMidpoint, Math::PlanePath::TerdragonMidpoint
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
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