HOME | DD
markdow β Three-cross DFT
by
Published: 2010-01-05 16:40:49 +0000 UTC; Views: 1695; Favourites: 12; Downloads: 122
Redirect to original
Description
Amplitudes of the discrete Fourier transform (DFT ) of a 3x3 greek cross recursively replaced with either itself or its complement.[Originally I had incorrectly described this in terms of checkerboards. Some of the comments reflect this error.]
The lowest spatial frequencies are represented at the center. The rounded square lines are formed by contiguous zero amplitude spatial frequencies, but I don't know the exact shape, or why they might be that shape. Nothing in the generating cross suggests these rounded squares. The shape might be a supercircle/squircle .
"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, itβs the exact opposite. The two are incompatible." Paul Dirac , criticizing J. Robert Oppenheimer 's interest in poetry
This image demonstrates a nested structure, not obvious to the eye, embodied by a recursive cross pattern. I don't know any precise reason or meaning of the structure, I just found a clear expression of its existence. Is it more art or science, visual poetry or diffraction pattern? I'm a mathematical Platonist -- I believe this pattern has a firm reality external to my perception or expression of one of its forms. I didn't make it up, I found it. Some people think that these patterns are only my construction, a metaphor for how I think about math or art. This difference of opinion makes it difficult to participate in this discussion of the "Delineations between Math and Art ".
Download for 2100 x 2100 px. version.
----------------------
This image is dedicated to the Public Domain . There are no restrictions on use. Claiming to be the originator or owner, explicitly or implicitly, is bad karma . A link (if appropriate), a note to markdow30@gmail.com, and credit are appreciated but not required.
Related content
Comments: 9
Have you seen "Seeing Forbidden Colors"? It is on page 72-77 of Scientific American Feb 2010. On page 76 it shows some transforms from retina pattern to brain response. I wonder if these are similar to the spacial DFT transform.
The article is *very* interesting.
--
Allen Brown [link]
If I had my life to live again, I'd make the same mistakes, only sooner. --- Tallulah Bankhead
π: 0 β©: 0
If you do a power of 2 rather than 3 do you get the rounded boxes? What about a power of 5?
--
Allen Brown [link]
"Anything worth doing is worth overdoing." --- me
π: 0 β©: 1
A power of 2 checkerboard is strictly periodic at a single spatial frequency, and so has a single non-zero amplitude (at the highest spatial frequency in both directions).
I don't know about power of 5, I've meant to check. I'll let you know.
π: 0 β©: 1
I think that is only true for an infinite checkerboard.
--
Allen Brown [link]
Yield to temptation -- it may not pass your way again. --- Robert A. Heinlein
π: 0 β©: 1
Yes, for an infinite checkerboard. The DFT gives the Fourier components of the periodic continuation of a finite pattern. In the case of an even checkerboard, the periodic continuation is an infinite checkerboard. An odd checkerboard's continuation has phase hiccups.
BUT, in trying to answer your question about powers of 5, I realized that I made a big mistake in the description of this pattern. It is NOT the DFT of a checkerboard, but that of a recursively replaced cross (on a 3x3 grid) -- replace each element of a 3x3 cross with either itself or its complement. It's still interesting (this and the 5x5 cross cases), but not as advertised here.
The system to produce these cross patterns is very close to that of checkerboards, and I'm just not a careful mathmatician. I'll send a clarification with some figures.
π: 0 β©: 1
OK, I couldn't remember if DFT did that or not. So a 4x4 in a field of black would not be representable with this technology. And that is what I was thinking of.
Your observation about the 3x3 being equivalent to a cross makes the 5x5 case even more interesting.
--
Allen Brown [link]
Every time Bush speaks, an English teacher dies. --- [link]
π: 0 β©: 0
I see a piled pattern with magnification step 3! 
π: 0 β©: 1
Yes, the checkerboard is periodic, but its construction is (or can be considered as) a replacement system with a magnification of 3 at each generation. It is surprising what patterns and DFT's have this kind of visually obvious piled structure.
π: 0 β©: 0