Ken’s Blog

Today Alex posted the following comment on this blog:

Thank you, but please allow me to ask one silly question. I think in many aspects of life, what scientist does is defining a problem, then finding out a solution. It seems like there is”solution” now. are there any problems we can solve with this fact, or it’s just a pure observation?

At first I was stumped. To me what is important about the relationship between the pyramid and the cube is that it is beautiful. It had never occurred to me that such beauty would require “defining a problem”.

And then I recalled something René Magritte once wrote, that his painting Le Modele Rouge (below) concerned “the problem of shoes”.

 
I do not know whether my joyous discussion about pyramids and cubes will end up solving any problems.

Meanwhile, it’s good to know that Magritte was thinking about the problem of shoes. Although I’m not entirely sure this makes him a scientist.

As requested by Alex, here is an old fashioned proof that packing spheres into a cube N on a side uses the same number of spheres as arranging them into an N-high hexagonal pyramid.

We already know that a 1³ cube uses just a single sphere — the same as a “pyramid” of height 1 — which is just a single sphere.

We just need to show that adding an Nth layer to a hexagonal pyramid uses the same number of spheres as incrementing from a cube of width N-1 to a cube of width N.

To get from an (N-1)³ cube to an N³ cube requires N³ – (N-1)³ more spheres. That’s N³ – (N³ – 3N² + 3N – 1) spheres, which simplifies to 3N(N-1)+1.

If we can show that a packed hexagon with N spheres on a side uses the same number of spheres, then we’re done. We do that as follows:

To go from our single sphere to a packed hexagon with 2 spheres on each side, we need to add 1 sphere for each of the hexagon’s six sides.

Similarly, to go from that to a packed hexagon with 3 spheres on each side, we need to add 2 spheres for each side of the hexagon.

So to build up to a packed hexagon with N spheres on a side, we need to add 1 + 2 + …. (N-1) spheres for each side of the hexagon.

The sum of the numbers from 1 through N-1 is N(N-1)/2 (here’s the proof on Wikipedia).

Putting it all together, we have six of those sums (because the hexagon has six sides), plus the one sphere in the center.

Which gives us 6×( N(N-1)/2 ) + 1, or 3N(N-1)+1.

As they say in latin class, QED. But I think the visual proof is a lot more elegant — and it didn’t require linking to the Wikipedia to explain picky details. 🙂

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?