Take a strip of paper; fold it; do it again and again. Sometimes up ("u") and sometimes
down "d". Label the sequence of folds "ududud" for instance. Unravel the paper.
It will have in this case 2^6 folds. Make each fold 90 degrees and plot the results.
Unexpectedly many had fun with this so I will make a couple of comments to all of you.
The curve uuuuuu is called the Dragon curve and there is quite a bit written about that. (I am not an expert.) I don't know anything about the other bit patterns other than the pictures including ududud which is a space-filling curve.
Questions that occur to me after seeing your pictures to which I don't have an answer
1) Is there a simple proof that the curve does not self-intersect and is space-filling?
2) If I follow a channel into the labyryinth of ududud what does it look like?
3) If I pretend the line is a string and start pulling on one end of the line does it come out nicely or does it get stuck on the rest of the curve. I am thinking of tigthly packed structures that are are in principle simple to unpack i.e. DNA.
4) How about 3-D generalizations. What if I take a rod and I fold it with possibilities up,down, or left and right?
A natural pattern to investigate would be uldruldruldr .... i.e. spiral.
Is the resultant curve also space filling? Is it self intersecting? If not for all patterns 4-symbol patterns. then for special values?
If it is space-filling and non-intersecting it seems it would be of relevance to the problem of packing dna.
If anyone knows the answers to these questions either by reading literature or your own investigations I would be interested in hearing about it.
Thanks for the replies!
ReplyDeleteUnexpectedly many had fun with this so
I will make a couple of comments to all
of you.
The curve uuuuuu is called the Dragon
curve and there is quite a bit written
about that. (I am not an expert.)
I don't know anything about the
other bit patterns other than the pictures
including ududud which is a
space-filling curve.
Questions that occur to me after seeing
your pictures to which I don't have an answer
1) Is there a simple proof that the curve
does not self-intersect and is
space-filling?
2) If I follow a channel into the labyryinth of ududud
what does it look like?
3) If I pretend the line is a string and start
pulling on one end of the line does it come out
nicely or does it get stuck on the rest of the curve.
I am thinking of tigthly packed structures that
are are in principle simple to unpack i.e. DNA.
4) How about 3-D generalizations. What if I take
a rod and I fold it with possibilities
up,down, or left and right?
A natural pattern to investigate would be
uldruldruldr .... i.e. spiral.
Is the resultant curve also space filling?
Is it self intersecting? If not for all patterns
4-symbol patterns. then for special values?
If it is space-filling and non-intersecting
it seems it would be of relevance to the problem
of packing dna.
If anyone knows the answers to these questions either
by reading literature or your own investigations
I would be interested in hearing about it.