Square Signals

Leveraged Making

Fri, 27 Apr 2012 08:48:16 -0700

When I was a kid, I spent much of my time making video games. Then I devoted my early teenaged afternoons to making an app for making art for video games. Now I make tools which can (among other things) help make apps for making art for video games.

My goals have shifted away from games over the years, but I still consider the same metaphor that excited me in eighth grade: that of leverage. It’s a useful principle no matter one’s goal, so I’d like to share a few examples.

Say you want the world to have amazing teddy bears. You could make them yourself, but you couldn’t make enough for everyone! If you spent your time teaching people how to make especially wonderful teddy bears—and if you did a very good job—the world might have more amazing teddy bears in the end. Your effort is effectively magnified, leveraged. You might also amplify your work by researching extra-fluffy stuffing and selling the results to teddy bear manufacturers.

So, no more video games: now the point of my work is to expand the reach of human knowledge. Computers and software are outrageously useful implements for that end. We’ve got a world of knowledge at our fingertips, along with the tools to explore, share, and expand it. Those tools give us leverage over knowledge. An hour at a library can become seconds with software like Wolfram|Alpha.

Presently, I make tools for making tools. On the occasional days when I do something that radically improves the quality and ease of creation of all these levers, the effect compounds into fantastically broad reach.

I’ve long mused that a teacher could have much greater leverage over the reach of human knowledge: after all, he trains the people who could go on to make tools for making tools for creating knowledge! Among all the other wonderful things his students might do.

In the last couple years, though, I’ve enjoyed considering the principle taken one step further: what if I created tools which allowed teachers everywhere to be radically more effective and efficient? Or which helped students teach themselves far more easily? Leverage abounds.

Top to bottom; bottom to top

Tue, 06 Mar 2012 09:19:54 -0800

Need I remind you: the human race now sends robots to Mars. That is a thing we do! We derive that absurd capability from generations of previous epistemological revolutions which we compound and extend to create new marvels.

I like to picture the tower of abstractions that enable this kind of progress. If you looked closely, you’d see its struts form an intimidating (and not altogether reassuring) lattice. Rocketry builds on chemical engineering and relativity but not pharmacology; flu vaccines build on chemical engineering and pharmacology but not (directly) relativity.

Now, we could examine the same data or questions from any position on this tower. One fun example: why is your hair the color that it is? Because you have a certain gene which expresses that phenotype! Or… because of the cellular machinery which expressed those genetics? The electrochemistry which drove those interactions? Optics? Cultural tropes influencing the mating habits which favored those genes?

One approach suggests that the observer can best understand a complex system by breaking it into composite parts, until she trips upon axioms, which are (perhaps temporarily) where truth lies. This is sometimes called reductionism. This word has other—incompatible!—definitions, but this is the one I’ll reference.

Meanwhile, others suggest that for many complex systems, the constituents hold only noise: real meaning emerges at the highest levels. That interpretation seems especially appealing when considering human consciousness and free will. After all, who wants to think of their being in terms of roiling, deterministic chemistry? I’ll call this approach holism (with the same lexicological caviats).

This is a figure from the “Ant Fugue” in Douglas Hofstadter’s Gödel, Escher, Bach. It contains letters made of other letters. At the highest level we read “MU.” The ‘M’ comprises “HOLISM” three times; the ‘U,’ “REDUCTIONISM.” Each letter of the former contains the latter word, and vice versa. Then, gleefully (and invisibly in this reduction), the smallest letters are themselves formed by repeated “MU”s.

Each level of abstraction in this figure encodes information (here, words) in its emergent properties. If you näively apply reductionism and interpret the smallest components, or take ten steps back as holism prescribes, you read only “MU.” In Chinese, “mu” means nothingness, often in a spiritual sense.

It’s not impossible to discern the middle layers of words from either of these positions, but it’s also not expedient! Independent access to each layer of abstraction delivers novel perspectives and understanding. I suspect the same is true for any complex inquiry.

Now then: how does one acquire that independent access to each layer? Which reduces to asking how one can build expertise in general—how can we learn the many facets of any complex field?

A key issue in education is that the student’s goals are often holistic. She wants to learn how to make a robot, for instance, and might not be interested in the complex analysis necesary to understand the physics necessary to understand the electrical engineering necessary to understand the servos and sensors necessary to make it move.

Two dangers lurk here. The first and more apparent: if the student plays only with pre-built motors and components (without understanding or considering the principles by which they’re constructed), she’ll be seriously limited in how she can compose them and will endure endless headaches diagnosing and fixing problems.

The second and more insidious: this naïvely holistic (vocational?) approach could appear to work for quite some time. A year or two later, when she abuts the edges of the epistemological landscape visible from her position, she’ll face enormous psychological barriers as she moves down the stack. She must unlearn parochialisms, summon humility in a field she thought she understood, muster patience for moving—temporarily!—more slowly relative to her ostensible goals.

But if she builds her tower of understanding from principles—brick by reductionist brick—boredom may dominate good intentions and drown her emerging motivation.

I propose a hybrid approach: derive motivation and intuition from top-down exploration while cementing understanding and that crucial universal access with a bottom-up climb. Critically, though, she can ground and inspire the lower levels by studying them in context of her experiences with their emergent behaviors.

Some schools will offer a physics class for physics majors and a physics class for engineers. But generally the latter distinguishes itself by watering down the concepts, skipping a great deal of material, and hand-waving over derivations—not by providing motivating context.

With this idea in mind, personalized education could provide enormous benefits not just in terms of keeping pace with the student’s progress and prior experience, but also in captivating him through critical foundational material which would otherwise never take root.

I’ll leave you with a personal anecdote. At Caltech, I spent two years studying not computer science (my major) but math, physics, chemistry, biology, astronomy, laboratory practice, etc—because I had to! Every student had the same core requirements, regardless of major: Caltech history majors have studied statistical thermodynamics.

At the time, I was incredibly bitter about this situation! I felt like I was wasting my time, so I didn’t take the material seriously. I was supposed to have learned linear algebra my freshman year, but I didn’t: I learned it after I bumped up against a brick wall in my studies of machine learning, which I’d practiced for a few years with misleading success.

So I went back to re-learn linear algebra in a real context. This time I cared, so this time it stuck. This time I understood how these abstractions might be applied, so they made sense. Which was nice, because years later they enabled me to (more or less on a whim) radically improve how iOS’s 3D “page curl” effect tracks a user’s finger.

I’ve had similar experiences with those physics, chemistry, and biology classes. They’re a means, not an end, but it wasn’t until the end was in sight that I could truly learn and build from their subject matter.

The discomfiting continuum of consciousness

Wed, 11 Jan 2012 22:32:00 -0800

Some folks have ethical objections to eating meat, but no one seems to mind savaging a carrot. Why? Both fish and ficus are living organisms, after all. I doubt vegetarians consider taxonomic kingdoms when making moral judgments: the decision comes more intuitively.

Douglas Hofstadter suggests that many are judging the presence of consciousness, a sense of self-hood, “inner light.” He argues that this attribute isn’t just an on–off switch but rather a continuum: humans seem to have a stronger sense of “I” than a dog than a fish than a fly. Your sense of species’ relative positions might be different from mine—I’ll dodge the question of what specifically we’re assessing—but surely you’d place, say, monkeys and ferns in different spots.

Now let’s zoom in on the “human” portion of that spectrum. Does a newborn occupy exactly the same spot as you? Infants don’t recognize themselves in the mirror, don’t appear to reflect upon their experiences, can’t plan, couldn’t pass the Turing test. They’re closer to you than a fish, sure, but how do they compare to a chimp?

Someone recently asked me if I’d ever been to Colorado. I said “yes, but I was nine-ish. Not sentient yet.” My reply was flippant, but it reflects some truth: my “inner life” at that age was far simpler than the one I’ve enjoyed in adulthood. My behavior was almost purely reactive; I integrated facts without understanding; I exhibited limited theory-building or subjunctive thought. I was closer to an “i” than to an “I.”

That made sense: consciousness seems to be a feedback system constantly evolving in response to perception, so it should grow in reach as it consumes more input.

But does our consciousness grow monotonically? Consider the last time you were quite drunk—after all, alcohol is beloved in part because it makes us less self-conscious! Other drugs make their users more self-conscious, while still others move one not exactly towards “I” nor “i”, but sideways into surprising alternative modes of awareness.

So perhaps consciousness grows monotonically when not adulterated by external chemical influences. But then what about internal chemical influences? How did your sense of self respond the last time you stayed up all night? I find (to my great alarm) that if I lose even an hour of sleep, most factors I associate with my consciousness suffer hugely!

Conversely, if I find myself unable to perceive, reflect, analogize, or emote, then a peaceful walk often restores these faculties somewhat.

This is all fairly terrifying: our incredibly intimate sense of self ebbs and flows by the hour with unseen tides. But if it’s going to happen, perhaps at least we can exert some control over it! Can we exploit the “peaceful walk” mechanism not only to restore these sentient faculties but to boost them? Longer-term, could certain hobbies or lifestyles lead to a more powerful sense of self? Which, while we’re introspecting, demands a further meta-question: would that even be desirable or useful?

Teach kids to code; watch them cure cancer

Wed, 16 Nov 2011 19:22:37 -0800

People list plenty of reasons why kids should learn how computers work. They could automate repetitive tasks; they’d be empowered to create in all kinds of media; they’d learn powerful new problem solving approaches. That last point (so innocent seeming!) has gripped my imagination most thoroughly: a mastery of abstraction offers stupendous power in practically every endeavor.

But perhaps the most important tower of abstraction—one whose structure we scarcely understand despite its supremely personal importance—is rarely mentioned in this context: life.

Say we cure cancer. Hooray for us! But we’ll face other diseases, then still more after we’ve addressed those. We address the analogues in software (bugs, security holes) at a totally different pace from those of life. Even the attitude is different. Software engineers know their software has bugs—and that it probably always will—but the industry nevertheless accelerates and thrives because when issues crop up, they’re generally resolved quickly and decisively.

We can fix software bugs quickly because we (mostly) understand the systems we’re addressing at all relevant levels of emergent behavior. We’ve established that understanding through composable abstractions about which we can reason in isolation, and by employing tools which enable us to rapidly test conjectures and gather data.

Personalized medicine applies some of these principles to healthcare: when a patient’s bacterial population evolves a resistance to a drug, we should be able to rapidly study that resistance and produce an appropriately targeted treatment. For this end, we’ll need diagnostic and fabrication tools, yes; but more fundamentally, we’ll need medical understanding at the many levels of emergent phenomena between organic chemistry and the consequent symptoms we observe.

Software engineering could provide a powerful microworld for biological engineering. A generation of children steeped as much in abstraction as in language would become a generation of adults extraordinarily well-equipped to understand our biology—and therefore to manipulate it. It’s difficult to comprehend the reach of explosive growth in this field. How many more generations will die of old age?

Feeding abstraction with understanding

Wed, 26 Oct 2011 22:11:00 -0700

Abstractions empower and accelerate. As usefully encapsulated nuggets of understanding, the creation of novel abstractions drives a field’s progress, but their invention is possible only with deep understanding of present ideas. So I declare: if we are to master a field, we must accept none of its abstractions as magic. Rather, we should yoke them as automations of what we already understand.

This concept separates the random, slowly-reinforced walk of genetic evolution from the directed, rapid exploration of scientific evolution. Understanding provides not only a compass bearing but also course correction, without which extended progress is impossible.

For instance, if you always cook from recipes (without understanding why you’re executing each step), you’ll be poorly equipped to fix a dish that’s gone wrong, improve an average one, or synthesize something new altogether. The same limits apply to parochial understandings of abstractions in musicianship (“minor chords sound sad”), writing (the five-paragraph essay), mathematics (the Pythagorean theorem), and engineering (caching systems).

Say you accept this idea. We’re then faced with a practical problem: how can you ever learn to do anything useful without endless prerequisites? Algebra classes are particularly afflicted.

TEACHER: Today we’ll learn how to factor quadratics of this form.
STUDENT: Why do I need to learn that?
TEACHER: You could use it to find where parabolas hit the x axis.
STUDENT: Why do I need to learn that?
TEACHER: You could use that to find the trajectories of point masses.
STUDENT: What?
TEACHER: Believe me, you’ll be glad you learned this someday.

To foster real understanding, we must format concepts so that students can derive solutions for themselves, not encourage them memorize some procedure. And of course, they must be motivated to find those solutions by a problem they actually care to solve.

Microworlds may form the basis of one solution. The idea is to create a tiny, self-consistent sandbox in which the student can explore some concept, given little instruction beyond what he already knows. The sandbox doesn’t hide detail so much as focus. One can imagine, then, constructing a successively expanding universe of these microworlds—each itself useful—so that hops between neighbors aren’t too vast.

Imagine trying to learn to play the keyboard. You could start with one which only allows the student to play in the pentatonic scale in C. Five keys aren’t so intimidating. The others are there, but dimmed out. You can still play all kinds of melodies. You can explore rhythm. You could play chords, though many won’t realize it. Then you could switch over to a hexatonic blues scale mode: also self-contained; also not too intimidating. Then you could start displaying the notes the student plays on a staff above the keyboard. And so on. The skills learned on these limited keyboards are not abandoned when moving to more complex ones—they’re fruitfully employed!

Each microworld will naturally engage different sorts of students, and that’s okay: they should be free to roam between any within reach. The goal is to provide a sandbox—not a syllabus—for experimentation and the formation of understanding. Understanding is marvelous because it’s so readily a feedback loop: we can use it to make more of it.

The air sings: sympathetic vibrations

Wed, 12 Oct 2011 23:14:00 -0700

Chords sound rotten on an out-of-tune piano. I’ve written about why we think that might be: harmonics of the detuned lower notes end up a small distance from those of the upper notes, rather than directly overlapping them.

My piano was tuned this weekend. Playing it after the technician had left, the sound’s transformation startled me: how rich and powerful my piano had suddenly become! I remembered, then, a less obvious but absolutely critical effect of correct tuning: sympathetic vibrations of a note’s upper harmonics.

When you play a middle C, the resulting wave carries pitches of higher frequencies according to the harmonic series: C₅, G₅, C₆, E₆, and so on. The wave, emanating from middle C’s string, will then excite the strings corresponding to the other pitches—providing the dampers on those strings are lifted (by pressing its key or the damper pedal). The sympathetic vibrations in these higher strings increase the volume of their respective harmonics, coloring the resulting tone.

In other words, not every C major chord composed of a C₄, E₄, and G₄ will sound the same. The resulting sound will depend on which dampers are lifted in the octaves above! If the piano is out in tune, though, the frequencies of the strings in the harmonic series won’t align, so there will be little sympathetic vibration.

Here’s a little video demonstrating this principle in action:

Download video as MP4.

Juggling exploration and exploitation in intuition

Tue, 20 Sep 2011 23:06:00 -0700

I spend my life building abstractions and automating processes. It’s (literally) my job, yes; but my head’s always whirring away to that end.

Say I’m tackling some piano piece. I transform the score’s markings into musical language. I reify that musical language into instructions: “play this key here,” “increase in volume,” etc. I execute those instructions via complex musculature. Reacting to the sound produced, I form an interpretation which (hopefully) maps emotion onto the score via a series of fine adjustments.

I’ve spent many years automating these processes, relegating always more to my subconscious. I find even now that my playing of a passage will never match the sound in my head until it’s memorized, and my hands navigate the keys unbidden. I build these automations by finding patterns and constructing abstractions. I might not be able to read a trio of notes at tempo if I address them individually, but I could recognize their collective shape on the page as an arpeggiated triad or an ascending scale, which I could then execute without further thought.

I suspect intuition is the automation of acts once executed consciously. Of course I’ve built most of this intuition unconsciously, but as I’ve begun to study studying, I’ve tried to exert conscious control of that growth. Cognizance has been the primary mechanism: constant reflection on and criticism of my actions informed by their outcomes.

Now I wonder: if I automate ever more layers of activity so I can focus my attention on higher levels still, can I ever stop scrutinizing the automated abstractions? What if a powerful new skill won’t emerge until I change the way I’ve always intuitively performed some basic task? Maybe the skill tree’s actually a maze, and I need to backtrack!

I’ve found new layers of understanding in my work¹ that required me to abandon fundamental concepts I’d automated long ago. And in my piano education, I’ve spent years unlearning bad habits I developed as a child. So how should we balance exploration and exploitation here? It seems we can trust nothing, yet if we spend all our time questioning assumptions, we’ll never draw any conclusions.

Including (software engineering alert!): fun with dependent, existential, and generalized abstract data types; invisible (to me) dangers in traditional asynchronous models; the ubiquitous and unformalized transactions and state machines in imperative logic; the value of immutable memory; knitting together the declarative, functional, and imperative; etc. Someday I’ll write technically about software engineering again…

In defense of "fusion" cuisine

Tue, 06 Sep 2011 22:38:00 -0700

San Francisco bursts with fantastic international food. Ethiopian, Thai, Japanese, and Peruvian food carts roam our streets every day, vending explosive flavor combinations that are often unknown to local cuisine—except, that is, in “fusion” restaurants, which (often unsuccessfully) derive novelty from culinary combination. Here I’ll advocate fusion, but of a different sort than these amalgamating restaurants practice: not of flavors, but of techniques.

For instance: we don’t need to eat overcooked meat anymore, ever (sous vide cooking has eliminated this problem), but you will never find a bowl of curry with meat that hasn’t been heated to at least 170°F.

Then there are the vegetables, so often mushy and flavorless, from the cheapest source available, their color pallid from imprecise cookery. I remember by contrast the first time I made the bœuf bourguignon from Thomas Keller’s Bouchon. He braises beef with aromatic vegetables, then replaces them with a fresh batch before serving—they’ve already lost all their flavor. He cooks each type separately and as it is best prepared: carrots glazed, mushrooms browned, potatoes simmered.

Salt, too, is fundamental: it increases the intensity of all other flavors. Meat without salt is like a desaturated photo; salt sprinkled on top of cooked meat will always disappoint because the interior remains unseasoned. In quality Western restaurants, meats are generally salted days in advance so that the center of the meat will satisfy. But that’s exceedingly rare in most non-European cuisines.

This weekend, I made a Thai green curry while minding the European cooking principles I’ve learned. For the curry, I infused coconut milk with lime, galangal, green chilis, coriander, cardamom, nama shoyu, shallot, and lemongrass. A typical blend, modulo soy for shrimp paste.

The rest of the curry, though, was prepared non-traditionally. Restaurants in California talk about being “ingredient-driven” as if that describes a cuisine, but it’s just an attribute—and one that should be applied to all cooking. So I got the best vegetables and meat I could find and prepared them with individual attention and precision. Or at least as much as my dilettantish skills could muster.

I packed some lovely duck breast in salt well in advance, then cooked it sous vide at 137°F with some of the curry paste. At service, I cross-hatched its skin and slowly crisped it. Then I sliced it thinly against the grain. No bones.

I sautéed oyster mushrooms over high heat with plenty of room to crisp them slightly. I salted them early and finished by deglazing the pan with a bit of rice wine vinegar. I drained them carefully so they wouldn’t be soggy or rubbery. I tossed cheddar cauliflower in oil, salt, and pepper, then roasted it over high heat until deep golden. I lightly browned golden squash in a hot pan with salt and rice wine vinegar, leaving the pieces with tooth and loads of flavor.

Then when serving the dish, I warmed all the components in the curry to marry the flavors. But each ingredient retained its nature: it was the first time I’d had squash in a curry that actually tasted like squash.

So: I know that I can visit Hapa Ramen to get ramen with perfectly cooked pork belly. The folks at Ad Hoc are making precise fried chicken. Now where am I going to find modern masala?

Subconscious Kindling

Wed, 24 Aug 2011 22:43:00 -0700

In the first chapter of The Beginning of Infinity, David Deutsch writes:

The role of experiment and observation is to choose between existing theories, not to be the source of new ones.

I have the most fun attacking questions with no satisfying theories in sight. So: how can we generate novel theories for a problem? It’s critical to recognize that the creation of a hypothesis demands a wholly distinct approach from its evaluation.

Sometimes I uncover a critical idea by consciously exploring possible avenues, but more often, the realization flashes unbidden, with my attention elsewhere—like a lightbulb out of a classic cartoon. In these cases, I suspect my mind has been churning ceaselessly at some tree of possibilities beneath the surface of consciousness, projecting potential futures, searching my memory for relevant assocations, and evaluating the relevance of its findings before bubbling any useful notions up to my awareness.

We can propel this part of our psyche along by seeding and predisposing it for this pursuit. By providing kindling. I like to imagine waking up little subconscious drone ants and sending them in every direction, hoping that one will return with something useful later.

I find that doodles, in particular, provide especially effective kindling—whatever that means in your field: instinctive improvisation at the piano, whiteboard sketches of systems in engineering, lists of assertions in writing.

This process sounds much like brainstorming or mind mapping, but I don’t think it is, exactly: those are aids to conscious invention and organization. I’m suggesting something more like rubber duck debugging, a technique in which the simple act of explaining a problem in great detail induces cognitive dissonance during the weakest parts of the explanation.

As a highly creative act, theory generation benefits from many of the same techniques that bolster artistic pursuits. Besides doodling, I’ve developed theories by exaggerating, by celebrating contrast, and by applying techniques from one field to another unrelated one. Whatever I can do to throw more wood on the fire.

Second-person self-awareness

Tue, 09 Aug 2011 19:46:00 -0700

In advice given to prospective students, my former piano teacher, James Boyk, suggests:

I’m not the teacher you want if you want to be passively turned into a pianist and musician instead of actively turning yourself into one.

This statement so concisely and wonderfully conveys my position on education and self-improvement. Its attitude absolutely informs the celebration of cognizance in my writing here, but so far, my suggestions in that regard have been for personal action. There’s a role for others in self-awareness, too: my teacher might expect his students to take ownership of their education, but that doesn’t mean he mentors passively!

The “1 on 1” meetings between employee and manager might strike you as a venue for second-person self-awareness, but these house tremendous opportunity for passivity: generally, the manager suggests how his employee might improve his performance. If only one could achieve mastery simply by following others’ instructions! All we’d have to do is carefully note what our bosses say in those meetings and then execute those directions.

No, I suspect level-jumping changes originate from within. The teacher’s role, then, is to inspire, to provide objectivity, to develop the student’s taste and understanding so he can find his own weaknesses, and to make sure he can find the tools to defeat those weaknesses once he understands them.

Many advocate personal professional journals: every day, write down what you’ve done, what you’re going to do tomorrow, what you could do better. I keep one, and I think they’re a good idea. But, to crib again from Jim, a second-person journal can provide incredible perspective and self-understanding. After every lesson, his students email him a detailed summary of what they’ve done, what they’re going to do, and reflections on the lesson’s contents. He then responds actively to the student’s self-analysis, calling out particularly effective or misguided ideas, and guiding their development. He’s published some examples about halfway down this page.

We exchanged tens of thousands of words in this manner over two years, and I progressed all the more for it. And critically: this practice helped me learn to teach myself—the ultimate goal, since I would be eventually moving away. Now, I just wonder how I can apply this technique to my present pursuits.

Executing my soup program with excellence

Tue, 02 Aug 2011 20:08:00 -0700

This past weekend I encountered the following sales pitch for an industrial food service product: “are you ready to execute your soup program with excellence?”

Corporatese hilarity set momentarily aside, I derived inexplicable comfort from the vision of a mythical fast food franchise owner—ardent and passionate about every aspect of his business—who, reeling with the possibilities of this product, would engage and debate the salesman with wide eyes. Who might share his discovery with his friends (naturally: a set comprised mostly of fellow franchise owners), zealously contending the relative merits of competing industrial soup products.

I don’t know if this owner exists, but even entry-level employees in my industry (software engineering) often behave like him. I suspect that’s due to—for better or for worse—the exaltation of purpose in one’s occupation: aspiring to do something objectively important, meaningful, or good during business hours.

I remind myself often how absurdly, outlandishly fortunate I am to have chosen an occupation with a supply–demand curve that makes purpose a conceivable job amenity. If I’d chosen another path, I’d be forced to spend those eight hours a day with utter personal indifference to my success or failure.

This celebration of purpose, though, comes with cost: if I think my work Matters with a capital ‘M,’ I’ve tied my self-esteem, temperament, psychic wellness to that undertaking. My day-time struggle with a nasty problem will seamlessly transition into detached night-time conversation, my frustrated, churning mind still back at my gray plastic desk under fluorescent light.

I now suspect the real danger is in a false sense of purpose: in accepting its bleed into after-hours mentally and physically, yoking it to consummate responsibility, equating personal success with the project’s success—all while the work itself does not tally the imagined objective import, meaning, or good.

Sometimes this behavior is easy to spot. I worry about new college graduates choosing a first apartment a block away from work for a convenient commute. I’m self-aware enough to sharply change course when all my conversations somehow return to software. But unstated attitudes and priorities are harder to catch.

Having worked twenty-seven of the last thirty days, I’ve found I needed this brisk reminder: to attribute only as much meaning and intrapersonal attention to my work as to balance the purpose it delivers.

Vegetable stock: my secret lover

Tue, 26 Jul 2011 11:02:00 -0700

After learning how to season food correctly, homemade stock is the single best thing anyone can do for his cooking. A stock is a liquid extraction of complementary flavors, a broad palette upon which sharper splashes of color may be painted.

Stock is about balance and intensity of flavor. Making a mushroom sauce? If you were to just purée some cooked mushrooms, the isolated flavor would taste harsh, but add some stock and it will gain balancing sweet and vegetal notes. Those carrots need more flavor? Simmer them in stock instead of water: stocks are distillations of elemental flavors. They can be diluted or concentrated at will.

I credit stock with much of my ability to impress in the kitchen. They’re as much cooking technique as they are ingredient. But, you keen, they’re too hard to make! They take too long! Nay. Sit down. Let me tell you about my favorite stock. I make it almost weekly, and it’s so easy that I am positively mystified why every cook in every household isn’t making this elixir regularly. Let me tell you about vegetable stock, and let me tell you how to make it.

I made nasturtium soup this weekend, and the subsequent dinner’s conversation remained imperturbably anchored to how unbelievably tasty it was. Yet it consisted of nothing more than nasturtium leaves puréed with vegetable stock, with a nasturtium flower for garnish.

Consider how else you might make this soup. You’ve got some nasturtium leaves: fine. How will you turn that into a soup? You’ll need to add some liquid, clearly. Looking around the internet, I see a minefield of misguided ideas. Steep the leaves? That will diminish their brightness. No, they need to be puréed uncooked. But with what? Cream? That will dull the leaves’ radish-y spice. Chicken broth or stock? The flavor of nasturtium is too delicate: you’ll overpower it. Thicken with potato? We don’t need filler here. Water? The result will taste thin and incomplete, lacking sweetness and umami.

Purée the leaves with vegetable stock and it will taste like a garden. It will taste like it smelled when you went back there to pick the nasturtium. It will taste like sitting in a park in spring. And then the horseradish spice of the leaves will give you a hearty, well-rounded slap and bring you back to the kitchen.

Making vegetable stock

Get three large carrots, an onion, two leeks, and a bulb of fennel. Peel the carrots and onion; discard all but the white and light-green part of the leeks. Wash the leeks and fennel. Chop everything up finely and put it in a big pot with a few glugs of oil.

Now you’re going to sweeten up these vegetables by cooking them over low heat. You don’t want to brown them—that will make a darker, richer variant of this stock, which should be bright and clean. Add a large three-fingered pinch of salt to draw out the vegetables’ juices and turn the heat to low. Cook, stirring occasionally, for about fifteen minutes. Taste an onion. It should be quite sweet.

Add enough water to cover the vegetables, then add that amount of water twice more. Bring the water to a bare simmer and then let it cook for forty-five minutes.

Strain the stock. That’s it. Store it in your fridge, but the stock will lose its brightness after a few days. That’s why, when America’s Test Kitchen set out to find the best grocery store vegetable stock, they concluded that all of them were unusable.

This stock provides a mostly neutral foundation for flavor. You can adjust for its intended purpose readily and improvisationally. Want grassier flavors? Add half a bunch of parsley. Heartier? Brown the vegetables while sautéing them. Asian flavors? Add ginger, a little lemongrass, and star anise. And so on.

How else you might use this stock

Reduce the stock by half, add salt and some fresh lemon juice; and you will have the best vegetable broth you’ve ever tasted. Add a bunch of fresh, perfectly-cooked summer vegetables immediately before serving and you will wow people. One friend, tasting in disbelief, asked how much honey I added to make it so sweet. I told him “three carrots.”

Make risotto with it. Cook pasta in it. Any vegetable which you would normally cook by simmering in water (carrots, potatoes, turnips, pearl onions…)? Cook them in this with a healthy dose of salt. Make a fantastic sauce for any dish by puréeing a complementary, perfectly-cooked vegetable with reduced stock. Season it and use it in place of store-bought chicken broth: that stuff tastes like industrial chemicals.

Once you’ve got this down, no one will have to persuade you to learn about chicken, veal, pork, fish, and mushroom stocks.

Musical tone and perceived dissonance

Tue, 12 Jul 2011 23:28:00 -0700

Why does a perfect fifth sound so satisfying when a brash tritone is just a half step away?

It seems we have a physiological correlate to these psychological effects. As sounds arrive, the basilar membrane vibrates like an oblong drum head, stimulating tiny hair cells attached to nerves. Each hair cell corresponds to a perceived frequency, working like little band-pass filters.

One hair cell will be stimulated over some small range of input frequencies: for instance, hairs for middle C (262 Hz), will also react to sounds between 230 Hz (roughly A♯₃) and 290 Hz (roughly D₄). We’d say, then, that the critical bandwidth at middle C is around 60 Hz. Researchers have found that the bandwidth of a given hair equates to around a whole step on the western musical scale, basically the whole way up and down the keyboard.

As for the tritone: when a sound vibrates two hairs with overlapping critical bandwidths (like C₅ at 523 Hz and C♯₅ at 554 Hz), we perceive it as rough and rattling. Why are we wired that way? No study has conclusively determined the answer, but I’d wildly speculate that such sounds share acoustic properties with noise from friction. Rustling leaves; shuffling paws? Or screaming? Maybe it’s just a warning sign when our brain can’t readily separate two tones.

Whatever it is, this effect alone can’t explain tritones’ impudence. Take C₄ (262 Hz) and F♯₄ (370 Hz): they’re several bandwidths apart, but they still sound dissonant when played together on my piano.

That’s because when you play a C₄ on the piano, you are also playing a bunch of other tones: 523 Hz (C₅), 786 Hz (G₅), 1048 Hz (C₆), and so on for the next three integer multiples of C₄’s frequency. The tone, or timbre of an instrument—what separates the sound of a flute from that of an oboe—largely derives from the relative strengths of these extra tones (harmonics) to the pitch you meant to play (the fundamental).

So, your C₄ (262 Hz) includes an inadvertent and somewhat quieter G₅ (786 Hz). Note two of the tritone—the F♯₃ (370 Hz) will include an extra bit of F♯₄ (740 Hz). These two harmonics have colliding tiny-hair-bandwidths, just like the minor second we discussed earlier: hence the rough sound.

Even fairly consonant chords, like a major third, feature these collisions. C₄ (262 Hz) and E₄ (330 Hz) will clash at the third harmonic of the former (C₅ at 1048 Hz) and the second of the latter (B₄ at 990 Hz). Those harmonics are much quieter than the fundamental and first harmonic, though, so the major third grates less than the tritone.

Remember, though: the harmonics for a given note are produced at different relative intensities from instrument to instrument. This means that the same interval, played on different kinds of horn, could sound substantially more or less dissonant! Moreover, musicians can control their instruments’ tone (i.e., in part, their harmonics’ amplitudes) at will while they play. Sing an “ah”, then raise and lower your soft palate. You’ll hear your voice’s tone change substantially. (And you’ll sound ridiculous.)

Did composers subconsciously account for instruments’ tonal effect on consonance when arranging their pieces? Is it possible that excellent musicians intuitively adjust the tone of their instruments to modulate the dissonance produced by the harmonies they form? Could this account for the sudden beauty of a chord in one ensemble’s rendition of a piece you’ve heard a thousand times?

Raven and Badger

Tue, 28 Jun 2011 20:41:00 -0700

People have asked me (quite nosily, I think): “Andy, why are the badger and raven in your paintings facing away from each other?”

“Well:” I say.

“Badger had spent all day preparing quite a serious dinner party. Or, rather, he had been looking fervently over the shoulder of his poor chefs, who were themselves preparing the quite-serious dinner party.

“Raven’s invitation had arrived—letterpressed and perfumed—two weeks prior, bringing with it unbridled excitement. Badger’s celebrations, you see, have developed a deserved reputation among refined animals. RSVP dispatched, our avian protagonist began to fret in earnest about choosing a gift for his host.

“Naturally, any edible gift should appropriately complement the dinner’s flavors, but Raven remembered from previous events that Badger’s excitable but indecisive palate had led to menu changes up to (again, remember the poor chefs) hours before showtime.

“It had to be something neutral. Broadly compatible. Chocolates were right out: what if Badger serves a chocolate cake for dessert? And cheese, well: Mouse has enough trouble keeping his rodential habits in control at the dinner table. Cheese would not help the matter. A basket of fruit? No, no, not fine enough.

“Cue Raven, minus a few feathers, at his favorite wine shop. Bottles opened, tastings poured, he found a Viognier that should have pleasantly accompanied whatever (always red) meat Badger’d chosen.

“The night of the party. Raven arrived with the usual ritual: warm handshakes, smiles all around, the bottle proffered, thanks returned (‘this will pair beautifully with the squab!’). The crowd sprawled vague and disjoint social circles, and conversation meandered effortlessly.

“Everyone found their seats at the dinner table (thanks to calligraphied place-cards). All guests now expectant and awaiting their soon-to-have-been-fabulous dinner, conversation fell into a brief lull.

“White leather gloves placed course one before each guest simultaneously. As silver lids revealed the contents of the covered bowls beneath, Badger’s beatific smile faltered. ‘Excuse me a moment,’ he said. ‘Please enjoy.’

“Some clamor ensued from a distant hallway (something about a lumpy quenelle?), but the party’s genial atmosphere soaked up concern before it shadowed the mood. Badger returned and scarfed a tepid slice of langoustine, silent.

“The ramp course elicited another grimace from Badger. His guests, now lubricated with several glasses of fine wine, and blissfully enjoying the room’s conversation (have you seen the new Picasso exhibition at the Antoni?), hardly noticed his departure or the crash of distant copper.

“Badger’s staff delivered the squab with some trepidation, then sighed in relief as he jubilantly attacked his dish. Now, as Badger engaged his guests in a fiendish brain teaser involving snowmen and a checkerboard, it was Raven’s countenance that cracked.

“What about his wine? The Viognier chosen with such care? Was it not good enough for Badger? Of course not—the snob! What a brute. Raven delicately inquired if his host might like to sample the gifted wine with his squab. Oh, but a stout Sauv had already been poured. And the guests would need to drive home yet.

“Now quite warm-bellied himself, Raven wondered aloud about the fitness of Badger’s constitution. His host, in no mood after the first two courses’ disappointments, pointed out the nearest fraternity for his guest. Which elicited from Raven a crack about smoking vests. Then from Badger a retort cleverly rhyming ‘molting’.

“The guests’ jocular laughter wilted into tense silence as the exchange continued then devolved into respective hissing and squawking from either side of the table. Time seemed to slow as Raven’s flailing wing (gesturing luridly) toppled the hand-crafted crystal wine glass that (until very recently) sat before him. And then as Badger pounced over the shards and tumbled with Raven across the floor.

“No one really talks about the events that ensued. As with everything in this kind of company, everyone pretends nothing happened (Badger: ‘oh, how is that Raven fellow? haven’t seen him in ages!’)—until you get them alone. Then, in whispers, you’ll hear about how Raven got that scar on the back of his neck and why Badger’s deep carpet needed shampooing. And a return visit for more shampooing.”

So that’s why they face away from each other. Just don’t tell anyone I told you.

A Surprising Advantage of Vinyl

Tue, 21 Jun 2011 20:21:00 -0700

I am emphatically not an audiophile. My home contains neither $500 ethernet cables nor “acoustic isolation platforms.” I am absolutely not here to convince you that vinyl recordings sound inherently better than the same data on CD.

But: there is one absolutely irrefutable advantage to many modern vinyl releases, and that virtue has nothing at all to do with the medium. It’s the data on the record that’s superior, and it’s a conscious choice made by today’s artists and producers.

Perhaps you’ve heard of the loudness war. Essentially, tracks are being mixed louder and louder in production—maybe because when played on a radio station, a slightly hotter recording will sound more intense than the track which preceded it; maybe because loud audio sounds better on cheap speakers. The volume’s been creeping up, up, up, far past the point where peaks got clipped against the dynamic “ceiling.”

Below are the left channels from Radiohead’s “15 Step”. On top, the vinyl’s audio; below, the CD’s:

If we were to scale the CD’s audio to the same average volume as the vinyl’s, we can literally see the missing data:

Of course, your eyes can’t be the judge here. Only your ears can choose the winner. Here are 15 seconds from “15 Step,” first from the CD, then the same sample from the vinyl:

Your browser doesn’t support the <audio> tag, or else you’re viewing via an RSS reader or the Tumblr dashboard. Here’s the raw audio: 15 Step Sample.

It’s not just Radiohead. This effect manifests on almost every modern mainstream release which had a vinyl edition: everyone from Arcade Fire to Ben Folds to Mars Volta. Maybe it’s time to hit up your local record store.

Scaling the mentor–apprentice relationship

Wed, 15 Jun 2011 08:37:00 -0700

Meet Practica Musica, an electronic tutor for music theory and ear training. Let’s not talk about its aesthetics; instead, I’ll use it to focus a couple topics broadly relevant in education.

From brush strokes to big picture

Practica Musica features a massive collection of interactive activities, organized by goal and ordered along appropriate difficulty curves. A few examples include: syncopated rhythm reading, chord progression ear training, correction of inscribed melody from audio playback, and rhythmic dictation. The publisher’s site hosts videos of some activities; the pitch error exercise is representative.

Now picture you’re some teenager with a guitar and inappropriately long hair. Your uncle gave you this app for your birthday, but see, you just want to write some rockin’ hooks. This blather about figured bass looks terrifically dull.

This stuff is not Hanon’s Virtuoso Pianist, though. It’s not an eating-your-vegetables proposition. These exercises aim to build intuition. You’re not going to be recalling what your music theory book says about harmonic progressions when you’re writing a hit single. Your internalized knowledge will puppeteer you toward natural results.

I applaud the diverse suite of exercises in Practica Musica, but it’s hit upon a common and decisive educational pitfall: a hazy path through the doldrums of boring intermediate goals to the student’s true desires on the other side. Inspiration is step zero. The student must understand how his hard work will pay off.

Practica Musica’s publisher mitigates the situation with a companion textbook that links the exercises into something like a narrated curriculum. But the textbook is utterly static, ignoring the incredible benefits explorable explanations can have in students’ understanding.

For instance: inline with its description of altered chords, the textbook could let the student drag a triad’s notes around on a staff to see and hear its variations. Or he could “scrub” through a progression by running his mouse over the chord letters “I IV V I”. Exercises which test concepts introduced in the book could be directly integrated in layout—or even made required to proceed—rather than passively referenced.

Responsive testing

One Practica Musica exercise tests aural identification of major thirds, minor thirds, perfect fourths, and perfect fifths. If you consistently misname minor thirds, it will play them more regularly. And it won’t let you proceed until you’ve answered a few correctly in a row.

Contrast this to an Algebra student who is being tested on his exponents but does not understand that 6² × 6⁴ = 6⁶. Regardless of whether he passes or fails the test, this represents a fundamental gap in his knowledge that may never be directly retested. Each subsequent exam will build upon this concept (less and less explicitly) until the student has either figured out his missing understanding or flunked out.

What if our tests were dynamically generated in response to each student’s strengths and weaknesses? How many struggling students missed just one fundamental concept—but so long ago that neither they nor their teachers understand the root of their confusion?

The Khan Academy has experimented with these ideas in classrooms. In his TED talk, its founder Sal Khan claims:

You see students who took a little bit of extra time on one concept or the other, but once they get through that concept, they just race ahead. And so the same kids that you thought were slow six weeks ago you now would think are gifted. And we’re seeing it over and over and over again. It really makes you wonder how much all of the labels maybe all of us have really benefited from were really just a coincidence of time.

We can respond to students’ progress even more effectively by exploiting spaced repetition and related techniques: present troublesome material repeatedly, but with increasing intervals between each exposure modulated by success rate. The exact curves remain controversial but could be approximated by machine learning.

From mentor to classroom

I’m convinced that mentor–apprentice education is the best way to learn most subjects. Unfortunately, it doesn’t scale at all. But if we understand what, specifically, makes this approach so effective, maybe we can emulate the results in classrooms.

A mentor will be exposed far more directly to his apprentice’s excitement and frustration. His student’s emotions, combined with his empathy, all but force him to paint the path I described through the doldrums of beginnerhood. To provide inspiration. A classroom teacher could do that, too; I just don’t think it’s recognized as a goal or component of their curricula.

Likewise, a student’s weak spots are much more apparent in one-on-one tutelage, and the format encourages a naturally flexible curriculum schedule. Exam metrics and automated reporting could enable classroom teachers to identify problem areas, and tools for responsive testing could address students’ individual issues by personalized spaced repetition.

Often I wonder: what would happen if just one classroom could have a dedicated software engineer who would shadow the teacher and build tools full-time to help him?

Presenting Without Comment

Tue, 31 May 2011 23:38:00 -0700

In the Vatican Museums, alongside myriad sculptures of gods, emperors, and scholars, stands this old fisherman. His singular presence among these great figures places this sculpture’s significance firmly in its statement rather than its subject. Given that the portrait took months or even years to create, the artist clearly felt some ferocious drive to immortalize an ordinary man—perhaps to glorify the everyday or the earthly. Stranger, since these artists generally depended on patrons, someone wealthy supported that message.

A little earlier in my trip, I saw at Pompidou a truly enormous canvas, painted entirely a rather pleasant shade of blue. Next to it, of course, was mounted a lengthy plaque bloviating the painting’s profound interrogation of audience expectations, art criticism, and the true definition of beauty. Which bloviating, fundamentally, mirrors my natural reaction to the old fisherman sculpture.

That led me to wonder: is it possible to present anything unusual (but true and without pretense!) without these reactions in its audience?

I suspect we can chase these levels of meta to arbitrary depth, if we account for the artist’s own awareness of this phenomenon. Say you decide to take a photo of yourself every day. So you’re about to take one, but you can just feel an ill-placed lock of hair across your forehead. Do you brush it aside? Or leave it be for a “natural” photo?

You decide to just take the photo, but wait: if you always look disheveled, aren’t you maybe making some other statement? Might you convey some kind of, like, hipster-disaffection-I-don’t-care-how-I-look-but-I-care-about-looking-that-way countenance? So maybe you’d better fix your hair… but only a little? Make sure not too many photos in a row are perfect? But wait—will people figure out that pattern?

Or, as Mills asked when discussing this with me: can I cover a wall with a massive monochromatic canvas just because I really like that color? And have anyone believe me when they ask?

I’m not sure I’d believe him. But maybe this is just some kind of social paranoia? “That you will become way less concerned with what other people think of you when you realize how seldom they do?”¹

Consider, though, as one closing point in this non-argument: on a Facebook wall, a self-portrait with an empty expression seems more affected than one with an enormous, impossibly celebratory grin.

c.f. Infinite Jest by David Foster Wallace. I know I’m years late to the party on this book; I beg your weary tolerance.

Leading photo of the Old Fisherman by Dennis Jarvis, licensed BY-SA.

The Little Addictions That Define Us

Tue, 24 May 2011 21:10:00 -0700

I returned from a magical jaunt through Europe craving vegetables.

In my own bed at last that night, I greeted my down pillow as a long-lost friend. The next morning, only great will could dislodge me from the piano bench.

A bit of a reverse culture-shock awaited my return: all the things I missed while on that incredible adventure!

Fan though I am of Italian cuisine, after a full day of carb-heavy, greens-light meals, I developed a headache not unlike that of coffee withdrawal. Which begs: could I be as addicted to veggies as I am to caffeine? This country’s people seem to enjoy their native diet.

I know a gym-going young man who feels bodily ill when he misses a day of exercise. Some of my coworkers suffer terrible anxiety when away from their email for more than an hour at a time.

So. Can I be addicted to stimulating conversation? Certainly my affect suffers if days pass without any. I’ve developed drug-like tolerance, too: as I meet increasingly interesting people, previously acceptable topics become banal.

Couldn’t I be addicted to—i.e. expect, demand, develop a tolerance for, require always more of—being surrounded by thoughtful design? Flattering lighting? My girlfriend’s touch?

People say that caffeine delivers a boost but also adjusts your baseline. That after a while, you’re by default less energetic, and the caffeine boost returns you to a once-normal level.

These other addictions (and “addictions”) function by the same principle: a shifted baseline or sense of the ordinary. The unquoted addictions arise, I suppose, when our acquired normalcy strays too far from reality’s, as with amphetamines. Though not so damaging, it can nonetheless harrow to glimpse the gap between our sense of normalcy and others’, our mild anomie. Used to socializing over beer? Ordinary in American colleges; literally criminal in Sudan.

Self-incited addiction can be useful. I’d love to have my friend’s “problem” with the gym! But, as in so many things, cognizance in this regard elucidates, illuminating both opportunity and concern.

An addiction to vegetables, of course, landing firmly with the former.

The Problem with Hard-with-a-Capital-H Problems

Tue, 10 May 2011 22:15:42 -0700

I sat down to my very first Caltech math problem set with the adorable exuberance that only a freshman could muster. Pencils sharpened, whiteboard ready, textbooks unwrapped, I read the first problem:

I read it again, hoping I’d missed something—additional constraints or guidance, perhaps? This was on page 15, and the few preceding pages yielded no tutorial for this type of problem. I blinked.

I did not attend a high school with an “of Math and Science” suffix like so many of my new classmates. We were taught math in the same way most high schools teach the subject: algorithmically. Apply this procedure to solve problems which look like that. Students had only to identify the type of the problem at hand, memorize the corresponding technique, and execute a mechanical process.

I was fairly certain I had not been taught a technique for problems of the Rational Set Membership Proof type. My notes had nothing of the sort. Subsequent lectures did not return to the subject.

When I visited office hours, the professor wouldn’t help me with the problem because its only challenge lay in discovering a critical insight which, once found, rendered it trivial. He solved some other simple proofs in front of me. I nodded along: each step was indeed true. But how on earth did he generate the steps? He shrugged and offered to do some more examples. Each seemed just as obvious as the last—but only in hindsight.

I later learned that my professor’s insights came from intuition. No one could explicate or decompose that sense for me, so I wrote these problems off. Who needs proofs to write software?

But I didn’t realize that these proofs are hard in the same sense as the most interesting problems in all fields: they have no obvious, closed-form solution. They’re Hard with a capital ‘H,’ and the contents of their toolbox apply just as well to, say, design problems.

Much to my dismay, I also learned at Caltech that a field’s greatest masters are likely not its greatest teachers. Pedagogy requires a different art entirely. So how can we teach insight, intuition, analytical thought?

My answer (as to most things): cognizance. Let’s articulate the inner monologue in solving a Hard (though fairly simple) analytical problem.

The Snowman Problem

You’ve got a 5×5 grid of Unicode snowmen at a tea party. When you blow a whistle, every snowman must move to a new space. But a snowman can only move to a cardinally adjacent square, and we can’t end up with two snowmen sharing the same square.

How should the snowmen move to do this? Or prove it’s impossible.

Start by trial and error. Draw arrows on a whiteboard. Tear bits of paper into ☃ stand-ins. Make little groups that can move successfully. But no matter how you arrange them, one seems left without nowhere to go.

By this point, we might suspect impossibility. But how can we prove it?

Let’s try decomposing the problem. How about a 2×2 grid? In that case, the solution is simple. Maybe I can use this simple solution to generate more complex solutions: for 4×4 grids, I can repeat this smaller scheme in each corner. But that only works for boards with even widths and heights.

So, generalizing, maybe odd-sided boards can’t be solved! But if we try the next-largest grid, we see that 2×3 can be readily solved with a slightly altered “looping” approach.

3×4 is just two 2×3 boards next to each other, so that’s solved, too. Studying composition a little more, you’ll find that we can combine 2×2 and 2×3 boards to solve any even-sided or even-by-odd-sided board.

But 3×3 seems just as hard as 5×5. Notice that if we could solve 3×3, we’d be done, since we could make a 5×5 from it by adding a 2×2 and two 2×3s. So we’ve just reduced the problem to a simpler one.

What makes 3×3 so hard? Intuitively, there’s always an “odd man out.” What else is different about 3×3 and 5×5 from all the other cases we’ve considered? Well, both side lengths are odd. The boards that worked have at least one even side length. And one of the properties of even numbers is that an even number times any other number is always even. Aha: “hard” boards have an odd number of squares.

How else does parity apply to this problem? Consider a snowman’s coordinates. A snowman at (1, 1) must move to (0, 1), (1, 0), (2, 1), or (1, 2). A snowman at (2, 2) moves to (1, 2), (2, 1) (3, 2), or (2, 3). Notice: either the X or Y coordinate must change parities when moving.

Let’s consider this result graphically. Overlay a checkerboard over our problem. Now it’s clear: a snowman on a black space must move to a white space, and vice versa. But when there are an odd number of squares, there will be one more square of one color than the other.

Since only one snowman can occupy each space, at least one snowman will have nowhere to go when the colors are uneven.

A 5×5 board has an odd number of squares, meaning it has more tiles of one color than another. Which will make the move impossible. QED.

I suspect that most who excel at Hard problems have no idea how they solve them. This example illustrated a number of key approaches: reduction, composition, (failed) induction, synthesis, comparison and contrast, and visualization. Once mastered, we could use these techniques to solve problems in any number of fields.

I realized recently that some of these techniques are a bit like playing Set! with the problem space. Which parameters restrict us most? Which parameters are similar? Are some irrelevant? I imagine we could devise more apparently innocent games to train lateral thinking.

Permission to Suck

Tue, 03 May 2011 19:38:00 -0700

You’re new to writing, but you’ve been working on a short story for weeks on end. When you meet up with friends, they ask what you’ve been up to; given all the time you’ve spent on your story, you bring it up each time you gather. And each time your expectations grow: now every word must be perfect. What else would be worth turning down all those nights out?

Instead, imagine that you joined a group of other novice writers. Each week, you all share a few pages of your work. After a session or two, you notice that your peers suffer from the same awkward diction. This one uses too many compound sentences; that one struggles with adverbs. And just like that, you’ve been given permission to suck.

Creative work often flops wildly on the desk, too slippery to shape to our will. It hides under stress; it makes funny faces when our backs are turned. A hint of despair quells the wildest brainstorm. I do my best creative work when I feel utterly safe, even though the ideas I’m considering might be anything but. It must be okay to fail.

As Merlin Mann suggests, don’t write one song by the end of the week—write five. Tell yourself not to worry: none of this is final. You won’t be showing anyone. Then marvel as you generate something raw but more passionate than the pristine work you polished carefully for weeks. And in no time.

My friend David told me about a charcoal drawing class he took years ago. Students were given two canvasses and instructed to fill both by the end of the period. Each student would draw his actual piece on one canvas; ideas too outlandish for his real output would go on the other. But: students were allowed to switch which canvas held their “real” drawing at any time. It shouldn’t come as any surprise that by the end of class, most of them did just that.