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:
- take each small line and replace it with two lines half the length, one at right angles to the other
- or take two small copies of the fourth graph, one a mirror image and put them together, the reflected one to the right of and above the other.
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:
- Stretching out part of the interval, and squishing up some of the rest to make room for it.
This will be continuous, meaning that points which start fairly close together end up fairly close together. Its inverse is also continuous.
- Taking two or more pieces, and swapping them round.
This will be continuous everywhere except at the endpoints of the pieces taken out and replaced.
- Taking one piece (or the whole interval) and turning it around.
Again continuous except at the endpoints of the piece you moved.
- Any combination of the above
If you apply one bijection, then another to an interval, the combined transformation is another bijection. So you can combine as many as you like of these three types (in practice one of each type is enough) to get more complicated transformations.
- Something more interesting
Some transformations can't be made from a combination of these three types. In particular, if you want the transformation to be discontinuous at more than a finite set of points. These are the ones I'm interested in...
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:
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.
choosing between the internet and the radio is like choosing between having arms and having legs.
the euclidean algorithm
i hate euclid.
i love it
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
High School Teachers
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annoying animated gifs
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
Description du Skyblog :[Bouna & Zied !! Friends who left us too soon. let's show them respect. all clichy is with you, guys.]
BOUNA ET ZIED !! D pote ki nou on kité tro to.
rendon leur hommage.
TOUT CLICHY EST AVEC VOUS LES GARS
De 78, posté le Samedi 05 novembre 2005 à 16:14
je connaissai pa ces 2 mec mé en tt k ce kil leur es ariver c degeulasse mem si je les connaissai pa g une pensée pour eu pour les familles et tous le 93*[i didn't know these two guys but anywaywhat happened to them is disgusting. even if i didn't know them a thought for their families and all of 93. i say that sarkozy and all these son of a bitch police we're gonna have them all to the last one. if we have to we'll f*ck up all of france. 78 will support 93 to the end]
moi jdi ke sarkozy et tous ces fils de pute de flic on va se les fer tous jusko dernier si il fo on nikera tte la france
le 78 soutiendra le 93 juska la fin
en tt le responsabl du blog a bien fé de mettr leur tof bel hommage
Hommâge au pti Zihed et Bouna, deux adolescents de 17 et 15 ans, originaire de Clichy sous Bois, morts électrocutés après s'être introduits dans l'enceinte d'un transformateur EDF ,alors qu'ils tentaient d échapper à la police.[A tribute to little Zihed and Bouna, two adolescents aged 17 and 15, from Clichy sous Bois, died electrocuted after going into the compound of a EDF transformer, while trying to escape from police. Rest in peace inch'allah]
Qu'il repose en paix Inchallah...
16h15 - Paris Le maire de Paris Bertrand Delanoë (PS) attend de l'Etat "qu'il garantisse, à Paris comme ailleurs, l'application de l'ordre républicain"[The mayor of Paris expects the State to "guarantee that the rule of law be maintained, in Paris as elsewhere"]
De flavie, posté le Lundi 31 octobre 2005 à 02:17
allah y rahmou franchemen c triste de voir des jeune partir ossi to a cose de flic kom dab.['allah y rahmou' frankly it's sad to see young guys go out like this because of cops like (?). anyway be brave miloud i know it's hard to lose a mate but tell yourself it's 'mektoub'(?) and there's nothing one can do about it. 'boussa' and respect to you for putting them in your blog. leave your comments everyone]
en tou cas courage miloud je sai ke c dur de perdre un pote mai di toi ke c lmektoub et kon y peu rien
boussa et respect pour toi de lé avoir mi dan ton blog
alor lacher vos coms tous
make my day
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..
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the blt i had for lunch
- malted brown bread (42%)
- sweetcure smoke flavoured bacon (17%)
- tomato (17%)
- iceberg lettuce (11%)
- seasoned mayonnaise (10%)
- butter (not stated but presumably 3% +/- rounding errors)