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.
3130 2726 5
   
3229/3328/2425 4
 
3534/3839/2322 1110 76 3
      
3637/4120/4021/1716/1213/98/45 2
     
50 4742/4619/4318 1514 32 1
    
49/5348/64 45/6544/68 69 01 <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
45

32

01
which then repeats in selfsimilar 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 76
   < repeat rotated +90
98,45

32

01
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 anticlockwise 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
16/3220/63

21/60 9/565/128/59
   
17/33 6/130/1/2/34/1519/35
   
10/577/1411/58 23/62

22/6118/34
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 returnundef
.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 + ($arms1))
.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 writeup
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 nonzero digit gives the turn
lowest non0 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^k5^(k+1)


2*5^k2*5^k


01*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 non4 digit.
lowest non4 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^k1)/2
A008776 count singlevisited 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^kk
A104743 E[k] boundary lengths, 3^k+k
A135518 1/4 * sum distinct abs(nother(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
HOUSE OF GRAPHS
House of Graphs entries for the R5 dragon curve as a graph include
19655 level=0 (1segment path)
568 level=1 (5segment path)
25149 level=2
25147 level=3
SEE ALSO
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::TerdragonCurve
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.