more bijections

an example of a bijection of the unit interval which is discontinuous at infinitely many points.
the animation shows the graph of this bijection as a limit of bijective functions which are discontinuous at 2n points. the function in question is continuous everywhere except points of the form j/2n, for any n and any j≤2n, ie the numbers with binary representations that terminate.

(you have to imagine the limit.)
the first few steps allow you to easily imagine the squares of which the line segments are the diagonals. notice that each piece of graph from one step, although it gets chopped up in later steps, stays inside the same square. this allows us to see that the function is continuous except at x=j/2n (the places where the graph gets chopped).

like most self-similar fractals, there are two different ways of describing how to get from one step to the next. to get from the fourth step to the fifth we can either:

our fractal is thus a shape which, if we apply the second procedure to it, stays the same.

if we draw a graph of a bijective function in this way, the inverse of the function is simply its reflection in the bottom-left/top-right diagonal (the line f(x)=x). so if we have a graph which is symmetrical along this line, it is its own inverse. by changing the way i drew the above graph slightly, i graphed a function which has the same points of discontinuity, and is its own inverse

i created these pictures with a program written in C, using the Allegro library for its bitmap handling routines. if anyone knows a simpler way of scripting image manipulations like these, i wish they would tell me.



thinking about bijections

..of the unit interval.
the unit interval is the segment of the number line between 0 and 1, although any segment comes out to pretty much the same thing, and a bijection is a transformation of the interval so that every point is goes to a unique point and each point has exactly one mapped to it. basically the point (npi!) of all this is that if it's a bijection, it has an inverse; given this transformation you can make another which sends every point back to where it came from. You can't do that to a transformation if two points go to the same point - because you don't know which one to send it back to, or if some point doesn't have any points mapped to it - because you've got nowhere to send it back to.

Ways of making a bijection:

The pictures show graphs of each type of mapping. The unit interval goes left to right along the bottom border and a copy of it goes up the left edge from bottom to top. The colours show where specific points on the interval are mapped to on the copy.

Other ways...



that's whaat they said

I wonder if becoming critical is destroying our creativity.

It just occurred to me that listening to Goldfrapp is like being hit on by a mid-thirties conceptual artist in an all-white, austere, ultra-hip London gallery.

I've been asked by several people why I support the war in Iraq even though I totally disagree with Bush's reasons for waging it.

I wish there was a word for that feeling… I guess it’s like… a talent crush. The feeling you get when you feel like you could fall in love with someone just based on their work (even though you know it would never really work out).




superhuman intelligence is inevitable, as is the end of the human era:
The Singularity
by Vernor Vinge. interesting. houellebecq also talks about the imminent obsolescence of the human race (by entities with a higher developed capacity for happiness and sexual pleasure, natch) at the end of 'atomised', his most best book.

Voices from The Hellmouth
Slashdot column (?) a few days after Columbine. By Jon Katz. touching. The title is a tribute to Joss Whedon's understanding of the high school environment. (Uh? -> Buffy writer, the hellmouth located in Sunnydale High is where most of Buff's enemies creep out from)

Caring for Your Introvert
by Jonathan Rauch. essential reading. Linked from Neal Stephenson's site. He is the author of slightly overlong, overwritten cryptography tech-thriller 'the cryptonomicon' that i didn't get around to finishing back in april or something. i wonder how many novels i could have read in the time taken to browse this much internet..

Jason Salavon has made a lot of art created by averaging colors and brightness over large populations of pictures. 100 Special Moments is deliciously cynical. Playboy centerfolds averaged over an entire decade are anatomically as well as sociologically intriguing. He does some other cool stuff too. I once saw a print of British prime ministers churchill to blair superimposed but i think that was by someone else.




my stereo has been making an annoying low buzzing noise and last night i figured out why: the wireless internet connection is interfering with it.
choosing between the internet and the radio is like choosing between having arms and having legs.




if i hear one more idiot say "it's time to have a public debate" about something, i'm going to resign my position as a member of the public.



the euclidean algorithm

the subject of a lecture this week for the fourth or fifth time in my university career. maybe i chose some of the wrong options this year.

i hate euclid.



i love it

the bright, grey, clear light from a completely overcast sky that you get in london on a winter's afternoon and nowhere else.
it make the whole city look grey and drained and yet more real, more like london than ever.
as if the whole summer had just been a dream.



don't ask me what i was googling for

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annoying animated gifs

last night at about 3am, i discovered that the sum to infinity of the nth Fibonacci number divided by two the nth power is .. 4!

I was quite interested and disappointed at the same time. Interested because I didn't know it (although I should have). Disappointed because if it had been less than four, it would have helped me out quite a bit.

The result is not too groundbreaking; if you know what a generating function of a sequence of integers is, you can work it out, plug in 1/2 and get 4.

You can also find it out some simpler ways; the way I found it is close to an obvious calculation you might make if you know a little about infinite sums.

I was trying to construct a variation on the Cantor set where instead of removing the middle third of each interval, you remove the third quarter:

Looking at it that way, it's reasonably clear that the black set will end up having the same length as the original interval.
If you like, the area of the white set will be
3/4 * 3/4 * 3/4 * 3/4 * ... = 0, since each subsequent white set is three-quarters the length of the previous one.

But if you group the removed (black) intervals by their size, there are Fn intervals of length 1/2n+2, on an interval of length 1. (Fn the nth Fibonacci number, of course)

So their total length is 1/4 times the sum to infinity of 1/2n * Fn, which is the same as the length of the interval, ie 1.

I'd been working this out in the hope that the white set would have a positive length, which would have been useful to me. But that's another story.



bouna and zied

all translations mine and approximate. *93 is the postal code of department 'Seine-Saint-Denis'. 78 is another Paris department postal code.


make my day

i'm trying to install linux wireless drivers on my new computer, Sonja, and it's not going well.
for some reason the Readme file is telling me to invoke 'make' with a option SUBDIRS=blahblahblah and make either isn't finding blahblahblah or isn't finding the files that should be there or thinks that something else is somewhere it isn't or just hates me.
reboot Windows, reboot Linux..



quite clever

a 'visitor' had this as his browser identification line
Default 0.00
Get an interest free line of credit of upto 100,000,000.00 CBD just for signing up and pay back whenever you want! (compatible; MSIE 6.0; Windows NT 5.1; SV1; .NET CLR 1.1.4322; Get an interest free line of credit of upto 100,000,000.00 CBD just for signi

what is a CBD?



the blt i had for lunch

gives the percentages of each ingredient used:
malted brown bread contains:
wheat flour; water; malted wheat flakes; wheat bran; malted wheat and barley flours; yeast; salt; wheat protein; spirit vinegar; vegetable fat; emulsifiers: mono- and di-acetyltartaric esters of mono- and di-glycerides of fatty acids, mono- and di-glycerides of fatty acids, sodium stearoyl lactylate; flour treatment agent: ascorbic acid.

sweetcure 'smoke flavoured' bacon contains:
pork belly; sugar; salt; 'smoke flavouring'; malt extract; emulsifier: sodium triphosphate: preservative: sodium nitrite

seasoned mayonnaise contains:
vegetable oil; pasteurised egg; non brewed condiment; salt; sugar; pepper; stabilisers: guar gum, xanthan gum, carob bean gum; mustard flour; preservative: sorbic acid

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