Provided by: libmath-planepath-perl_125-1_all bug

NAME

       Math::PlanePath::R5DragonCurve -- radix 5 dragon curve

SYNOPSIS

        use Math::PlanePath::R5DragonCurve;
        my $path = Math::PlanePath::R5DragonCurve->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This path is a "DDUU" turn pattern similar in nature to the terdragon but on a square grid
       and with 5 segments instead of 3.

                    31-----30     27-----26                                  5
                     |      |      |      |
                    32---29/33--28/24----25                                  4
                            |      |
                    35---34/38--39/23----22     11-----10      7------6      3
                     |      |             |      |      |      |      |
                    36---37/41--20/40--21/17--16/12---13/9----8/4-----5      2
                            |      |      |      |      |      |
           --50     47---42/46--19/43----18     15-----14      3------2      1
              |      |      |      |                                  |
           49/53--48/64  45/65--44/68    69                    0------1  <-Y=0

              ^      ^      ^      ^      ^      ^      ^      ^      ^
             -7     -6     -5     -4     -3     -2     -1     X=0     1

       The name "R5" is by Jorg Arndt.  The base figure is an "S" shape

           4----5
           |
           3----2
                |
           0----1

       which then repeats in self-similar style, so N=5 to N=10 is a copy rotated +90 degrees, as
       per the direction of the N=1 to N=2 segment.

           10    7----6
            |    |    |  <- repeat rotated +90
            9---8,4---5
                 |
                 3----2
                      |
                 0----1

       Like the terdragon there are no reversals or mirroring.  Each replication is the plain
       base curve.

       The shape of N=0,5,10,15,20,25 repeats the initial N=0 to N=5,

                  25                          4
                 /
                /           10__              3
               /           /    ----___
             20__         /            5      2
                 ----__  /            /
                       15            /        1
                                   /
                                  0       <-Y=0

              ^    ^    ^    ^    ^    ^
             -4   -3   -2   -1   X=0   1

       The curve never crosses itself.  The vertices touch at corners like N=4 and N=8 above, but
       no edges repeat.

   Spiralling
       The first step N=1 is to the right along the X axis and the path then slowly spirals anti-
       clockwise and progressively fatter.  The end of each replication is

           Nlevel = 5^level

       Each such point is at arctan(2/1)=63.43 degrees further around from the previous,

           Nlevel     X,Y     angle (degrees)
           ------    -----    -----
             1        1,0         0
             5        2,1        63.4
            25       -3,4      2*63.4 = 126.8
           125      -11,-2     3*63.4 = 190.3

   Arms
       The curve fills a quarter of the plane and four copies mesh together perfectly rotated by
       90, 180 and 270 degrees.  The "arms" parameter can choose 1 to 4 such curve arms
       successively advancing.

       "arms => 4" begins as follows.  N=0,4,8,12,16,etc is the first arm (the same shape as the
       plain curve above), then N=1,5,9,13,17 the second, N=2,6,10,14 the third, etc.

           arms => 4
                           16/32---20/63
                             |
           21/60    9/56----5/12----8/59
             |       |       |       |
           17/33--- 6/13--0/1/2/3---4/15---19/35
                     |       |       |       |
                   10/57----7/14---11/58   23/62
                             |
                   22/61---18/34

       With four arms every X,Y point is visited twice, except the origin 0,0 where all four
       begin.  Every edge between the points is traversed once.

   Tiling
       The little "S" shapes of the N=0to5 base shape tile the plane with 2x1 bricks and 1x1
       holes in the following pattern,

           +--+-----|  |--+--+-----|  |--+--+---
           |  |     |  |  |  |     |  |  |  |
           |  |-----+-----|  |-----+-----|  |---
           |  |  |  |     |  |  |  |     |  |  |
           +-----|  |-----+-----|  |-----+-----+
           |     |  |  |  |     |  |  |  |     |
           +-----+-----|  |-----+-----|  |-----+
           |  |  |     |  |  |  |     |  |  |  |
           ---|  |-----+-----|  |-----+-----|  |
              |  |  |  |     |  |  |  |     |  |
           ---+-----|  |-----o-----|  |-----+---
           |  |     |  |  |  |     |  |  |  |
           |  |-----+-----|  |-----+-----|  |---
           |  |  |  |     |  |  |  |     |  |  |
           +-----|  |-----+-----|  |-----+-----+
           |     |  |  |  |     |  |  |  |     |
           +-----+-----|  |-----+-----|  |-----+
           |  |  |     |  |  |  |     |  |  |  |
           ---|  |-----+-----|  |-----+-----|  |
              |  |  |  |     |  |  |  |     |  |
           ---+--+--|  |-----+--+--|  |-----+--+

       This is the curve with each segment N=2mod5 to N=3mod5 omitted.  A 2x1 block has 6 edges
       but the "S" traverses just 4 of them.  The way the blocks mesh meshes together mean the
       other 2 edges are traversed by another brick, possibly a brick on another arm of the
       curve.

       This tiling is also found for example at

           <http://tilingsearch.org/HTML/data182/AL04.html>

           Or with enlarged square part, <http://tilingsearch.org/HTML/data149/L3010.html>

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::R5DragonCurve->new ()"
       "$path = Math::PlanePath::R5DragonCurve->new (arms => 4)"
           Create and return a new path object.

           The optional "arms" parameter can make 1 to 4 copies of the curve, each arm
           successively advancing.

       "($x,$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 $n gives an X,Y position along a straight line between the integer
           positions.

       "$n = $path->xy_to_n ($x,$y)"
           Return the point number for coordinates "$x,$y".  If there's nothing at "$x,$y" then
           return "undef".

           The curve can visit an "$x,$y" twice.  The smallest of the these N values is returned.

       "@n_list = $path->xy_to_n_list ($x,$y)"
           Return a list of N point numbers for coordinates "$x,$y".

           The origin 0,0 has "arms_count()" many N since it's the starting point for each arm.
           Other points have up to two Ns for a given "$x,$y".  If arms=4 then every "$x,$y"
           except the origin has exactly two Ns.

       "$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, 5**$level)", or for multiple arms return "(0, $arms * 5**$level +
           ($arms-1))".

           There are 5^level segments in a curve level, so 5^level+1 points numbered from 0.  For
           multiple arms there are arms*(5^level+1) points, numbered from 0 so n_hi =
           arms*(5^level+1)-1.

FORMULAS

       Various formulas for boundary length, area, and more, can be found in the author's
       mathematical write-up

           <http://user42.tuxfamily.org/r5dragon/index.html>

   Turn
       At each point N the curve always turns 90 degrees either to the left or right, it never
       goes straight ahead.  As per the code in Jorg Arndt's fxtbook, if N is written in base 5
       then the lowest non-zero digit gives the turn

           lowest non-0 digit     turn
           ------------------     ----
                   1              left
                   2              left
                   3              right
                   4              right

       At a point N=digit*5^level for digit=1,2,3,4 the turn follows the shape at that digit, so
       two lefts then two rights,

           4*5^k----5^(k+1)
            |
            |
           2*5^k----2*5^k
                     |
                     |
            0------1*5^k

       The first and last unit segments in each level are the same direction, so at those
       endpoints it's the next level up which gives the turn.

   Next Turn
       The turn at N+1 can be calculated in a similar way but from the lowest non-4 digit.

           lowest non-4 digit     turn
           ------------------     ----
                   0              left
                   1              left
                   2              right
                   3              right

       This works simply because in N=...z444 becomes N+1=...(z+1)000 and so the turn at N+1 is
       given by digit z+1.

   Total Turn
       The direction at N, ie. the total cumulative turn, is given by the direction of each digit
       when N is written in base 5,

           digit       direction
             0             0
             1             1
             2             2
             3             1
             4             0

           direction = (sum direction for each digit) * 90 degrees

       For example N=13 in base 5 is "23" so digit=2 direction=2 plus digit=3 direction=1 gives
       direction=(2+1)*90 = 270 degrees, ie. south.

       Because there's no reversals etc in the replications there's no state to maintain when
       considering the digits, just a plain sum of direction for each digit.

OEIS

       The R5 dragon is in Sloane's Online Encyclopedia of Integer Sequences as,

           <http://oeis.org/A175337> (etc)

           A175337    next turn 0=left,1=right
                        (n=0 is the turn at N=1)

           A006495    level end X, Re(b^k)
           A006496    level end Y, Re(b^k)

           A079004    boundary length N=0 to 5^k, skip initial 7,10
                        being 4*3^k - 2

           A048473    boundary/2 (one side), N=0 to 5^k
                        being half whole, 2*3^n - 1
           A198859    boundary/2 (one side), N=0 to 25^k
                        being even levels, 2*9^n - 1
           A198963    boundary/2 (one side), N=0 to 5*25^k
                        being odd levels, 6*9^n - 1

           A052919,A100702  U part boundary length, N=0 to 5^k

           A007798    1/2 * area enclosed N=0 to 5^k
           A016209    1/4 * area enclosed N=0 to 5^k

           A005058    1/2 * new area N=5^k to N=5^(k+1)
                        being area increments, 5^n - 3^n
           A005059    1/4 * new area N=5^k to N=5^(k+1)
                        being area increments, (5^n - 3^n)/2

           A125831    N middle segment of level k, (5^k-1)/2
           A008776    count single-visited points N=0 to 5^k, being 2*3^k
           A146086    count visited points N=0 to 5^k

           A024024    C[k] boundary lengths, 3^k-k
           A104743    E[k] boundary lengths, 3^k+k

           A135518    1/4 * sum distinct abs(n-other(n)) in level N=0 to 5^k

           arms=1 and arms=3
             A059841    abs(dX), being simply 1,0 repeating
             A000035    abs(dY), being simply 0,1 repeating

           arms=4
             A165211    abs(dY), being 0,1,0,1,1,0,1,0 repeating

SEE ALSO

       Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::TerdragonCurve

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde

       This file is part of Math-PlanePath.

       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/>.