and accordingly this works on Kandalov's site with the same
axiom:
A => AFFFB-F-FB+F+A
B => B+F+AF-F-AFFFB
oh, now i know what the problem is, D is a drawing command in fractint; i had too much D, the opposite of the usual problem. so this works:
hcurve { ; by oneearedrabbit.net. See
; <https://news.ycombinator.com/item?id=38029945>. Based on "an obscure
; paper by Rolf Niedermeier, Klaus Reinhardt, and Peter Sanders that
; introduced a rather peculiar curve with an unfortunate name: H-curve
; [1]. The paper mentions that H-curve preserves better locality
; properties compared to Hilbert curve. It fills the space with H-like
; shapes, hence the name. Also, like the Moore curve, it generates a
; loop."
; <https://www.sciencedirect.com/science/article/pii/S0166218X00003267>
Angle 4
Axiom A
A=BF-F-BFFFX-F-FX+F+BF-F-BFFFX-F-FX
B=BFFFX-F-FX+F+B
X=X+F+BF-F-BFFFX ; C and D are reserved in Fractint
}
but i like `simplified` above better
incidentally the paper seems to call it 'h-indexing'
incidentally, this slight variant l-system, which is probably what you meant
a where
a -> afffb-f-fb+f+a
b -> b+f+af-f-afffb
has the property that the axiom is a proper prefix of the axiom's expansion,
which turns out to be equivalent to the property
that every generation is a proper prefix of the following generation;
this means that in a sense each one is "approaching a limit"
of a single infinite string,
in the sense that it's a successively longer prefix of that string.
this infinite string,
called a 'morphic word',
is a fixed point of the mapping
the l-system does each generation
this is literally a numerical approximation if you treat the string
as a fractional number in some base, e.g., base 10 with a=1, b=2, f=3, +=4, -=5
with that interpretation, the first approximation 'a' is 0.1, the second approximation 'afffb-f-fb+f+a' is 0.13332535324341, the third approximation 'afffb-f-fb+f+afffb+f+af-f-afffb-f-fb+f+af-f-afffb+f+afffb-f-fb+f+a' is 0.133325353243413332434135351333253532434135351333243413332535324341, and so on.
the thue-morse sequence can be generated in the same way with the l-system
0 where 0 -> 01 and 1 -> 10
although the so-called fibonacci word is slightly simpler
a where a -> ab and b -> a
all of the above morphic words are aperiodic, though it's trivial to design a periodic morphic word
a program to output the infinite morphic word of movement commands for the h-curve of a single triangle is
queue = ['a'], []
d = dict(a='afffb-f-fb+f+a', b='b+f+af-f-afffb')
while True:
for item in queue[0]:
for c in item:
n = d.get(c, c)
yield n
queue[1].append(n)
queue = queue[1][::-1], queue[0] # amortized constant time
queue[1].clear()
this recipe does work in fractint to produce the triangular shape:
and based on that, this produces the whole cycle: and accordingly this works on Kandalov's site with the same axiom: oh, now i know what the problem is, D is a drawing command in fractint; i had too much D, the opposite of the usual problem. so this works: but i like `simplified` above betterincidentally the paper seems to call it 'h-indexing'