Provided by: libmath-planepath-perl_129-1_all 
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
HOUSE OF GRAPHS
House of Graphs entries for the R5 dragon curve as a graph include
<https://hog.grinvin.org/ViewGraphInfo.action?id=19655> etc
19655 level=0 (1-segment path)
568 level=1 (5-segment path)
25149 level=2
25147 level=3
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, 2018, 2019, 2020 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/>.
perl v5.32.0 2021-01-23 Math::PlanePath::R5DragonCurve(3pm)