I just read Ben's post on an article recently published at the Columbia Journalism Review, and also agree at the neatness of the quote as highlighted in the title of this post. It just brought me to mind of like minded graphic I saw during a presentation by David Orr, whilst he was here at UBC in January.
This graphic, I think, illustrates the same point in an even more potent manner. Here, you can see the interface of the "unknown" simply increases (and in a exponential manner at that), as knowledge is acquired. It simply nails home the notion that, in reality, the more that becomes known, the more becomes unknown.
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"Exponential"? Please! Accepting that diagram as worthwhile, and taking "interface of the 'unknown'" to mean "circumference of the circle", then this interface grows linearly with the radius of the known (C = 2Ïr) and as the square root of the area of the known: A = Ïr2, so C = 2Ï(A/Ï)1/2.
I don't get it. The second one. Not the original.
Blake is right. My bad. For some reason, I had in my head a sphere thing going on, whereby the interface is represented by the surface area. This increases in an exponential manner. The squigglies in his comment are suppose to be the symbol "pi."
Didn't something similar happen with the Hubble when it first went up?
Moloch damn it, the π symbols came out looking like the letter "pi" when I hit Preview. My apologies for puzzling everyone with squiggles.
I suppose it's a little too much to study such a figure so explicitly, 'cause who's to say the "known" lives in a 2D space, a 3D space or some exotic manifold in higher dimensions my brain can't directly comprehend? After hearing too many people yap about the Technological Singularity, though, backing up their Rapture-esque techno-mysticism with some very shaky curve-fitting, I get ansty around claims that something grows "exponentially".
alright. i'm back with you. what you guys said.
Just a suggestion. Exponential is ok if you are using the hyperbolic disc model. I vaguely remember that William Thurston used the model in some network (?) problems.
Actually, I would argue that this is a better representation:
http://www.gapingvoid.com/Moveable_Type/archives/000945.html
Hey Anandaswarup. Just did a quick wiki search of some of the things you mentioned (the stuff about William Thurston is especially interesting). Is there a book, you know Brian Greene-ish, Feynman-type book out there on this stuff. I'm kind of intrigued now, in a "don't really follow but would love to try" sort of way. This way, maybe we can get Thom Yorke to devote his CD on this kind of stuff.
And Sara: Hugh's stuff is great. He even let me use some it once at the SCQ.
Sorry; I have been visiting too many sites, forgot the name of this site and could not get back. Thurston's students have written some popular articles. I will try to find out and get back in a week or so.
The reason I thought of hyperbolic geometry: geodesics diverge exponentially and the cirmumference of a circle of hyperbolic radius r is 2 \pi sinhr. Of course, the applicability depends on the hypotheses invoved. Googlng for references seems easier than enquiring. Here are some references, I found them googling under 'networks and hyperbolic geometry'. I have not read any of these and you may find more suitable ones by searching. One relating to Thurston's work:
http://www.isl.uiuc.edu/Publications/alice8way.pdf
Some network related problems and hyperbolic geometry;
http://www.iis.ee.ic.ac.uk/~rick/research/pubs/bubbletree-chi2000.pdf
http://orworld.uni-paderborn.de/deliverables/wp5/wp5-hypertree.pdf
http://www.techfak.uni-bielefeld.de/ags/ni/publications/media/OntrupRit…
I have retired and am now trying to learn some biology and economics and so may not follow up on any of these papers. Good Luck.