Kevin Ryde > Math-PlanePath > Math::PlanePath::DekkingCurve

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NAME ^

Math::PlanePath::DekkingCurve -- 5x5 self-similar edge curve

SYNOPSIS ^

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

DESCRIPTION ^

This is an integer version of a 5x5 self-similar curve per

F. M. Dekking, "Recurrent Sets", Advances in Mathematics, volume 44, 1982, pages 79-104, section 4.9 "Gosper-Type Curves"

The base pattern is N=0 to N=25. It repeats with rotations or reversals which make the ends join. For example N=75 to N=100 is the base pattern in reverse, ie. from N=25 down to N=0. Or N=50 to N=75 is reverse and also rotate by -90.

    10 |             123-124-125-...      86--85
       |               |                   |   |
     9 | 115-116-117 122-121  90--89--88--87  84
       |   |       |       |   |               |
     8 | 114-113 118-119-120  91--92--93  82--83
       |       |                       |   |
     7 |     112 107-106 103-102  95--94  81  78--77
       |       |   |   |   |   |   |       |   |   |
     6 |     111 108 105-104 101  96--97  80--79  76
       |       |   |           |       |           |
     5 |     110-109  14--15 100--99--98  39--40  75          66--65
       |               |   |               |   |   |           |   |
     4 |  10--11--12--13  16  35--36--37--38  41  74  71--70  67  64
       |   |               |   |               |   |   |   |   |   |
     3 |   9---8---7  18--17  34--33--32  43--42  73--72  69--68  63
       |           |   |               |   |                       |
     2 |       5---6  19  22--23  30--31  44  47--48  55--56--57  62--61
       |       |       |   |   |   |       |   |   |   |       |       |
     1 |       4---3  20--21  24  29--28  45--46  49  54--53  58--59--60
       |           |           |       |           |       |
    Y=0|   0---1---2          25--26--27          50--51--52
       +----------------------------------------------------------------
         X=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15

The curve segments correspond to edges of squares in a 5x5 arrangement.

     +- - -+- - -+- - 14----15  ---+
     |     |     |     |  v  |>    |
        ^     ^       <|     |
    10----11----12----13- - 16   --+
     |              v        |>    |
     |>       ^           ^  |
     9-----8-----7 -- 18----17   --+
        v  |     |     |>          |
     |        ^  |>    |        ^
     +- -  5-----6 -  19    22----23
           |          <|     |    <|
     |    <|  ^        |    <|     |
     +- -  4-----3    20----21 -- 24
                 |       v        <|
        ^     ^  |>    |     |     |
     0-----1-----2  -- + -- -+-   25

The little notch marks show which square each edge represents. This is the side the curve expands into at the next level. For example N=1 to N=2 has its notch on the left so the next level N=25 to N=50 expands on the left.

All the directions are made by rotating the base pattern. When the expansion is on the right the segments go in reverse. For example N=2 to N=3 expands on the right and is made by rotating the base pattern clockwise 90 degrees. This means that N=2 becomes the 25 end, and following the curve to the 0 start at N=3.

Dekking writes these directions as a sequence of 25 symbols s(i,j) where i=0 for left plain or i=1 for right reverse and j=0,1,2,3 direction j*90 degrees anti-clockwise so E,N,W,S.

Arms

The optional arms parameter can give up to four copies of the curve, each advancing successively. Each copy is in a successive quadrant.

    arms => 3                |
                      67-70-73       42-45                  5
                       |              |  |
             43-46-49 64-61 30-33-36-39 48                  4
              |     |     |  |           |
             40-37 52-55-58 27-24-21 54-51                  3
                 |                 |  |
                34 19-16  7--4 15-18 57 66-69               2
                 |  |  |  |  |  |     |  |  |
                31 22 13-10  1 12--9 60-63 72               1
                 |  |              |        |
        ...--74 28-25  5--2  0--3--6       75-...     <-- Y=0
              |        |
             71 62-59  8-11                                -1
              |  |  |     |
             68-65 56 17-14                                -2
                    |  |
                50-53 20-23-26                             -3
                 |           |
                47 38-35-32-29                             -4
                 |  |
                44-41                                      -5
                             ^
         ... -5 -4 -3 -2 -1 X=0 1  2  3  4  5 ...

The origin is N=0 and is on the first arm only. The second and subsequent arms begin 1,2,etc. The curves interleave perfectly on the axes where the arms meet. The result is that arms=4 fills the plane visiting each integer X,Y exactly once and not touching or crossing.

FUNCTIONS ^

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

$path = Math::PlanePath::DekkingCurve->new ()
$path = Math::PlanePath::DekkingCurve->new (arms => $a)

Create and return a new path object.

The optional arms parameter gives between 1 and 4 copies of the curve 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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, 25**$level), or for multiple arms return (0, $arms * 25**$level).

There are 25^level + 1 points in a level, numbered starting from 0. On the second and third arms the origin is omitted (so as not to repeat that point) and so just 25^level for them, giving 25^level+1 + (arms-1)*25^level = arms*25^level + 1 many points starting from 0.

FORMULAS ^

X Axis Segments

In the sample points above there are some line segments on the X axis. A segment X to X+1 is traversed or not according to

    X digits in base 5

    traversed        if X==0
    traversed        if low digit 1
    not-traversed    if low digit 2 or 3 or 4
    when low digit == 0
      traversed      if lowest non-zero 1 or 2
      not-traversed  if lowest non-zero 3 or 4

In the samples the segments at X=1, X=6 and X=11 segments traversed are low digit 1. Their preceding X=5 and X=10 segments are low digit==0 and the lowest non-zero 1 or 2 (respectively). At X=15 however the lowest non-zero is 3 and so not-traversed there.

In general in groups of 5 there is always X==1 mod 5 traversed but its preceding X==0 mod 5 is traversed or not according to lowest non-zero 1,2 or 3,4.

This pattern is found by considering how the base pattern expands. The plain base pattern has its south edge on the X axis. The first two sub-parts of that south edge are the base pattern unrotated, so the south edge again, but the other parts rotated. In general the sides are

           0 1 2 3 4
    S  ->  S,S,E,N,W
    E  ->  S,S,E,N,N
    N  ->  W,S,E,N,N
    W  ->  W,S,E,N,W

Starting in S and taking digits high to low a segment is traversed when the final state is S again.

Any digit 1,2,3 goes to S,E,N respectively. If no 1,2,3 at all then S start. At the lowest 1,2,3 there are only digits 0,4 below. If no such digits then only digit 1 which is S, or no digits at all for N=0, is traversed. If one or more 0s below then E goes to S so a lowest non-zero 2 means traversed too. If there is any 4 then it goes to N or W and in those states both 0,4 stay in N or W so not-traversed.

The transitions from the lowest 1,2,3 can be drawn in a state diagram,

               +--+
               v  |4                           base 5 digits of X
               North  <---+    <-------+       high to low
             /            |            |
            /0            |4           |
           /              |            |3
    +->   v               |       2    |  
    |   West             East  <--- start lowest 1,2,3
    +--   ^               |            |
    0,4    \              |            |1
            \4            |0           |or no 1,2,3 at all
             \            |            |
               South  <---+    <-------+
               ^  |0
               +--+

The full diagram, starting from the top digit, is less clear

               +--+
               v  |3,4
        +--->  North  <---+
       3|    /  | ^  \    |3,4
        |   /0  1 |  2\   |              base 5 digits of X
        |  /    | |    \  |              high to low
    +-> | v     | |     v |   <-+
    |   West 2---------> East   |        start in South,
    +-- | ^     | |     ^ |   --+        segment traversed
    0,4 |  \    | |    /  |    2         if end in South
        |   \4  | 3  2/   |
       1|    \  v |  /    |0,1
        +--->  South  <---+
               ^  |0,1
               +--+

but allows usual DFA state machine manipulations to reverse to go low to high.

          +---------- start ----------+
          |       1    0|   2,3,4     |         base 5 digits of X
          |             |             |         low to high
          v       1,2   v   3,4       v
    traversed <------- m1 -------> not-traversed
                      0| ^
                       +-+

In state m1 a 0 digit loops back to m1 so finds the lowest non-zero. States start and m1 are the same except for the behaviour of digit 2 and so in the rules above the result for digit 2 differs according to whether there are any low 0s.

Y Axis Segments

The Y axis can be treated similarly

    Y digits in base 5  (with a single 0 digit if Y==0)

    traversed        if lowest digit 3
    not-traversed    if lowest digit 0 or 1 or 2
    when lowest digit == 4
      traversed      if lowest non-4 is 2 or 3
      not-traversed  if lowest non-4 is 0 or 1

The Y axis goes around the base square clockwise, so the digits are reversed 0<->4 from the X axis for the state transitions. The initial state is W.

           0 1 2 3 4
    S  ->  W,N,E,S,S
    E  ->  N,N,E,S,S
    N  ->  N,N,E,S,W
    W  ->  W,N,E,S,W

N and W can be merged as equivalent. Their only difference is digit 0 going to N or W and both of those are final result not-traversed.

Final state S is reached if the lowest digit is 3, or if state S or E are reached by digit 2 or 3 and then only 4s below.

X,Y Axis Interleaving

For arms=2 the second copy of the curve is rotated +90 degrees, and similarly a third or fourth copy in arms=3 or 4. This means each axis is a Y axis of the quadrant before and an X axis of the quadrant after. When this happens the segments do not overlap nor does the curve touch.

This is seen from the digit rules above. The 1 mod 5 segment is always traversed by X and never by Y. The 2 mod 5 segment is never traversed by either. The 3 mod 5 segment is always traversed by Y and never by X.

The 0 mod 5 segment is sometimes traversed by X, and never by Y. The 4 mod 5 segment is sometimes traversed by Y, and never by Y.

        0       1       2       3       4
    *-------*-------*-------*-------*-------*
        X       X    neither    Y       Y
      maybe                            maybe

A 4 mod 5 segment has one or more trailing 4s and at +1 for the next segment they become 0s and increment the lowest non-4.

    +--------+-----+-------+                 
    |  ...   |  d  | 4...4 |   N   == 4 mod 5    X never
    +--------+-----+-------+                     Y maybe

    +--------+-----+-------+
    |  ...   | d+1 | 0...0 |   N+1 == 0 mod 5    X maybe
    +--------+-----+-------+                     Y never

Per the Y rule, a 4 mod 5 segment is traversed when d=2,3. The following segment is then d+1=3,4 as lowest non-zero which in the X rule is not-traversed. Conversely in the Y rule not-traversed when d=0,1 which becomes d+1=1,2 which in the X rule is traversed.

So exactly one of two consecutive 4 mod 5 and 0 mod 5 segments are traversed.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::DekkingCentres, Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve

HOME PAGE ^

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

LICENSE ^

Copyright 2011, 2012, 2013, 2014, 2015 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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