Kevin Ryde > Math-PlanePath > Math::PlanePath::R5DragonCurve

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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. In the current code the smallest of the these N values is returned. Is that the best way?

@n_list = $path->xy_to_n_list ($x,$y)

Return a list of N point numbers for coordinates $x,$y. There can be none, one or two N's for a given $x,$y.

$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 and area 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

    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

    A008776    count single-visited points N=0 to 5^k
                 being 2*3^k

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

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