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Module Version: 116   Source   Latest Release: Math-PlanePath-119

# 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 is the R5 dragon curve by Jorg Arndt,

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 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

This replication is similar to the TerdragonCurve in that there's 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 for example

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.

# FORMULAS

## 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.

## Boundary Length

The length of the boundary of the curve points N=0 to N=5^k inclusive is

B[k] = 4*3^k - 2
= 2, 10, 34, 106, 322, 970, 2914, ...

The boundary follows the curve edges around from the origin until returning there. So the single line segment N=0 to N=1 is boundary length 2, or the "S" shape of N=0 to N=5 is length 10.

4---5
boundary              |        boundary
B[0]=2               3---2    B[1]=10
|
0---1                 0---1

The first "S" shape is 5x the previous length but thereafter the way the curve touches itself makes the boundary shorter (growing just over 3x as can be seen from the power 3^k in B).

The boundary formula can be calculated from the way the curve meets when it replicates. Consider the level N=0 to N=5^k and take its boundary length in two parts as a short side R and an inner curving part U.

R          R[k] = side boundary
4---5        U[k] = inner curve boundary
R | U
3---2        initial R[0] = 1
U | R              U[0] = 3
0---1
R

The curve is shown here as plain lines but becomes fatter and wiggly at higher replications. Points 1 and 2 are on the right side boundary, and similarly 3 and 4 on the left side boundary, so in this breakdown the points where U and R parts meet are on the boundary. The total is

B[k+1] = 4*R[k] + 2*U[k]

The curve is symmetric on its left and right sides so R is half the total boundary of the preceding level,

R[k] = B[k] / 2

Combining these two equations gives

2*R[k+1] = 4*R[k] + 2*U[k]
R[k+1] = 2*R[k] +   U[k]

When the curve replicates to the next level N=5^k the boundary length becomes,

R
*---5
R | U       R       R           R[k+1] = 2*R[k] +   U[k]
*---*   *---2   *---*         U[k+1] =   R[k] + 2*U[k]
U | U |   | U |   | R
4---*---*---*---*---1         # eg. 0 to 1 on the right for R[k+1]
R |   | U |   | U | U           #     0 to 3 on the left for U[k+1]
*---*   3---*   *---*
R       R       U | R
0---*
R

The expansion for R[k+1] is the same as obtained above from symmetry of the total. Then U from 0 to 3 gives a second recurrence. Eliminate U by substituting the former into the latter,

U[k] = R[k+1] - 2*R[k]                       # from R[k+1] formula

R[k+2]-2*R[k+1] = 2*(R[k+1]-2*R[k]) + R[k]   # from U[k+1] formula
R[k+2] = 4*R[k+1] - 3*R[k]

Then from R[k]=B[k-1]/2 this recurrence for R becomes the same recurrence for the total B,

B[k+1] = 4*B[k] - 3*B[k-1]

The characteristic equation of this recurrence is

x^2 - 4*x + 3 = (x-3)*(x-1)     roots 3, 1

So the closed form is an a*3^k+b*1^k, being 4*3^k - 2. That formula can also be verified by induction from the initial B[0]=2, B[1]=10.

## U Boundary

The U length above can be calculated from the R[k+1]=2*R[k]+U[k] formula above,

U[k] = 2*3^k + 1
= 3, 7, 19, 55, 163, 487, 1459, 4375, 13123, 39367, ...

## Area

The area enclosed by the curve from N=0 to N=5^k inclusive is

A[k] = (5^k - 2*3^k + 1)/2
= 0, 0, 4, 36, 232, 1320, 7084, 36876, 188752, ...

A[k] = 9*A[k-1] - 23*A[k-2] + 15*A[k-3]

4
generating function  x^2 * -------------------
(1-5x)*(1-3x)*(1-x)

1/2      1      1/2
= ----- - ------ + ---
1-5*x   1-3*x    1-x

This can be calculated from the boundary. The R5 curve encloses unit squares in the same way as as the dragon curve per "Area from Boundary" in Math::PlanePath::DragonCurve, so 2*N = 4*A[N] + B[N], giving

2*5^k = 4*A[k] + 4*3^k - 2
A[k] = (5^k - 2*3^k + 1)/2

The 5^k term can be worked into the B recurrence in the usual way to give the A[k] recurrence 9,-23,15 above, and which can be verified by induction from the initial A[0]=0, A[1]=0, A[2]=4. The characteristic equation is

x^3 - 9*x^2 + 23*x - 15 = (x-1)*(x-3)*(x-5)

The roots 3 and 5 become the power terms in the explicit formula, and 1 is the constant.

Another form per Henry Bottomley in OEIS A007798 (which is area/2) is

A[k+2] = 8*A[k+1] - 15*A[k] + 4

## Area by Replication

The area can also be calculated explicitly by replications in a similar way to the boundary. Consider the level N=0 to N=5^k and take its area in two parts as a short side RA to the right and an inner curving part UA

RA          RA[k] = side area
4---5        RA[k] = inner curve area
RA | UA
3---2        initial RA[0]=0,RA[1]=0  UA[0]=0,UA[1]=0
UA | RA
0---1        A[k] = 4*RA[k] + 2*UA[k]
RA

As per above, point 1 is on the right boundary of the curve. Area RA is the region between the 0--1 line and the right boundary of the curve around from 0 to 1. This boundary in fact dips back to the left side of this 0--1 line. When that happens it's reckoned as a negative area. A similar negative area happens to UA.

___   <-- negative area when other side of the line
/   \
0----/-----1
\  /          line 0 to 1
--           curve right boundary

The total area is the six parts

A[k] = 4*RA[k] + 2*UA[k]

The curve is symmetric on its left and right sides so RA itself is half the total area of the preceding level,

RA[k] = A[k-1] / 2

Which gives

RA[k+1] = 2*RA[k] + UA[k]

When the curve replicates to the next level N=5^k the pattern of new U and R is the same as the boundary above, except the four newly enclosed squares are of interest for the area.

R
*---5                         square edge length sqrt(5)^(k-2)
R | U       R       R           square area = 5^(k-2)
*---*   *---2   *---*
U | U |   | U |   | R
4---*---*---*---*---1
R |   | U |   | U | U
*---*   3---*   *---*
R       R       U | R
0---*
R

The size of the squares grows by the sqrt(5) replication factor. The 25-point replication shown is edge length 1. Hence square=5^(k-2).

The line 0 to 1 passes through 3/4 of a square,

..... 1
.    /      line dividing each square
.   | .     into two parts 1/4 and 3/4
.   / .
*..|..*
.  /  .
. |   .
/    .
0 .....

The area for RA[k+1] is that to the right of the line 0--1. This is first +3/4 of a square with a further two RA on its outside, then -3/4 of a square with a UA pushing out (reducing that negative).

RA[k+1] = 3/4*square + 2*RA[k] - 3/4*square + UA[k]
= 2*RA[k] + UA[k]

This is the same recurrence as obtained above from the symmetry RA[k] = A[k-1]/2.

The area for UA[k+1] is that on left of the U shaped line 0-1-2-3,

UA[k+1] = -3/4*square + UA[k] + 3/4*square
+ 2*square + UA[k] + RA[k]
UA[k+1] = RA[k] + 2*UA[k] + 2*5^(k-2}           # square = 5^(k-2)

Notice for RA that the first 3/4 square has the left side of that square dipping in. For RA it's counted as a full +3/4 being the right side of the centre line. Then in UA on the left it's -3/4 which gives a net area of just what's between the left and right curve boundaries.

UA is eliminated by substituting the RA[k+1] recurrence into the UA[k+1]

UA[k] = RA[k+1] - 2*RA[k]      # from the RA[k+1] formula

RA[k+2]-2*RA[k+1] = 2*(RA[k+1]-2*RA[k]) + RA[k] + 2*5^(k-1)
RA[k+2] = 4*RA[k+1] - 3*RA[k] + 2*5^(k-1)

Then from RA[k] = A[k-1]/2 the total area is as follows,

A[k+2] = 4*A[k+1] - 3*A[k] + 4*5^k      # k>=2

This is the same as boundary calculation above but an extra 4*5^(k-2) which are the 4 squares fully enclosed when the curve replicates.

## Single Points

The count of single-visited points N=0 to N=5^k inclusive is obtained from the boundary in the same way as "Single Points from Boundary" in Math::PlanePath::DragonCurve,

S[k] = B[k]/2 + 1
= (4*3^k - 2)/2 + 1
= 2*3^k
= 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, ...

The double-visited points are the same as the area (also as per the dragon curve) and the total singles and doubles is 5^k+1

Singles[k] + 2*Doubles[k] = 2*3^k + 2*(5^k - 2*3^k + 1)/2
= 5^k + 1
being points N=0 to N=5^k inclusive

## Right Boundary Segment N

The curve segment numbers which are on the right boundary are

RN = N, in ascending order, which in base 5 with 1s deleted
does not have any of the following eight digit pairs
22, 23, 24,
30, 32, 33, 34,
40

= decimal 0,1,2,3,4,  5, 6, 7, 8, 9, 10,11, 16, 21,22,23,24, 25,...
= base5   0,1,2,3,4, 10,11,12,13,14, 20,21, 31, 41,42,43,44, 100,...

This characterization is obtained by considering the boundary in four parts

4-------5
|    E          E[k] = 4...
| D             D[k] = 3...
|    C          C[k] = 2...
3-------2       R[k] = 0...
|
| R
|
0-------1
R

The values in each part R[k] etc has k many base-5 digits. The two R parts are the same, since points 0, 1 and 2 are all on the right boundary. C is those points starting digit 2 which are on the boundary. Point 3 is not on the boundary so there are fewer segments in C than in R.

The curve expands as follows

*-----5                              R -> 0R, 1R, 2C, 3D, 4E
|   E                                C -> 0R, 1C
|D                                   D ->     1D
|   C          R           R         E ->     1E, 2C, 3D, 4E
*-----*     *-----2     *-----*
|E   C|     |E   C|     |
|     |     |     |     | R
/  D  \     \  D  /     |
4----/ /---\ \---\ \---/ /----1
|     /     \     \     /   E
|     |     |     |     | D
|     |     |     |     |   C
*-----*     3-----*     *-----*
| R
|
0-----*
R

2 to 3 is section C and it expands to an R and a C. The further parts of C are not on the boundary. So in section type C a digit 0 leads to an R and a digit 1 leads to a C, each of the preceding expansion level.

The boundary N values are then determined by starting from state R and making transitions to state R, C, D or E according to the digits from high to low per the expansions shown.

It can be seen that a digit 1 always leaves the state unchanged. So any digit 1s in N can be ignored. With that done the digit determines the state, since the transitions are always 0->R, 2->C, 3->D and 4->E. The disallowed state transitions therefore become disallowed digit pairs.

Since the D part only leads to another D part it can be seen that once a digit 3 is seen the only permitted digit below there is 1. There can be zero or more such 1s. The N=3 segment is zero 1s, then N=16 = base-5 "31" has a single 1 below, etc. The "3" giving D state can be reached from either R or E, but once there that state is D always.

For computer calculation it works equally well to consider digits high to low or low to high. A state variable can maintain the preceding digit and suitable table entries can leave the state unchanged to skip 1 digits. For high to low the initial state is equivalent to a 0 digit. That can be thought of as a 0 above the highest of N. For low to high a special initial state "no digit seen yet" is required. That state skips low 1 digits until a non-1 is reached.

## Right Boundary Segment N Lengths

The number of values in the C, D and E sections after k expansions is

C[k] = 3^k - k  = 1, 2,  7, 24, 77, 238, 723, 2180, 6553, ...
D[k] = 1
E[k] = 3^k + k  = 1, 4, 11, 30, 85, 248, 735, 2194, 6569, ...

These are obtained from the relations

R[k+1] = 2*R[k] + C[k] + D[k] +   E[k]
C[k+1] =   R[k] + C[k]
D[k+1] =                 D[k]
E[k+1] =          C[k] + D[k] + 2*E[k]

initial R[0]=C[0]=D[0]=E[0] = 1

Since D[k+1]=D[k] and initial D[0]=1 it is simply D[k]=1 always. C[k+1]=R[k]+C[k] makes it a cumulative R[k], so

C[k] = R[k-1] + R[k-2] + ... + R[0] + C[0]
k-1
= 1 + sum 2*3^k - 1
i=0
= 1 + 2*(3^k-1)/(3-1) - k
= 3^k - k

Then substituting R, C and D into the first equation gives E

E[k] = R[k+1] - 2*R[k] - C[k] - D[k]
= 2*3^(k+1) - 1 - 2*(2*3^k - 1) - (3^k - k) - 1
= 3^k + k

The total C,D,E is the U boundary length,

C[k] + D[k] + E[k] = 2*3^k + 1 = U[k]

# OEIS

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

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

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

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