--- name: "Math::PlanePath::AlternatePaperMidpoint" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::AlternatePaperMidpoint" --- # Math::PlanePath::AlternatePaperMidpoint(3pm) ## 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 \ (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 \ ## 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 \.