Monthly Archives: June 2013

mathimage #19: cloudy terrain

A mathematical memory of those summer evening showers we used to have when I was a kid.

This image was an exercise on “evoking things” more than explicitly painting them.

The reason for it needs some explanation and context: thing is all these mathematical images I’m post every few days are actually developed in a very unusual way for me, in that instead of using powerful programming languages and hardware to create the most complex graphics possible, I am instead using the web browser as a platform. That means that, unlike in my other personal projects, I cannot add the amount of detail or physical fidelity I wish to these weekly images.

This, in turn, means that I have to cheat more than usual in order to get this images to render at all. For example, in this particular case, the trees are evoked by green (volumetric and transparent) noise, which performs poorly in the still image but works just fine when the camera is in movement, as our eyes and brain get fooled easily when things move. The cloud shadows on the trees are not actual shadows but dark blobs that roughly follow the shape of the clouds, which again, the average eye cannot distinguish from a real shadow. Etc etc, hack after hack, the image is formed in the web browser.

symmetrically wrong

it has just been brought to my attention that i still have my xmas tree in my living room.

i tend to forget it can look wacky to still have it. but the thing is that, besides being too lazy to store it again, i actually like having it around.

but in any case, since it’s past June 21st, it means currently xmas 2013 is already closer than xmas 2012 is distant. so, if only for symmetry in my wrongness, i’ll keep it around for the rest of the year as well.

plus, did i say already i like it a lot?

oh c’mon, really…?

Today the Bart train was in strike, something that came as a surprise to many of us who commute from SF to the east bay every day. The Transbay bus being the only alternative, there was a huge line that had been growing for 45 minutes in length, despair, tension and, for some reason, anger. When the bus finally arrived and unloaded an equally desperate crowd commuting the other way, our line suddenly broke in a orderless mess of animalized people fighting for a place in the bus quite as if their very survival depended on it. It wasn’t at the “titanic is sinking – find a boat!!” level really, but it got quite unpleasant, and needless. So… if this what we humans are ready to get into these days for a mere one hour delayed commute, then I truly am very disappointed.

mathimage #18: flower

Somebody asked me, “would you make a flower in your next mathematical image?”. I liked the request. And I like flowers. So, “Sure, I’ll give it a try”.

I’m taking these as quick exercises, so yet again, I wished I had had more time and to a real cool flower which doesn’t look so plastic-y. This particular flower bloomed in about 2.5 to 3 hours.

First a white sphere popped out of an equation. Then it got squeezed, stretched and bent (literally) to look like a petal. Three more petals grew from this one by means of a translation and a rotation. Then they got multiplied by two symmetries. Every petal got its own extra subtle rotation, such that no two petals would look the same. Then, a new sphere grew, a yellow one, there right in the middle of the petals. And from it, one cylinder. The cylinder reproduced itself by means of a periodic domain distortion and a stretch. Incs and colors started to develop over the surfaces along radial lines from the center of the new born flower, and some polynomial noisy functions printed in polar coordinates created interesting hue patterns. Then, there was light. First shooting straight in the flower from the left, pretty yellow, probably from the sun. Then, some extra soft light from the right to fill the darks, some blue light from above, perhaps from the sky. Some extra modulated lighting made the petals look thin and somehow translucent, and some subtle highlights completed the bath of light shaping. Dot products and square roots were the best brushes for this task. Lastly, some mathematical juggling was needed in order to bring shadows and shades. And after all of that, the flower was there.

(click in the title of the image to jump to the code)

10 seconds

ouch!, it seems my right shoe laces got loose.

i’m late, and despite i know it’ll take me a mere 10 seconds to tie them back, i somehow have the feeling those 10 seconds are going to be decisive in being on time for the train or missing it. in any case, there’s not much that i can do about it. so i tie the laces, jump in the bike and start pedaling. it’s an amazing evening again – the air is warm, the sun hasn’t set completely yet, and everything is and feels just right.

a few minutes later i arrive to the train station, and get off my bike. right then i hear the noise of an actual train stopping. dman, it’s going to be a tight one! i put the bike on my shoulder, and run upstairs to the platform two steps at a time, pretty fast, but not enough – when i reach the platform and turn left i see the doors of the train cars closing. then, the train resumes movement and leaves the station.

i missed this train. just as feared. just by 10 seconds.

luckily the air is warm and the sun hasn’t set yet in the station. so i smile, and relax. i then look down and i see the laces of my right shoe are untied, again.

horrifying

i’m sorry, but i can’t get used to it. i don’t want to get used to it. i’m speaking of the idea that having kids grow up thinking that war and killing is somehow a cool (manly?) thing to do (and i say manly cause back then women were taught to stay in the kitchen).

i don’t care how much Vietnam and how much cold war the US was undergoing, there’s no excuse to have kids being indoctrinated in the wonders of killing enemies or shooting to the air with a replica of a killing machine. i think this is, no matter the angle you look at it from, a horrifying abomination.

nasty singulars and breaking rules

Speaking of simplifying language…

As we know the use of an “-s” as an ending letter for a word is usually reserved for plurals. But not always indeed. For example, “penis”, “clitoris” and “anus” are all singular words ending in “-s”. In those infrequent cases, plurals are formed by appending an extra “-es”, just as with all the other regular words. Hence, we do speak of multiple “penises”, “clitorises” and “anuses”. Easy. However, I’m suspecting that there’s something fishy hidden in this linguistic orgy. Because…

… these three happen to be Latin words, and there’s this rule in English by which latin words ending in “-us”, such as “radius” or “torus”, form their plural by replacing “-us” with “-ii”, as in “radii” and “torii” (just as in Latin). Therefore, I’m thinking, shouldn’t two or more “anuses” in fact have been two or more “anii”…?

So I’m speculating somebody must had at some point in the past decided to break the stupid rule and simplify things. If that is the case, well done! Well done, and keep it up! What about fixing the remaining words and getting rid of the rule altogether? I guess that will take a while, for people seem to be far more interested on anuses than in “radiuses” and “toruses”.

the entropy of English

You know, English speakers believe (and naively claim!) that theirs is a very irregular language, without realizing that in fact they have one of the most regular languages ever!

And this is an ongoing trend. It seems to me English (at least american English) is pretty good at self polishing and cleaning, getting rid of spelling redundancies or grammatical irregularities. Quite as in a physical system, by the force of the use of language by its speakers, English is lowering its entropy. Which is a good and a natural thing to happen.

on half-marathons and ergodicity

once i run a half-marathon. pretty hard, yeah, i was 18.

however i never thought of it as a great feat, and never bragged about it. don’t get me wrong, though, it’s not that it was an easy challenge or anything, quite the contrary! i suffered like an animal for one hour and forty-five minutes, and that’s why i’ll never do it again.

yet, the reason i’ll never be especially proud of it is that “half marathon“, with that “half” in it, always sounded to me like an incomplete thing. a bit like it was the marathon of the paralympics, something you do when you are not good enough to do the real task, something that when finished you are given a chocolate medal. i think that even if you end first, still, you are the best only among those pussies who cannot run but a half marathon, or better said, half a marathon.

now i have an intriguing question. is running long distances in massive races an ergodic process? or in other words, being first in a half marathon, and being half way in the list of athletes completing a full marathon, are those anyhow equivalent?

i think no. to me, a half-marathon still sounds like, indeed, only half meaningful. in fact, it’s only 21 kilometers long and we do know that anything that aspires to be relevant in the universe has to be 42.

prefijo incorrecto

desfallecer debería significar nacer, o si acaso, resucitar. pero no, claramente desfallecer es más como un mini-fallecer.

word wanted: “toweek”

in english there are the words yesterday, today and tomorrow. talking about any other day needs to be done through a combination of words. in spanish the day before yesterday has also a special word for it, although oddly enough the day after tomorrow doesn’t (not officially, although there’s slang for it, so chances are some day we will have one). seems linguistic awareness of time never spans more than a handful of days.

in other (but similar) news, today i really missed a word to refer to “this week”. basically, in the same way we have the word “today”, i want to have the word “toweek”.

mathimage #17: guts

a very, very quickly made image (25 minutes) that looks pretty disgusting, but i enjoyed a lot improvising.

in this case i played again with line integrals over a (divergence free) noise field, and did toyed with the different parameters in order to create the feeling of volume, lighting and translucency. needles to say there’s nothing 3d here, it’s all massaging math expressions.

as usual, click the play button to see it in movement, and follow the link i the title to jump to the code/maths that generated the image.

this shouldn’t be happening

that awkward moment when a graphics programmer has no choice but admit that his/her only plan of attack left is to systematically try all permutations of the following operations:

* multiply the whole thing by -1 (especially if “the thing” is a cross product or a determinant)

* transpose the matrices involved (especially if you like writing maths the standard way but you work with a strange API, ahem…)

* change the order of the matrix multiplications (just because you never know)

* replace Y with Z (especially when swimming between camera and world spaces – because some people got traumatized at an early age by the Y-up thing at school or something)

this shouldn’t be happening, this shouldn’t be happening

was it really the greatest thing?

speaking of sliced bread, what’s up with all this idiom thing about “the greatest thing since sliced bread”?

was it really such a huge thing at the time, did it have that big of an impact in americans live in the 40s? like, if you ask elders about it, do they go like “yeah yeah yeah, we invented cars, and television, and stuff, no big deal. well, okey, those were great and all, whatever… but when the sliced bread arrived, OH BOY, that was so rad! son, there’s never truly been anything like the sliced bread

why this is not a solved problem yet

toasters. machines designed to toast sliced bread. easy job, you’d think.

however, in a world where humans are about to start exploring other planets like Mars, in a world where we can talk to people in the other side of the planet and instantly send them live pictures of our happy vacations, the task of “toasting a slice of bread” proves to be challenging for our scientists and engineers. especially the “without burning the toast” bit of it. what i’m saying is that for some reason, now matter how hard you try not to, when using a toaster you’ll always burn half the slices you bought.

so yes, apparently there’s something with the toasting technology that we keep failing at, still in 2013. that, or companies producing sliced bread have secret agreements with the toaster manufacturers!