OFFSET
0,3
COMMENTS
The space filling curve P corresponds to the midpoint curve of the alternate paperfolding curve and can be built as follows:
- we define the family {P_k, k > 0}:
- P_1 corresponds to the points (0, 0), (1, 0), (2, 0) and (2, 1), in that order:
+
|
|
+----+----+
O
- for any k > 0, P_{n+1} is built from four copies of P_n as follows:
+
|A
+ |
C| +----+ |
A B| ---> |C B| |B C
+-------+ + | +----+-+
O C| | C|
A B| A| A B|
+-------+ +-+-------+
O
- the space filling curve P is the limit of P_k as k tends to infinity.
We can also describe the space filling curve P by mean of an L-system (see Links section).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..4095
Joerg Arndt, L-system corresponding to P
Robert Ferréol (MathCurve), Courbe de Polya [in French]
Kevin Ryde, Iterations of the Alternate Paperfolding Curve
Rémy Sigrist, Colored line plot of the first 2^14 points of the space filling curve P (where the hue is function of the number of steps from the origin)
Rémy Sigrist, Colored scatterplot of the first 2^20 points of the space filling curve P (where the hue is function of the number of steps from the origin)
Rémy Sigrist, PARI program for A334576
EXAMPLE
The first points of the space filling curve P are as follows:
6| 20...21
| | |
5| 19 22
| | |
4| 16...17...18 23
| | |
3| 15 26...25...24
| | |
2| 4....5 14 27...28...29
| | | | |
1| 3 6 13...12...11 30
| | | | |
0| 0....1....2 7....8....9....10 31..
|
---+----------------------------------------
y/x| 0 1 2 3 4 5 6 7
- hence a(9) = a(12) = a(17) = a(26) = a(27) = 5.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, May 06 2020
STATUS
approved