Math::...ngCurve(3pm)
💖 Math::PlanePath::DekkingCurve 💖
okay bestie, welcome to the Math::PlanePath::DekkingCurve page from the Perl user-contributed docs (that's your 3pm manual energy ✨). It's a 5x5 self-similar edge curve and it's lowkey iconic, no cap.
✨ NAME ✨
Math::PlanePath::DekkingCurve — a 5x5 self-similar edge curve 🌀
🌸 SYNOPSIS 🌸
use Math::PlanePath::DekkingCurve;
my $path = Math::PlanePath::DekkingCurve->new;
my ($x, $y) = $path->n_to_xy (123);literally three lines and you're on the curve, periodt 💅
🦋 DESCRIPTION 🦋
So this is the integer version of a 5x5 self-similar curve, and it comes from:
F. M. Dekking, "Recurrent Sets", Advances in Mathematics, volume 44, 1982, pages 79-104, section 4.9 "Gosper-Type Curves"
It's also a horizontal mirror image of the E-curve from:
Douglas M. McKenna, "SquaRecurves, E-Tours, Eddies, and Frenzies: Basic Families of Peano Curves on the Square Grid", in "The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History", Mathematical Association of America, 1994, pages 49-73, ISBN 0-88385-516-X
ngl the citation energy is giving academia but make it art 🎀
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 15The 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 + + 25The 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. cute little breadcrumbs fr fr 🍓
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. the way it just... rotates 💫
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.
💜 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. main character behaviour, honestly 🌈
🔥 FUNCTIONS 🔥
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes. (that's the shared lore, iykyk 📚)
$path = Math::PlanePath::DekkingCurve->new ()$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. no points for you, bestie 💀
🌟 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. the math is mathing, periodt 🧮💅
🎀 FORMULAS 🎀
okay bestie, this is the part where we get into the why. the actual math. the receipts. it's giving theorem-core and honestly? we love that for us. 💅
✨ X Axis Segments ✨
so like, in the sample points above (iykyk) there are some line segments just vibing on the X axis. a segment from X to X+1 is either traversed or not, according to this whole situation:
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,...ok so look at the samples: the segments at X=1, X=6 and X=11 are traversed because their low digit is 1. their preceding segments X=5 and X=10 have low digit==0 and their lowest non-zero digit is 1 or 2 (respectively), so those get traversed too. but at X=15? the lowest non-zero digit is 3, so not-traversed there. sorry babe. 💀
in general, in groups of 5 there's always an X==1 mod 5 that's traversed, but its preceding X==0 mod 5 is traversed or not depending on whether the lowest non-zero is 1,2 or 3,4.
so where does this pattern even come from?? glad you asked. it's 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 are rotated. in general the sides go like this:
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,Wstarting in S and taking digits high to low, a segment is traversed when the final state is S again. that's it, that's the whole trick. 🌟
any digit 1,2,3 goes to S,E,N respectively. if there's no 1,2,3 at all then you just start in S. at the lowest 1,2,3 there are only digits 0,4 below. if there are no such digits then it's only digit 1 which is S, or no digits at all for N=0, and it's traversed. if there's one or more 0s below then E goes to S, so a lowest non-zero 2 means traversed too. if there's any 4 then it goes to N or W, and in those states both 0,4 stay in N or W — so, not-traversed, bestie, sorry.
the transitions from the lowest 1,2,3 can be drawn in a state diagram, and honestly it's a little slay:
+--+
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 lowkey 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 it allows the usual DFA state machine manipulations to reverse it and go low to high. we love a glow-up. ✨
+---------- 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 right back to m1, so it finds the lowest non-zero. states start and m1 are literally the same except for how digit 2 behaves — which is why, in the rules above, the result for digit 2 differs depending on whether there are any low 0s. the way that one digit carries the whole personality of the machine… iconic. 👑
🌸 Y Axis Segments 🌸
the Y axis can be treated similarly, bestie:
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. periodt.
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,WN 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 anyway. so, same energy, honestly. 🙈
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. simple as that. 🦋
💖 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, and the curve doesn't touch. no collisions, no drama, we love a clean handoff. 🫶
you can see it 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 maybea 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. look:
+--------+-----+-------+
| ... | d | 4...4 | N == 4 mod 5 X never
+--------+-----+-------+ Y maybe
+--------+-----+-------+
| ... | d+1 | 0...0 | N+1 == 0 mod 5 X maybe
+--------+-----+-------+ Y neverper the Y rule, a 4 mod 5 segment is traversed when d=2,3. the following segment is then d+1=3,4 as the lowest non-zero, which in the X rule is not-traversed. conversely in the Y rule it's 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. alternating. the duality. the serve. 😭💖
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,...👀 SEE ALSO 👀
go read these next, bestie, they're all in the same family and they all eat:
Math::PlanePathMath::PlanePath::DekkingCentresMath::PlanePath::CincoCurveMath::PlanePath::PeanoCurve
🌈 HOME PAGE 🌈
the link, the legend, the main character's house:
<http://user42.tuxfamily.org/math-planepath/index.html>
📜 LICENSE 📜
Copyright 2011, 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/>.
💜 heads up: this license is the real deal — read it properly if you're actually shipping this. i'm just the hype woman.
and that's a wrap on the whole page, babe! 🎀 this bad boy came from perl v5.32.0 and is dated 2021-01-23, and you found it under the name Math::...ngCurve(3pm) in the User Contributed Perl Documentation. go forth and curve. slay. 💅✨
📦 where this page comes from
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