It is not clear it counts as research, since I did it purely for fun, but I wrote a recursive code to build this kind of space-filling curve. It is slow, but it works for any image (array) that is 2n×2n. If it has any research application, it is in image compression, but I officially don't care about that subject (this paper notwithstanding; it isn't about image compression; it is about information!).
I am so proud of this; check out how scale-free it is: The top-left 4×4 block has the same two-dimensional pattern for the ones digit as the whole chart has for the sixteens digit, but with the pixels
being sixteen-cell blocks.
Sir,
ReplyDeleteFrom my viewpoint, what I have come to call quasic space. This sort of grid has wide applications that makes clear many concepts of physics and math.
The recursive aspects (as in fractals- and btw what space are we filling) gives us some more insights to the projective plane and other magic square like things in multiple dimensions.
There is a certain invariance of the general, that is more general pattern of things that also sheds light on such patterns in numbers, dimensions, and group theory.
Also it raises the questions of just what we mean by singularities and how we translate among the various bases from one position grid which involves physical values which are their own inverse and so on...
This arose from my models of 4D chess in 67. I also am only a candle maker.
I am a long time reader of your research blog:
L. Edgar Otto (the pesla.blogspot)
Is the author of the blog able to impress its readers by publishing the graph for n=8? ...or, dare I say, n=16?
ReplyDeleteGeert: Yes, they are easy to compute; in what form do you want them? Or you can just have my code. Send email!
ReplyDeleteSir,
ReplyDeleteI wish I had the abilities to program such things. It takes so long by hand. It is rough working with assembly code levels. n can equal very large dimensions, 2^n and a lot depends on the ordering of the entries whatever the base.
From my view this simple graph with the issues of grey code paths to fill space is a matrix or an n-dimensional n-symmetric brane(s). But I have not yet looked at this particular structure which seems to divide the parts into two paths (is this usual, the same thing would be reflected in three space).
The n=8 is already in this graph if read a certain way. What is the 16 x 16 anyway but a very terse representation of the 240 eight dimensional sphere in close packing or even the Monster group symmetry?
I know it is easy to do but do you have a code for the chain codes for these fit well into the theory also? Thank you, LoversOfWisdom@yahoo.com The problem is to show if these are unique in this world where your sixteen hypercubes in sixteen hypercubes have the depth as well as span of space structures.
Even the way we read these things in the left and right and up and down influences our perceptions or actual measures. Not only can we compress and retrieve image information I foresee we can vastly speed up some computations.
The PeSla