Sierpinski carpet
April 3rd, 2008 by WaltThis is a test post to see what’s involved in uploading images.
This is of course the Sierpinski carpet. What’s interesting to me is that many objects that, in a previous age dominated by a picture of the physical world as a continuum, seemed deeply pathological, have straightforward computer-language descriptions. For example, you can check whether or not a point in the plane is on the Sierpinksi plane by looking at the ternary expansion of its coordinates, which is a couple of lines of computer code. From the point of view of the computer, the Sierpinksi carpet is not much more complicated than a parabola. I suspect that the popularity of fractals marks a change in the popular imagination of the dominant metaphor for mathematics, from mathematics as mechanics to mathematics as computer program.
April 4th, 2008 at 8:05 am
That’s dangerous talk. Next you’ll be raving about a New Kind of Science.
April 4th, 2008 at 11:33 am
Of course, the algorithm for determing whether a point is in the Sierpinski carpet can only tell you definitively that a point is *not* in it. If the point IS in the carpet, the algorithm never halts. It’s not clear to me whether that makes it a good description or not.
April 4th, 2008 at 1:55 pm
The carpet has an infinite level of detail, which of course the computer can’t handle. So the program produces something that is good enough for an engineer, but not for a mathematician.
April 4th, 2008 at 1:56 pm
What is all that “captcha” junk at the end of my post?
April 4th, 2008 at 1:58 pm
I see now - need to fill in the anti-spam text.
April 5th, 2008 at 11:21 am
Bill: The “captcha junk” at the end of your post was a miniature Turing test, asking you to “prove you’re a human being.”
When you failed, you were (temporarily but officially) not human. How did it feel?
Was it a “blooming, buzzing confusion,” or just a blank?reCAPTCHA WP Error:incorrect-captcha-sol
April 5th, 2008 at 11:22 am
What is all that captcha junk at the end of my post?
April 5th, 2008 at 11:27 am
Mad Patter: That’s true, but it’s also true for a straight line on the plane, so I wouldn’t hold it against the Sierpinski carpet too much.reCAPTCHA WP Error:incorrect-captcha-sol
April 6th, 2008 at 5:52 pm
arXiv:0804.0517
Title: Singular integrals on Sierpinski gaskets
Authors: Vasilis Chousionis
Subjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)
We construct a class of singular integral operators associated with homogeneous Calder\’{o}n-Zygmund standard kernels on $d$-dimensional, $d
April 9th, 2008 at 4:42 am
“dominant metaphor for mathematics, from mathematics as mechanics to mathematics as computer program.”
. . . just when the dominant paradigm for computing itself is changing from computing as information processing to computing as social interaction. See:
http://www.agentlink.org/roadmap/
April 10th, 2008 at 9:16 am
Sierpinski Cookies
http://www.evilmadscientist.com/article.php/fractalcookies