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Today I wanted to write more about giant clams and their astonishing ability to “farm” algae within their body (and then live off of the sweet sugars which the algae produce).  I still want to write about that, but it proving to be a complicated subject: giant clams mastered living on solar energy a long time ago, and we are still trying to figure out the full nature of their symbiotic systems.

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Today, instead we are going to look at the phenomenon which gives the mantles of giant clams their amazingly beautiful iridescent color. It is the same effect which provides the shimmering color of hummingbird feathers and blue morpho wings, or the glistening iridescence of cuttlefish.  All of these effects are quite different from pigmentation as generally conceived:  if you grind up a lapis lazuli in a pestle, the dust will be brilliant blue (you have made ultramarine!) but if you similarly grind up a peacock feather, the dust will be gray, alas! This is because the glistening reflective aqua-blue of the feather is caused by how microscopic lattices within the various surfaces react with light (or I suppose, I should really go ahead and call these lattices “nanostructure” since they exist at a scale much smaller than micrometers). These lattices are known as “photonic crystals” and they appear in various natural iridescent materials—opals, feathers, and scales.  Scientists have long studied these materials because of their amazing optic properties, however it is only since the 1990s that we have begun to truly understand and engineer similar structures on our own.

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Physicists from the 19th century onward have understood that these iridescent color-effects are caused by diffraction within the materials themselves, however actually engineering the materials (beyond merely reproducing similar effects with chemistry) was elusive because of the scales involved.  To shamelessly quote Wikipedia “The periodicity of the photonic crystal structure must be around half the wavelength of the electromagnetic waves to be diffracted. This is ~350 nm (blue) to ~650 nm (red) for photonic crystals that operate in the visible part of the spectrum.”  For comparison, a human hair is about 100,000 nanometers thick.

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The actual physics of photonic crystals are beyond my ability to elucidate (here is a link to a somewhat comprehensible lay explanation for you physicists out there), however, this article is more to let me explain at a sub-rudimentary level and to show a bunch of pictures of the lovely instances of photonic crystals in the natural world. Enjoy these pictures which I stole!

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But, in the mean time don’t forget about the photonic crystals! When we get back to talking about the symbiosis of the giant clams, we will also return to photonic crystals!  I have talked about how ecology is complicated.  Even a symbiotic organism made up of two constituent organisms makes use of nanostructures we are only beginning to comprehend (“we” meaning molecular engineers and materials physicists not necessarily we meaning all of us). imagine how complex it becomes when there are more than one sort of organism interacting in complex ways in the real world!

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Today, Ferrebeekeeper ventures far far beyond my comfort zone into that most esoteric and pure realm of thought, mathematics.  But don’t worry, we are concentrating on topology and geometry only for long enough to introduce a beautiful, intriguing shape, the torus, and then it is straight back to the real world for us…  Well, hopefully that will prove to be the case–the torus is anything but straight.  It is, in fact, very circular indeed, and, as we all know, it has at least one big hole in it….

Wikipedia defines a torus as “a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle.” That’s hard for me to wrap by head around but the meaning becomes much more comprehensible in the following illustration.

So a torus is a circle wrapped around in a circular path.  You can find a variety of other ways of mathematically representing the torus here, but the simple definition suits our purpose.

One ring toroid them all....

I admire torus shapes because I think they are very beautiful.  Just as the golden ratio and the Fibonacci sequence are aesthetically appealing, there is a pleasure to merely beholding or touching a torus: ask anyone who has contemplated a ring, a golden diadem, or a cinnamon donut.  Talk to an indolent adolescent sprawled on an inner-tube bobbing on the surf, and you will immediately grasp the hold that toroids have on humankind.

And beyond humankind....

In addition to its obvious aesthetic merits, however, there is a mysterious aspect to the torus, a hint at hidden dimensions, negative space, and infinity.  Kindly contemplate the old eighties video game Asteroid (you can play the game here if you are too young to remember: amusingly, your ship is an “A” which reminds me of Petrus Christus’ enigmatic painting).  If you pilot your ship to the far left of the screen you emerge on the right side of the screen: the flat screen represents a cylinder. However, to quote Bryan Clare from Strange Horizons, “the bottom of the screen is connected to the top as well. This has the same effect as if the screen were rolled into a cylinder, and then bent again to glue the two circular ends together, forming the familiar donut shape.” So, when you play asteroids you are trapped in a miniature toroid universe which appears 2-dimensional. Try to blast your way out of that!

The following famous math problem further illustrates the nature of the torus.  Three utility companies need to connect their respective lines (gas, water, and electric) to three different houses without ever crossing the lines.

Connect each utility to each house. Don't cross the lines.

The problem is impossible on a two-dimensional Euclidean plain and even on a sphere, however topologists realized that if you poke a hole through the plane or the sphere (thereby making it a torus) the lines can be connected.

To finish the article here is a movie of a torus being punctured and turned inside out. The result is a torus of the same dimensions but with reversed latitude and longitude.  It’s hard not to love such a funny shape.  But it is hard for me to wrap my mind around the larger implications.  I think I’m going to stop trying and head off for some donuts.

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