This 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.
That’s dangerous talk. Next you’ll be raving about a New Kind of Science.
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.
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.
What is all that “captcha” junk at the end of my post?
I see now - need to fill in the anti-spam text.
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
What is all that captcha junk at the end of my post?
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
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
“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/
Sierpinski Cookies
http://www.evilmadscientist.com/article.php/fractalcookies