- question: “what does it mean that a country is french speaking?”

- my thoughts: “errrr, are you retarded, or fucking kidding with me??”

So yes, you guessed, the picture below are 316 points (brush strokes) selected to have the right size, color and position in order to make the portrait. These “points” are little gaussian curves (remember the “Gaussian Bell” thing from school?), and I wrote a program that, given a desired target photograph, it would find the optimal set of such gaussian points that would resemble the photograph the most when drawn all together.

Indeed this is not one of my typical mathematical images. For I didn’t “paint” with maths as I usually do, but just wrote an algorithm to “express” the photograph mathematically.

The twist here is that, the fact that this can be done at all shouldn’t come with a be surprising to most people, because, believe it or not, ALL of the pictures you see in the internet, and all of the videos, sound and even your voice when you talk on your phone, are being mathematically expressed by the computer/device when it comes to storage, transmission, reception and display.

Yes you heard well, everything we see and hear in all these gadgets around us is a mathematical expression/representation of the actual photographs, music and voice. Usually, it’s not done by using tiny little gaussian dots as I did in this experiment, but cosine functions. But the idea is the same: find a mathematical description that approximates the actual data we want to work with.

If you want to see how, indeed, an image cab be just a small formula and a bunch of numbers, simply click here https://www.shadertoy.com/view/4df3D8 and scroll down the code window. Then select lines 200 to 235, and delete them. Then press the little triangle in below the code, and see how the image changes: you just deleted 36 of the gaussian dots. You got less numbers to be stored (that would mean a smaller picture file size), but the image got uglier and further from the true picture.

but i do have deeper fear to the ideas that motivate those bombs, wars, murders and violence.

i take a seat and lean the bike on my right leg as i usually do, still smiling. i notice an old couple looking at me from the seat in front. they must be in their 70s, i’d say. they are smiling at me so i look and (keep my) smile back to them. then i break the eye contact and get ready for my relaxed contemplative train ride of the morning. however, right before doing so, i look back once more to the elderly couple. they’re now staring and smiling at each other, conspiratorially. both’s eyes are shimmering with sparkles. and i think, “they are the most charming, cute, lovable, endearing and adorable couple ever”

unless it’s played by Yann Tiersen (and probably only Yann Tiersen)

then everything changes

so yeah, this was a quick but fun one: i painted Mike Wazowski with formulas and mathematics. well, where “painting” means sculpting, shaping, colorizing, texturing and lighting, as usual.

i wanted to have Mike himself pose and model for me, but apparently he was quite busy at the moment preparing the premier of his new movie (in theaters June 21st, folks). so instead of having him just for me for a few hours, i had to resort to the way less glamorous alternative of visiting Google Images, searching for his name and usinig the first picture as reference for my formulas.

the four hours of work were very productive, cause, as you can guess, Mike is in fact relatively easy to describe mathematically. his pretty much a pinched sphere (deform the domain in which you define your sphere, et voilà!). the mouth is what you get when you subtract an ellipsoid form the body’s sphere, and the teeth are what you get when you add a periodic (in the x axis) domain application of a small white sphere. its extremities are cones, which get attached to each other and to the body with some quadratic blend magic. eyelids are some radial cubic range distortion of the body. shading is a linearly interpolated noise pattern, and the eye is pretty much a fractal brownian motion indexed in polar coordinates.

as always, the code/formula is live online for free, ready to be played with, used, broken and tortured: https://www.shadertoy.com/view/MsXGWr

it’s interesting (or alarming) realizing how i get to be labelled so differently depending who i’m being analyzed by. needles to say that, despite i’m in love with the place i was born, despite i like dancing, despite i like developing and sharing my ideas, and despite i was educated in a society that naturally cares about one another, those people presuming those identities of mine are all wrong!

i wonder how much of that same thing i do regarding other people when i meet them for the first time. i believe not much, but i’m obviously not the best person to tell.

now, i’ve no idea what the store will be about once it’s complete. but i can already tell i’m gonna like it!

my first reaction is thinking “good for them”. then, i like looking and inspecting the much weirder faces people put while staring at the newcomer. it’s really entertaining to analyze their reactions: surprise, unease, curiosity, disapproval, envy.

so next time somebody weird looking joins in a public place, don’t stare at him/her, but look at the people around. it is priceless.

i ask to them, “how was it to not be born yet?”

no, but seriously. why would it be any different? well…

i don’t drink milk anymore, it’s being 6 years since last time, but making a fractal before going to bed feels almost as pleasant and sweet.

somtimes, you don’t need a reason to make a fractal

then there is a succession of thankyous and sorrys, don’t worrys and launghts between the five persons involved in this long chain of domino events.

I had just played the piano a few seconds before, as I do every morning right before leaving for the office. I always count with most neighbors being off at work by then, but seems some unfortunate soul like this mans must have been suffering those three daily minutes.

Still, I’m glad that we talked about other stuff than the music on our way out together. Once outside, when about to jump in my bike, I asked him his name. “Bill”, he said. Before I could tell him my name and introduce myself he told me: “You know, there’s only one more Inigo that I know of”.

At this point I realized that I must have been annoying the shit out of this poor man a lot lately with my piano thing, if he already knew my name by now. Then I expected to be told about “The Princess Bride”, but instead the man proceeded with “There was an Inigo living in Great Britain in the times of Shakespeare, who was a designer and architect working for the royalty, and founder of the classical British architecture”.

We talked a little bit more, and then I finally started pedaling uphill California St thinking about what could have possibly been the adventurous story of a Basque name ending up serving a British man.

as usual in these weekly experiments time was a limiting factor in development. sure enough with more time i’d have made a better formulanimation (and i’d have iterated in improving the shape of the creature and its colors/textures). but then again, the essence of these weekly images is to prevent me from getting stuck in a single idea for a long time, and go to something new the coming week. that way i can explore lots of stuff without getting lost in the endless refining of the final product. the more i try, the clearer it is in my mind what i’m doing next. it’s an exciting process, and i’m very glad i decided to undertake. in the meanwhile i do have other projects where i can put or my attention to detail anyway.

so yeah, this was a couple of three hours sessions. quite an investment, i must say, i don’t think i’ll spend this much time in a single image anytime soon. i must say however that during these six hours i spent most of the time fighting the technical difficulties set by the web browsing technology on which the maths run, not on making the actual images. sad, somehow wasted time. i guess that’s the prize you pay for accessibility and ease of distribution (as opposed to an installable demo application). as result of this lack of actual production time, the sky is flat, the shape of the insect way too simple, the textures are broken, the terrain has random black dots, and the movement of the creature is too simple and lineal.

BUT, despite all of these, the result is goo enough as to stop here, call it a new successful image for my exercise, and move on. i think i’ll do another creature/character this time, i have enjoyed doing something that is alive.

for the curious, the body of the insect is an ellipsoid plus some cosinuses, the legs some thick conical line segments plus a cosinus, and their smooth attachment to the body some exponential.log based function. if you want to see it moving in your own computer, or to explore more the mathematical details of the making, just follow this link: https://www.shadertoy.com/view/Mss3zM

i also rendered this video below over night, which higher quality than what the web browser can give you out of the box. hopefully you’ll find yourself saying “creeeeeepie!”. that means the math is doing its thing

This is the second such bizarre (but gratifying) interaction in less than a week, and I’m not used to it (but I like it). It is the spring that makes people happier and keen to talk to strangers?

w[i] = s*(1.0f-s) * slen * 0.04f * 0.66f * 2.0f * 1.5f * 1.5f * 0.5f;

Code-wise this is horrifying. But then again, this is not programming

unless i move it in a bit, as soon as i finish cleaning these dishes, i just have to finish with this few plates, then i’ll do it. yet, i know that that glass will somehow manage to fall and break eventually.

still, it would be great if i moved it right now cause then i’d had more room to let this dishes i’m cleaning dry. i’ll do it now, in a moment. in fact, you know what, it’s decided already – two more plates, and i’ll move it. cause, oh boy, i can totally see it falling and breaking, it’s gonna happen for sure.

in the other hand, now that i think about it, it’s being there for a while… so wouldn’t it be really bad luck if it fell and broke now that i have only one plate left? the pile seems to be pretty stable after all. yet… i know the glass is gonna fall and break. which is a pity cause i really like this particular glass.

well, never mind, i have just finished washing the last plate and it’s all fine, nothing broke. but i’ll move the glass anyway just in case. i just have to put this last plate here… and now take the glass and…

too late. the glass starts sliding as soon as leave the plate and, finally, falls. and breaks.

just as expected

Today I was doing some real-time mathematical image. In my oldschoolness, I thought I might attempt to speed up the rendering by replacing some trigonometric functions with something approximate but cheaper that would “look” the same in screen. This time I went for my well known and multi-purpose parabola f(x) = 4·x·(1-x). Here I’m scaling the domain from 0..Tau (2PI) to 0..1. It’s easier to work with normalized frequencies (Tau’s) than weird radians. Now, because the parabola (sort of) gives you half the sinus, you gotta duplicate it and put the copy upside down to get the full cycle. You can do that by multiplying the domain by two and renormalizing it to the unit segment by taking its fractional part. Then of course you want to make your function periodic so you end up having something like where I’m using the hat symbol to express the fractional part of a number. This results in something like this:

where the red curve is the real sinus and the blue one is the periodic parabola. Not bad. The problem is probably that there are two fractional part computations (one for 2x, one for x), which are expensive. I thought I might be able to do better that this.

My second attempt was therefore to use (again) a polynomial to generate a full cycle at once, then make that one periodic with a single fractional part computation. Since the curve has to pass by zero, one half and one, such polynomial could simply be g(x) = k·x·(x-0.5)·(x-1). In order to get k such that the function goes from -1 to 1, one has to make sure that g(x)=1 for the value of x that makes the derivative g’(x) = k·(3x² – 3x + 0.5) equal zero, resulting in k = 36/√3, making the complete function . The nice thing about this curve is that g’(0) = g’(1) = k/2, which means that the function is not only periodic, but it’s also smooth at the tiling points. It’s the blue curve in the following graph:

The function is a bit off the sinus, but it’s perfectly fine to be used in pattern generation for mathematical images/textures/animation. So, when I found it I copied it to the code and got my pretty image indeed. I was expecting to see my performance boost, call the whole thing a good move, and continue with my image design. However I got the very disappointing surprise of realizing that g(x) wasn’t faster, but actually much slower than using the original sin(2PIx) that I had been trying to avoid.

And that’s when I realized that I had to unlearn what I once knew to be true. Indeed, these days trigonometric functions are not 110 times slower than arithmetic instructions as they used to be back when they finally implemented them in computers, but they are in fact, basically as fast. And I knew this, yet… So, I wasted about 15 minutes of my evening. The good thing is that I have un-learned a good lesson. I hope this time is for real.

(more often than not both sides end up agreeing in little more than just acknowledging that they’ll never agree anyway, but even then it’s a bilateral resolution)

(other things that i can think of which cannot be unilaterally terminated either: sex and war)

i was late and there was a huge line for getting the bart ticket. but i didn’t mind, cause the temperature was warm and the weather beautiful, and there was a band of 4 bohemian musicians playing live, with a piano that they (somehow) brought and planted it in the middle of the station.

so there i was in the line, smiling distracted and dancing a bit, when i heard a voice telling me “you seem to be a good person”. i turned in the direction of the voice and i saw a man smiling and handing a bart ticket to me. i looked to the ticket and asked “oh… you don’t need it?” he said, “no, it’s all yours”. i thanked him, got the ticket, and run to the train, which i jumped into just on time for its departure. i sat, and smiled again thinking how lucky i had been, for i’d be a little bit less late than expected, and then i got distracted again in random thoughts. what i didn’t know yet is that the ticket i was holding in my pocket wasn’t a regular one way ticket, but a $60 one. that, i would discover only a few hours later.

of course it’s all very soft and tame for today’s standards, and i doubt anybody would consider those big undergarments erotic these days. but i think that the rest of the attire and bondage holds pretty well, 60 years later (not that i’m an expert in knobs, though), which is pretty amazing!

i’m waiting at the red traffic light, looking up to California street. i usually turn right here and avoid the hill (the HiLLmalayas, as i call them in SF). but as i’m looking up there, while i have my eyes posed in that point in the very top where it gets tangent with the very sky, an idea crosses my mind.

i know. climbing these three blocks in front of me with my single speed bike seems like a suicide act. but what fantastic views i can enjoy if i get up there. the sun, the bay, the bridge, the city…

the traffic light goes green. but i don’t turn right. instead i start pedaling straight fiercely. thankfully for me this is the back side of California Street (as known to the tourists, that is), so it’s not as painful as one might think (insert joke on back sides not being as painful as pop culture claims them to be). indeed, despite my heart beats fast, all of my focus is the pedaling and my eyes don’t move from the destination, i make it alive. and not only i make it alive to the top, but i make it with enough energy to smile on my arrival to the summit. and indeed, views are grand and inspiring.

i stop for a little bit to enjoy the moment. but not much, for i’m still just commuting to work after all. so i resume pedaling thinking that’s it, this has been the magical moment of the day. however i’m wrong.

as i begin to sliding down California St by its front side, i start realizing the special thing i’m about to experience. in my focused climbing effort up the hill i hadn’t payed attention to the fact that, for some reason, there’s no traffic nor cars around me. it must be the time of the day (no wonder i’m pretty late, once again). for whatever reason, thing is i’m at the cross with Mason St looking down from and i cannot spot but a few cars in all the way to the ocean ten blocks from here. in fact, the city feels dormant. all i can see is a cable car, and a beautiful long street stepping down to the sea surrounded by light and shadows cast by the buildings, the blue sky, and of course, the warm soft breeze that i love so much. my heart starts beating again, not because of any physical effort, but because of the expectation of what’s coming.

i give one impulse to the bike, and let gravity do the rest. slowly i start sliding. the air is warm, and i can see the bridge in front of me and the city below me. i then speed up, and at some point, i make a jump over the rails of the Powell-Hyde cable car line at its cross with California St. in my next block i draw S-curves with my bike as if i were skiing in the city. it feels like flying. then i have to slow down a bit, but that’s perfect cause that allows me to enjoy the views for a longer time. pass Stockton i get into China Town, and from there, right when the Bay Bridge finally disappears below the horizon, i arrive to the financial district and its life and buildings and noises and people. it’s all so beautiful right now that i want to cry. i slow down and do the last four blocks at the slowest speed i can, hand in pockets, with the biggest grin my face can accommodate.

i arrive to my destination. get of my bike slowly, and look up to the blue sky before bringing the bike up to my shoulders. this was a fucking amazing ride. and as with most good things, it was totally free.

so indeed this wasn’t the fastest, shortest or easiest way to work. but for sure it was the bestest.

not that anything has happened to me (or… has it?), but i thought it might be a good advice to give anyway

Now this was the really shocking part to me. People in general (both Americans and non-Americans, I think) seem to have no sense and measure of where science is, at all. They’re pretty clueless, as their predictions proves, which to me feels a bit like when in the renaissance people would dream of a future of flying machines in form of bikes with wings and feathers. And, while dreaming is a necessity for progress, these things are in the very least just too naive, if not alarming (in a time when education was supposed to provide one some sort of understanding of science, which is obviously not the case yet, although I don’t want to discuss that now).

Thing is, as of today we can barely teach a computer to tell the difference between a red square and a blue circle yet (unless we’re under very clean and controlled lighting conditions). We can’t even make a smartphone understand the words we are saying really, not to mention the actual “meaning” of our words. Reality is that, by 2030, we won’t be even remotely close to have robots cleaning our houses. If anything, we’ll have faster Facebook. And that’s pretty much it.

So, yeah. People’s ignorance is discouraging. But at the same time their naive hopes are somehow adorable.

still the execution was improvised of course – not that i have done a mathematical piano before (or anybody, for the matter). also, due to technical issues in the current web browsers i could only go that far in the amount of detail. same goes for lighting – browsers are not powerful enough to do global illumination in realtime. lastly, composition wise, some very intense and saturated red roses in the left part of the frame (over the piano) would have made an super cool image. BUT, part of the game in these weekly challenges is to only sketch things (starting from scratch from a white canvas), then stop wait to the following weekend for a new challenge. still, i think i like the result.

as usual, i put the code online accessible for everybody to play with, and do changes to it live and see the results right in the browser: https://www.shadertoy.com/view/ldl3zN

the body of the piano is made of 4 boxes with round corners. the keys are one white box made periodic over the x axis, and a black box made periodic over the same axis except for multiples of 2 and 6. the feet are one cylinder (made the formula symmetric over x to get 2 of them for the prize of one) with a vertical sinus distortion. the music sheet is a thin box pushed out with a symmetric parabola. the bench, a box with a sinus deformation on top. walls and floor, planes. the window doesn’t actually exist – i just indicate its presence by a soft-rectangular shaped formula with a cross in the middle which gets projected over the scene. the piano texture is some noise with horizontal patterns for subtle coloring. the broken paint, a grey coloring based on the local curvature of the piano formula (which sort of detects convexity, always a good indicator of exposure and hence age). the music is written with a vertical sinus to make the lines multiplied with a second sinus of one tenth of the frequency of the former, which creates the big white gaps that separate the paragraphs. the notes themselves don’t exist – instead some very high contrast noise was used, modulated by the underlying lines themselves, in order to suggest there actually are notes there (one of the tricks in mathematical imagery is to simply “suggesting” things and let the brain of the viewer interpret the thing and do the rest of the job for you). the wood tiles in the ground are sinuses in x with a varying phase which is a function of z. lighting is done with my usual tricks for softshadows and fake occlusion (documented in my website), and so on.

i’ve always been fascinated by the beauty of compacting theories.

but not only theories. even in the simplest interactions with maths you encounter constructions where the somehow magical powers of compaction manifests itself. when it does, it usually means you could have done something in a simpler way, but the fact that the thing arrives by itself anyway is fascinating, and almost mysterious.

like when you are describing something messy with your favorite notation, then you expand it into its elements, and start applying some standard techniques for simplifying things or rearrange terms, or simply move things around in the hope that some inspiration will work out some magic for you. then, most likely, you start slowly seeing things cancel out each other, patterns and symmetries appear, and eventually the whole thing reduces to a single, powerful expression that describes the original object or situation in a clean, concise, elegant way. and when things are seeing under this new self-uncovered optics, the result which was supposed to be the end of your mathematical journey, leads to a higher ideas, understanding and hence, more questions.

the fact that the results can often be described concisely means that the abstractions are higher of course. there’s little to no abstraction in “2+2=4″, it’s the simplest form of conceptualization possible. symbols are quantities, operations or statements – there’s little to no context needed to give it a meaning. however something like “E=mc²” or contains as many symbols, yet the ideas expressed are far more profound.

indeed, you need a bigger context to understand what those formulas mean, because the symbols compresses more meaning in them than a 100 page book can probably describe. yet, for the specialist, they are concise, simple, and describe beautiful relationships between the things around that matter to us. and in that sense, they become beautiful. and the compaction of a multiplicity of ideas and a variety of models in a simple expression, when it can be done, becomes the ultimate act of crafting beauty in knowledge.

this of course happens as a cascade of simplifications. when a new symbol or concept is invented for a set of ideas that got compacted into a single one, there’s a second level of ideas that fly around that the mathematician will immediately try to understand, find a structure to and eventually compact in a single idea, reaching that way the third level of abstraction. and so forth. from what we know after few thousand years of compacting ideas at an exponentially rate, this seems to be and endless process.

or perhaps there’s an end, and some day physicians are able to unify all forces, particles and micro effects together with the macroscopic world, and call it all with one final formula. perhaps something like “T(U)=0″, where T is some sort of operator (much more complex than any current laplacian or anything we have now), and U is the universe (something more complex than just a tensor or anything we have now). i don’t know what that would mean to us humans, though.

the lesson is that when doing maths, not only you can doodle and start without any defined target, but you can also start driving the process with a very strong intention yet end up somewhere completely unexpected indeed. and that’s part of the fun!

for the nerds, these are line segments connected to each other, which were made thick (like capsules) then deformed both in domain and range with cosinus functions to break the linear nature of the segments and make it all feel organic. there’s some cheap (fake) ambient occlusion and cheap (fake) subsurface scattering going on that in fact work really good. everything else is massaging color, shaping gradients and in general putting a bit of love everywhere.

the making was about 4 hours, from a blank canvas, which definitely makes it my 5th image of the week for my “one realtime mathemagical image a week for a whole year”. you can inspect the code, and change it and explore through the painting process here: https://www.shadertoy.com/view/Xdl3R4

intuitions

now that i think about it, my main point has always been the same: that during school, maths are formalized way too early. or perhaps, alternatively, that we never let kids use maths, play with them, and discover things by themselves before things get formalized. therefore they never get the chance to build intuitions. and trying to do maths without intuitions is like trying to find your way in a new city without a map.

my motivation to believe so comes from the observation that language learning, which usually is much smoother than math learning, happens very differently. basically, by the time kids are introduced to the formalization of language, to grammar and its syntax and structure, they already know how to speak. furthermore, they can perfectly read and write. not surprisingly, telling them that “a verb is an action”, “a noun is an object” and “an adjective is a qualifier” suffices to teach them the terms we collectively agreed on for speaking of language, because they already know intuitively what a noun or verb is. indeed, in language learning formalization happens to be trivial, painless, smooth, and natural, for kids are already familiar with language. hence the last step of organizing the knowledge within a framework is actually the easy part.

with maths, however, that’s no the case. that one terrible day the teacher arrives and announces “kids, we are going to learn trigonometry today” and introduces the notion of a cosinus and sinus, it is also the very first day kids ever get to examine angles and circles for the first time in their lives. crazy. no wonder this super-premature formalization of cosinus and sinus functions and their interrelations (described by weird formulas) alienates students to death. how wouldn’t it.

to make an analogy, the situation is pretty much like that of pretending to teach musical theory and what a major 5th or a minor 3rd sound like the very first day in their lives that the students see and listen to a piano. of course, it makes a lot more sense to let them first play music, often for many years, and only afterwards teach them its structure formally. any other approach to music teaching would be condemned for making no sense at all. obviously! yet, that’s exactly what we are doing with mathematics, throwing the whole thing at once to the students.

so, instead, what if kids had a chance to play with sinus waves, angles, circles, projections and shadows during 4 or 5 years? perhaps, that would allow them build their own intuitions and naturally understand these things. this could happen by means of visual problem solving, sound exercising, programming, drawing or any other method of direct manipulation of these objects. and what if after all of this playing, and only after the playing, we’d go for the formalization of the subject. my bet is that maths wouldn’t be any more difficult to kids than it is to learn language.

i believe that there’s a fundamental flaw in the current teaching method, and that it is one of the main reasons that prevents most people from grasping what maths actually are. and this is not a random believe, i’d say i can prove it. from my own studying experience during my engineering masters, which was heavily based in an “intuition construction” approach (including programming, osmosis, sleeping and relating things to other easier subjects), to all the people i ever met who i saw explaining the most abstract topics by using of very simple analogies and examples (topics that otherwise would have been considered difficult). such close examples have repeatedly proved to me the power of intuition and its role as the basis on which formal knowledge should be built upon.

so, taking into account that of course every person is different, and acknowledging the fact that every student has a different way to built intuitions – as some people are musical, others visual, others need spatial movement, others systematization, others narrative, others discovery – i think that a period of “playing with maths” is something that would benefit the grand majority if not all of the students. so, my suggestion would be to allow the students play for a few years indeed, and delay the formalization to later.

to make things even sadder, lets point to the fact that we are living in a world of computers, digital images and sounds, where not only we finally have the medium to make this new “intuitively playing with maths” paradigm possible and almost unavoidable. yet, we are avoiding it. seems to me digital devices are perceived only as a recreational gadgets. which is a pity, cause, perhaps, all those kids texting each other during the long math class are in fact holding the answer to the problem we are dealing with here right in their hands. this probably doesn’t need more explanation, for we all know that if there’s anything we humans have ever invented that is visual, dynamic, touchable, audible, interactive, intuitive, and well, a great platform for playing, that’s the smartphone (/computer). we do have the best intuition-making tool in our hands and pockets.

beauty

a second but not less attention worthy issue is that of aesthetical teaching. i usually refer to this as “sexiness in teaching”. i have the feeling that no matter how good the content of your message or story is, the format you present it in matters just as much in terms of effectivity.

these days we all expect to see beauty everywhere. not only we all want to look good or even be sexy (to the point people can get disappointed if they think they are not perceived as following the current standard of beauty), but we actually project the beauty as a value into the places we visit, items we consume and objects we own. indeed, the best selling devices and gadgets are sold for their design and beautiful lines not for their easy of use or technical features. cars are designed with very specific lines too which certainly don’t follow efficiency or any other technical criteria. every single movie and magazine undergoes heavy composition and color modifications to optimize their visual attractiveness. at more individual level, we no longer upload a picture to our online photo albums without passing it through color filters in an attempt to make it look “cool”. we as a society and as individuals, expect beauty everywhere, regardless of the quality of the object, subject, content or idea we are dealing with. a famous architect was putting it this way when talking about sustainability: the brutal truth to make sure an architectural piece is sustainable, in that it stays there for longs periods of time, is not to make it better solar powered or more heat efficient only – instead, the answer lays in making it beautiful, to be eye catching so it never gets aestehtically obsolete, that will ensure it stays there forever.

others forms of beauty are engagement and humor. of course no good speech can be expected to not start with a joke, then show strong passion and heart, and do explicit use of verbal repetition and reiteration of the key ideas in the speech. that, we probably could indeed call a beautiful speech.

so, coming back to the topic of teaching maths, my second idea for today is that the realizations of the importance of beauty in the format of whatever content you are dealing with regardless of its quality, probably should apply to teaching too. now, if these principles of beauty should be applied just as a direct and simple design issue in the material used for teaching, or if there are actually more profound ways to understand beauty in the teaching process is something i cannot foresee yet. what i do know is that a set of exercises in a blackboard or an online website are certainly not the sexiest way to present the otherwise beautiful ideas that math teachers aspire to teach.