Ken's Blog

It’s amazing how much easier it is to make something clear in a visual proof — using pictures rather than equations. Just for comparison, I wrote down the proof I described yesterday using equations instead of pictures, thinking I would post that today.

I encountered two problems: (1) Most people who read this blog won’t be able to follow the equations. (2) Even if they did, there are too many steps.

The biggest difference is at the end, when I showed how those three faces of the cube (which I colored red, green and blue) magically flatten out into three parallelograms to form the hexagon (after you add that one sphere in the center).

From the picture, it’s immediately obvious how this works. But not from the equations. So a story that is simple and elegant when told in pictures becomes a lot less simple and elegant when told with equations.

I wonder whether there might be some sort of hybrid way of describing such things — something half way between equations and pictures, so that you can get the best of both.

The other day I mentioned how if you arrange spheres into a pyramid of hexagons, you get the same number of spheres as if you’d arranged them into a cube. Today I thought it might be nice to prove it.

Well, supposed we have just one sphere. We can think of this either as the tip of a pyramid or as a 1×1×1 cube. What we need to show is that every time we add a layer of hexagons to the base of the pyramid (to make a bigger pyramid), we get the same number of spheres as if we’d arranged the spheres into a cube.

So first, let’s ask how many more spheres there are in the N³ arrangement below right than in the (N-1)³ arrangement below left:

What we need to show is that this difference is the same as the number of extra spheres needed to add an Nth layer to a pyramid of hexagons. In the image below, the new layer is highlighted in gold:

Going back to the cubes, let’s color all of those extra spheres in the N³ cube:

Notice that we get three extra walls — each with N×(N-1) spheres — plus one extra sphere in the corner. But that’s exactly how many spheres we need to make a hexagon with N spheres on each side:

So there you have it. We just showed that each additional hexagon layer contains exactly the difference between an (N-1)³ cube and and an N³ cube. So now we can be sure that a pyramid of hexagons, stacked N high, will always add up to exactly N³ spheres!

The following Venn diagram seems simple enough. Two sets intersect to create a set that is smaller and more restricted:

Yet when it comes to people, this logic is often completely wrong. The place where Lennon and McCartney worked creatively together was far larger than the creative space each would ever claim on their own. The same goes for Gilbert and Sullivan, Abbot and Costello, Ashman and Mencken, Siegel and Schuster, Greenwich and Barry, Dannay and Lee, Rodgers and Hammerstein and many others.

And so we have what seems like a contradiction: When it comes to human creativity, the set of intersecting talent can be far larger than the set of talents possessed by either participant.

This week I learned a wonderful thing: If you assemble spheres to make hexagons of different sizes (up to N spheres on a side), and then stack up those hexagons to make a pyramid, you get exactly the same number of spheres as if you made a cube with N spheres on a side.

I was so delighted by this that I wrote a little computer program to visualize it.

Below you can see the cases where the hexagons contain up to two, three or four spheres on a side, respectively. Next to each picture you see spheres arranged into a cube. In each case (and in fact in every possible case), the number of spheres is the same on the left and on the right.

How cool is that!

There is an animated Japanese film in town that I’ve really wanted to see, and a friend has also really wanted to see it — so we’ve decided to go later this week.

The film is being offered in both dubbed and original (subtitled) version. When we compared notes we found out that we both greatly prefer the original version. Which is interesting, because neither of us speaks Japanese, so we will both be reading subtitles to follow what’s going on.

Of course the case for seeing the original version is most compelling for a live action film. In that case, the people you see up on the screen are, literally, the voice actors. It is understandable that we may prefer to hear Alain Delon or Sophia Loren speak their own lines. I once inadvertently rented a dubbed version of “Et Dieu… créa la femme”. Every time Brigitte Bardot opened her mouth, a horrible brassy American voice spewed forth. I had to rent it again in the proper language, just to restore my sanity.

Yet who is to say that one language is preferable to another for an animated film?

I think my preference comes down to the following: Japanese and American culture are very different (“Lost in translation” is more than just the name of a Sophia Coppola film). The intention of the writer and director, with their particular cultural sensibilities, will likely be accurately reflected in the original voice acting. This is much less likely to be the case in dubbing sessions half a world away by voice actors who grew up in America.

This is particularly an issue in the case of Japanese animation, where subtlety of emotion is paramount. One of the wonderful things about Japanese animation for children is that it starts with the assumption that children are very intelligent and emotionally sensitive, and can appreciate complex and contradictory relationships.

I’ve seen American kids in screenings of Japanese films. It’s like seeing food given to someone who has been starving. It’s clear that the children in the audience are amazed and delighted that somebody has made a film that understands and trusts in their intelligence.

By comparison, American animated films — even the best ones — treat kids pretty much as idiots. This is not a criticism of American animation, but rather a reflection of the patronizing way American culture acts toward its children in general. I suspect most Americans are not even aware it is possible to make an animated film that does not talk down to children.

GIven all that, and given the privilege of seeing a good Japanese animated film, who would want to take the chance on viewing it with American voices?

One of the readers of this blog sent me a very nice and friendly email today, pointing out that the site has an extremely retro design, and that perhaps it might make sense to update to something more modern.

I thus find myself on the horns of a dilemma. On the one hand, I enjoy these sorts of retro things. For example, the smartest phone I am currently using looks like this:

and the telephone I use at home is this charming replica of a candlestick phone from the 1920s:

When are old fashioned things charming, and when are they just old? People tend to like my candlestick phone, presumably because it evokes a lost world of long ago. Yet my Samsung flip-phone seems to inspire something closer to bemusement.

In our modern cyber world, so relentlessly focused on the latest and greatest, is it valid to employ a retro web page design?

Or is it, in the immortal words of Seth MacFarlane, “still too soon”?

Computer games have a concept of “easter eggs”. These are hidden puzzles that are not officially part of the game. If you make a move in a certain way, or open a cabinet at just the right moment, something surprising might happen — a hidden message perhaps, or a spectral visitation from the game’s creator.

There is an entire sociology around easter eggs. Fans share them with each other, webpages are devoted to their secrets and mysteries, and occasionally a highly inappropriate easter egg pops up that was never intended for the release version of the game (much to the embarrassment of the game’s publisher).

I wonder whether it would be possible to design a computer game solely around easter eggs. Could one build a game for which the entire reward structure consists of finding arcane hidden messages and surprises?

If such a thing already exists, I suspect somebody reading this will helpfully point it out.

Yesterday I wrote, somewhat tongue in cheek, about how easy it would be for future wearable augmented reality technology to maintain the gulf between the “haves” and “have nots” of this world. In my dystopian scenario, high-tech wearables would augment the reality of economic inequality.

But suppose we could rethink the premise of augmented reality, from the ground up. Is there a way we could set it in a direction that would actually promote greater economic equality?

The largest potential power-up I can think of is in education. Perhaps inexpensive wearable devices could make it easier for high quality education to reach a greater portion of the world’s population. After all, wearable augmented reality has the potential to put teachers and students together, face-to-face, across large distances, or to simulate playful environments for learning and exploration that might be cost-prohibitive to build with laboratories of bricks and mortar.

Doesn’t the true potential of a child reside in what his or her mind can learn to do? Just as the increasing fluidity of information has helped to bring down dictatorships and repressive regimes, perhaps another leap in technology will make it possible to bring down the scourge of inadequate education.

What could possibly be more important to the world’s economy?

Let us fast forward a bit to a time when those of us in the developed world are wearing our forthcoming variants of Google Glass. Bear with me here.

Rather than staring down at iPhones as we walk blithely into oncoming traffic, we will be looking forward, heads held high, while augmented reality displays give us power-up knowledge of the world around us, and unobtrusive bone-conductance microphones whisper messages that only we can hear.

In this brave new world, our personal video cameras will be recording every passing moment — buildings, stores, cars, passers-by — and helping us to interpret the many things that pass into our field of view.

Yet we will not be alone. For our fashionable head-mounted device will be connect to the internet, and the internet is connect to the world.

Somewhere far away, where the average wage is perhaps one hundredth our own, some brilliant individual — perhaps a young person of unbounded promise but with no real future — will be paid three dollars a day to interpret what we see, to whisper in our ear, to point out objects in our view, suggest paths we may take, opportunities for purchase, ways forward that we ourselves may never have thought of.

To this young person the opportunities will seem like a dream. For a wage far greater than anything she could have imagined, she will see sights beyond the dreams of anyone in her third world village. A glimpse into a fabulous world where everyone can afford a car, where a movie costs more than a month’s salary, and where universal education is guaranteed.

We, of course, will not quite be aware of this person’s existence. We will simply chalk it all up to the magic of technology. “How,” we might ask ourselves, “did my augmented reality device know to send me to this particular store on this particular day? It is truly marvelous what they can do with computers these days!”

I am a big fan of Ray and Charles Eames. In the world of design they were perhaps the ultimate power couple, originating one ingenious idea after another. The Eames Chair has deservedly become an icon of modernist design.

Similarly, their 1961 interactive museum exhibit Mathematica: A World of Numbers … and Beyond is the best advertisement for the sheer delight of mathematics this side of Vi Hart’s videos — and a very high bar for the Museum of Mathematics to strive for as it continues to mature.

The there is one thing for which they unfairly get too much credit. Just today a colleague referred to their iconic work Powers of 10, a lovely short film that first zooms out from human scale to the universe, and then zooms in to the atomic world (and which was subsequently released as the book “Powers of Ten” by Philip and Phylis Morrison).

I say they get too much credit because the entire concept for this film was borrowed from Cosmic View, a 1957 illustrated essay by Kees Boeke.

Not surprisingly Boeke was a pioneer in many other ways. For one thing, he originated the concept of a Sociocracy in education. You could look it up.