Random ideas and objects that piqued my interest...

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Metamaterials, evanescent waves and microscopy

Telecentric Lens

An illusion created using a Telecentric Lens. Photo cred:


The term "Glory" describes an atmospheric phenomenon. A Glory looks sort of like a small, round rainbow, but it is caused by different and more complicated reasons. Glories are caused by the "backscattering" of light from very fine droplets of mist-- the more uniform the sizes of the droplets, the more distinct the rings of the Glory. The water droplets are usually about 4 - 25 micrometers in length. Not very much is known on why or how Glories form, but there are two main theories that speculate the cause of the effect: the Mie theory and the Debye theory. These theories allow the Glory effect to be simulated on a computer. Some scientists think that Glories may be caused by classical wave tunneling, a process that actually involves Evanescent waves! For more information about this, read this excellent Scientific American article.

Glories can only be seen if the observer is standing at the antisolar point, which means that a straight line can be drawn connecting the sun, the Glory, and the observer. Often, the Brocken Spectre is observed as well. The Brocken Spectre is the projection of the viewer's shadow onto the Glory; the name derives from rumours of mysterious, ghostly sightings of such phenomena on Brocken Peak, the highest mountain of the Harz Range in Germany. No doubt observers once thought the strange shadows to be ghosts!

Below are some pictures of the effect, and also some links to useful websites.

Photo cred:

In the above image, the Brocken Spectre can be observed. What's interesting is that if three people are all observing the Glory from slightly different viewpoints, they will only see their shadow on the Glory, not the shadows of all three observers. I guess this is because the viewer must be directly in line with the sun and the Glory to see it, and if three people are in different places, the lines are different, if that makes sense.

Photo cred:

The above image shows a Glory seen from an airplane. Apparently, such sightings are fairly common, though in the days before flight Glories could only be glimpsed by mountaineers braving the slopes of mist-shrouded peaks. Below is an excerpt of a 1748 mission's account of one of the first recorded sightings of a Glory:

    “A cloud that covered us dissolved itself and let through the rays of the rising sun.… Then each of us saw his shadow projected upon the cloud.... What seemed most remarkable to us was the appearance of a halo or glory around the head, consisting of three or four small concentric circles, very brightly colored.... The most surprising thing was that, of the six or seven people who were present, each of them saw the phenomenon only around the shadow of his own head, and saw nothing around other people's heads.”

    -- scientificamerican0112-68.html

I can only imagine seeing something like that, in a time when science was interwoven with magic in people's minds, and wondering which of the two I had seen...

Lego Diffraction Grating Projector

This is the coolest thing ever! I want to make this so much! Someone made a diffraction grating projector that (as it sounds) projects really awesome diffraction patterns onto walls! Here is the link; watch the video: Click me!

Update, 7/23/12: My brother and I made the projector (a simpler version) out of Legos. You can see it here.

Moiré Effect

The moiré effect occurs when two patterns overlap each other and create points of increased and decreased intensity. One can observe it by putting two screens on top of one another. My journal page goes into some more detail about this effect.

  • Here someone has made moiré patterns appear in mid-air with an empty cigarette lighter and a laser pointer. It looks pretty cool!

  • Here is an article that studies a method of cryptography, namely stenography, using moiré patterns. Basically, when the "wheel" is spun at the right frequency, the image appears. It's really amazing: Image Hiding Based on Circular Moiré

  • The moiré effect makes quite a nuisance of itself in photography. An effect called "aliasing" occurs-- see my journal for more-- and many image editing softwares have methods of eliminating the colorful stripes produced when a camera tries to image a very fine pattern at high resolution. Here is an online article that discusses the moiré effect in various cameras.

  • A useful source for understanding aliasing and moiré patterns in cameras: Click me!

  • These are some really interesting examples of moiré patterns. I can't wait to try modeling them on Mathematica!

Some other papers:

Moiré Patterns and Metal Fatigue

When I talked to a neighbour over the weekend about project ideas, he suggested that I think about moiré patterns and topography (see some of the papers listed above for more on that). One possible application is using moiré patterns to detect metal fatigue in a non-invasive way. Moiré patterns are already used to detect minute amounts of strain: there is a setup in the lab right now that demonstrates this in a memorable way. A laser beam is sent through a beam-splitter. One part of the beam is projected directly onto the wall, and the other part is bounced around by several mirrors before it recombines with the first one. Because of this, the beams are slightly out-of-phase with one another, and what's projected onto the wall are moiré fringes. These moiré fringes can show extremely small amounts of stress. For example, if you put a rubber band over one of the mirrors and pull it very slightly, the moiré fringes shift quite perceptibly. You can even bang on the table and see a shift! Because of the tiny details that moiré patterns show, perhaps they could also be used to detect fatigue and stress in metals, maybe through surface contouring or something. It's an idea that I unfortunately don't have time to pursue any longer, but perhaps another student will one day take up the idea and go with it!

Below are some useful links to follow if you're interested in this:

  • This site provides a really nice explanation of metal fatigue, and it's easy to follow. Check out the references mentioned in it as well!

  • Holographic interferometry for monitoring and controlling laser shock peening. This is a patent for something sort of similar to metal fatigue... maybe. It's hard to understand because of the formal language required.

  • This is a google book link that has lots of useful papers and references in it... Unfortunately, you can't see the papers in their entirety because the book preview does not allow it. Further search for the papers revealed that you have to pay for them everywhere, but still, maybe some way of viewing them can be found.

  • A low cost shadow moire device for the nondesctructive evaluation of impact damage in composite laminates. This is a report that has pertinence to metal fatigue and moiré patterns. It seems that the author focused on composite laminates rather than metal, but perhaps the ideas can be extended?

Quasiperiodic Structures and Quasicrystals

Quasiperiodic structures are structures that have an order to them, but are not periodic. That means that they do not have "translational symmetry"-- if you were to put one quasiperiodic structure on top of another one identical to it and move the top structure from left to right, there would be no other place where the structures were in perfect alignment. In contrast, consider a perfectly periodic structure, say, a picket fence or a sine wave. The structures would be in alignment at every interval of the period.

One famous example of a quasiperiodic structure is the Fibonacci sequence. Clearly, the Fibonacci sequence is not periodic, but one can call it quasiperiodic because it has a definite structure to the way it is formed. In mathematics, quasiperiodicity comes up in tiling-- Robert Berger constructed a large set of tiles (20,000)that could tile a plane in a non-periodic manner after his teacher, Hao Wang, postulated that if a set of tiles can tile a plane, they must also do it in a periodic fashion. Later on, Robert Penrose reduced 20,000 to just 2, creating non-periodic tiling called "Penrose tiling."

Quasiperiodicity can also be observed in various crystals, called "quasicrystals." The first quasicrystal was observed in 1982 by Dan Shechtman in Aluminum-Manganese alloys. He observed this by looking at the strange diffraction patterns that the crystal produced. Many other compounds that have quasicrystaline structure have since been found. Actually, because of the existence of quasicrystals, the definition of a crystal had to be re-worded to include the concept of periodicity that had previously always been taken for granted.

This is the electron diffraction pattern of a Ho-Mg-Zn quasicrystal. Photo cred:

Above is an example of Penrose Tiling. Photo cred: same as previous.

The only known naturally forming quasicrystal on earth was found in Russia, in the Koryak mountains. Scientists speculate from its composition that it came from a meteorite!

Quasicrystal believed to be from space. Photo cred:

One thing I would like to find out: can moiré patterns be made with quasiperiodic structures? I have only seen them produced with periodic structures, but what would they look like if they were made with patterns that had quasiperiodicity? Here is a patent that describes eliminating moiré patterns (in the sense of aliasing, I believe) by using quasiperiodic structures. Does this mean that moiré patterns can't be created? Hmm... I hope not.

"Humanizing" Sound

Computer-generated music has always sounded a little mechanical to the human ear, mostly because humans can tell that the computer is playing exactly on the beat. In contrast, humans play music slightly off the beat, and though the beat is only off by a few milliseconds, our brains can tell the difference! Sound editing softwares have sought to counteract this by introducing "white noise" into computer-generated recordings: essentially introducing slight random deviations from the beat in order to make the music sound more realistic. This is known as "humanizing" the music. Researchers have wondered, however, whether a human's deviations from the beat are truly random or if there is actually some kind of correlation or mathematical law behind our inexactitude. Recently, the July 2012 issue of Physics Today came out, and one article reported that in fact, human deviations from the beat follow a power law! I will summarize the article below, but the full version can be found here.

The authors recorded a professional drummer from Ghana for about five minutes while the drummer played to the beat of a metronome. The mean deviation from the beat was about -16 ms, which means that on average, the drummer anticipated the metronome by a little. Below are the graphs from the article:

Photo cred:

The above graph shows 2 seconds of the recording in greater detail. The red lines represent the drummer's beat, and the green, the metronome. One can see that around beat 284, the drummer anticipated the beat, but that he did so less and less as time went on until he played nearly on the beat at 288.

Photo cred:

This graph shows beat deviation (dn) as a function of beat index. There were 1030 total beats. From the graph, it is clear that the drummer oscillated between anticipating the beat and hitting the drum after the beat. This trend implies that rather than being random, a human's deviations follow a law. In fact, after further analysis, the beat deviations appear to follow a power law.

In some places in the graph, it can be seen that the drummer is extremely off the beat (relatively speaking), perhaps in anticipation of the metronome click. Immediately following those points, however, the drummer does not suddenly snap back to the beat. Instead, the deviation persists for many more beats, and the drummer slowly makes his way back to the metronome click, and then beyond it. This implies that one beat deviation has a lot of influence (for a limited amount of time) on the successive beat deviations. According to the article, this shows that a human's brain has a good memory for temporal intervals; the same phenomenon has also been seen with spatial intervals, as anyone who as ever tried to reproduce an inch with their fingers will know!

To analyze graph b, the authors tried to characterize how quickly the correlations between beats vanished-- in other words, how long does it taks before one beat deviation no longer has any influence on the rest? To put this into context, if a deviation has no influence on any following deviations, then the deviations are purely random. This is the white noise effect that recorders have been employing for some time. If, on the other hand, a deviation has some effect on the successive deviations, there must exist some sort of correlation between the beats; the deviations cannot be purely random. The authors figured out the correlations by expressing graph b as the sum of various sine and cosine functions of varying frequencies, a Fourier transform. A Fourier transform switches the time axis with a frequency axis, in order to better see some correlations. From that, a power spectrum was created. The power spectrum was found to have the form

S ∝ 1/fα

where α ≈ 1. That's a power law! As α approaches 0, the frequency, f approaches 1, and equal power is generated at each frequency. There is no correlation between the deviations. From 0 < α < 1, correlation decays more slowly. The times series has long-range-correlations (LRCs) if α is between 0 and 2.

I would like to explain more about what a power law is, because they seem to crop up everywhere in nature, but this has already gone on for long enough! Perhaps another time. Suffice it to say that there strongly seems to be a correlation between a drummer's beat deviations, and that from now on, audio editing software will have a new and better way to "humanize" music.

To listen to some samples of humanized music, click here.