Math::PlanePath::DekkingCurve -- 5x5 self-similar edge curve
use Math::PlanePath::DekkingCurve; my $path = Math::PlanePath::DekkingCurve->new; my ($x, $y) = $path->n_to_xy (123);
This is an integer version of a 5x5 self-similar curve from
F. M. Dekking, "Recurrent Sets", Advances in Mathematics, volume 44, 1982, pages 79-104, section 4.9 "Gosper-Type Curves"
This is a horizontal mirror image of the E-curve of McKenna (1978).
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 and which it 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 reversal and j=0,1,2,3 direction j*90 degrees anti-clockwise so E,N,W,S.
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.
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 $n < 0
then the return is an empty list.
($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.
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 XsegPred(X) = 1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,0,1,0,... =1 at 0,1,5,6,10,11,16,21,25,26,30,31,35,36,41,...
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.
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 YsegPred(X) = 0,0,0,1,0,0,0,0,1,0,0,0,0,1,1,0,0,... =1 at 3,8,13,14,18,19,23,28,33,38,39,43,44,48,...
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.
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.
XsegPred(X) or YsegPred = 1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,... =1 at 0,1,3,5,6,8,10,11,13,14,16,18,19,21,23,25,...
Math::PlanePath, Math::PlanePath::DekkingCentres, Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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