Doc Madhattan: June 2012

Multiscale gigapixel photography

Sometimes, using the Android application Wondershare Panorama, I try to shot some panoramical photos. The result is good, but it could be better, like everything: indeed the actual camera (also in our smartphones) work near the fundamental limit of 1–10 megapixels for millimetre-scale apertures⁽¹⁾. In theory⁽²⁾ we are able to upgrade this limit. If we define $SW$ like the upper limit for the number of data channels which can be handled in parallel⁽²⁾, after some calculations, Adolf W. Lohmann found that without aberration, it would increase quadratically with the scaling factor $M$⁽²⁾. This result is different from the purely geometry result that states $SW$ indipendent from scaling.

That result cannot be realistic, otherwise, very long focal length lens systems would be fairly useless. In practice, the apertures of these long systems are reduced, more or less according to the empirical rule⁽²⁾

Now a team of researchers develop a new photographical device that can resolve at 50 gigapixels: AWARE-2 camera^{(1, *)}.
And below there are some examples of its capabilities⁽¹⁾:

Sunland

In the image the solar power generated by country (via wolframalpha)

A quick tour in LHC

(via cosmiclog)

Turing patterns in coats and sounds

One hundred years ago was born Alan Turing. He was known essentially for his role during the Second World War: he encrypted the Enigma machine. But He is also a brillant mathematichian and today I would try to describe one of his better model, that today biologists are applying to their research field.

A vibrating object tends to vibrate at certain preferred frequencies, called natural frequencies. These frequencies depend on properties such as the density and tension of the vibrating object. Mathematicians and physicists can determine the natural frequencies of an object when they know the values for these other properties. This article describes new work being done to solve the reverse problem - calculation of properties such as density when the natural frequencies of the object are known.

Elizabeth Veomett about Good Vibrations by Joyce R. McLaughlin. American Scientist, July - August 1998

Cymatics was the study of the waves' patterns. The first interested in this subject was Galileo Galilei:

As I was scraping a brass plate with a sharp iron chisel in order to remove some spots from it and was running the chisel rather rapidly over it, I once or twice, during many strokes, heard the plate emit a rather strong and clear whistling sound: on looking at the plate more carefully, I noticed a long row of fine streaks parallel and equidistant from one another. Scraping with the chisel over and over again, I noticed that it was only when the plate emitted this hissing noise that any marks were left upon it; when the scraping was not accompanied by this sibilant note there was not the least trace of such marks.⁽¹⁾

Some years after Galilei (1680), Robert Hooke

was able to see the nodal patterns associated with the modes of vibration of glass plates.

In 1787 Ernst Chladni repeated Hooke's experiments and published his results in Entdeckungen über die Theorie des Klanges (Discoveries in the Theory of Sound). Finally in 1967 Hans Jenny published Kymatik (Cymatics), a book based on Chladni's work, and cymatics became an interesting science, in particular for artists! For example, Jeff Volk, poet, writes an interesting article about Jenny and the pattern of sound: From Vibration to Manifestation (pdf). In particular he presents an image from Alexander Lauterwasser's Water Sound Images

Pay attention: following Lauterwasser and Volk we could explain the pattern of leopard's coat, but the first explanation come from one of the Alan Turing's paper The Chemical Basis of Morphogenesis⁽²⁾. In this paper Turing is interested in the formation and development of path in biology (the so called phenomenon of morphogenesis).

Any pattern or shape observed in nature, even though governed by genetics, is most likely produced by an unknown mechanism. Thus, determining these mechanisms that generate pattern and shape in organisms is an important goal of theoretical biologists.⁽³⁾

The most used model for this type of systems is the reaction-diffusion system, described by the following formula: \[u_t = d \Delta u + f (\gamma, u)\] where $u$is the position of the gene, $u_t$ the diffusion speed, $d$, $\gamma$ real constants. We can write two similar formulas for every morphogene in the system.

Reaction-diffusion models are particularly compelling with regard to their ability to capture complex evolving patterns.⁽³⁾

Similar equations are really complex to analyze, due to their local and general phenomena. In an intuitively way, we can see the pattern formation like a challenge between reaction mode and diffusion mode. In his paper, Turing

suggested that a system of reacting and diffusing chemicals (morphogens) can interact to produce stable patterns in concentration (Turing patterns).⁽³⁾

Or in a more simple way: we can imagine the presence of an activator molecule of the morphogenesis. This molecule will be produced more and more thanks an autocatalysis process, but the activator will produce also an inhibitor, that will limit the production of the activator. The dynamics between activator and inhibitor will generate the pattern observed in nature (for example tigers' stripes). Tipically the two diffusion velocities are different, so we can explain the great variety of patterns.

The shield of Captain America

After the releasing of the movie Captain America: The first Avenger in 2011 by Paramount Picture, Suveen N. Mathaudhu, the Program Manager responsible for Synthesis and Processing of Materials at the U.S. Army Research Office in Durham, NC, written a brief article, The Making of Captain America's Shield (pdf), where he try to understand if today we have the ability to construct the Captain America's shield. Thanks to Lynne Robinson⁽¹⁾ and the Avengers movie, Mathaudhu and his little review returned to the attention of people.
First of all we try to resume the story of the Cap's shield. The first comics shield was a triangular shield but starting from Captain America Comics #2 (april 1941), Cap was equipped by the most famous circular shield:

A concavo-convex metal disc approximately 0.76 m in diameter, it is virtually indestructible and has remained his most constant shield over the decades.

Following Captain America #255 (march 1981), the shield was presented to Rogers by president Franklin Roosevelt⁽⁶⁾.

It is created by the scientist Myron MacLain during some experiments with vibranium, an extraterrestrial metal introduced in Fantastic Four #53 with the ability to absorb vibrations⁽⁶⁾. A useful utilization of the vibranium was made by Thor in Avengers #68 in order to contain the explosion of Ultron-6⁽⁵⁾.

During the same saga (started on Avengers #66), MacLain presented for the first time the adamantium⁽⁶⁾. This metal was created (or founded) by MacLain some years after the creation of Cap's shield: this last was made by an alloy of vibranium and steel with an unkown catalyst; so MacLain try to reproduce that experiment and he accidentaly created adamantium, like the same metallurgist telled in Captain America #303.

The Graham Bell's tetrahedronal shed

Tony Smith realized an interesting shed that it seems inspired by the tetrahedron, a particular polyhedron, but following Tropolism, this idea was just used by Alexander Graham Bell:

The tetrahedron is in general a polyhedron constituted by four triangular sides. Now, if we describe every vertices of every sides with the vector $(x_i, y_i, z_i)$, where $i = 1, \cdots, 4$, then the volume of the tetrahedron is given by: \[V = \frac{1}{3!} \begin{vmatrix} x_1 & y_1 & z_1 & 1\\ x_2 & y_2 & z_2 & 1\\ x_3 & y_3 & z_3 & 1\\ x_4 & y_4 & z_4 & 1 \end{vmatrix}\] And if the tetrahedron is regular, we can relate in one beutiful formula, the volume $V$, the area $\Delta$ of the triangles and the radius $R$ of the sphere outside the tetrahedron (or the circumsphere)^{(1, *)} \[6RV = \Delta^2\] The regulartetrahedron is also the platonic solid $P_5$

(...) with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces.

His symmetries are a bit complex, with 12 rotational symmetries, and the tetrahedral group is isomorphic with the symmetric group $S_4$, i.e. the group of all permutations of 4 elements.

The tetrahedron is also the basic idea to the Four corner project, developed by the artist David Barr in 1976, with the idea to realize a Erth-size tetrahedron in order to span our planet.
Finally we can find tetrahedron also in chemistry: methane, xenon tetroxide, perchlorate ion, sulfate ion, phosphate ion and others.
It's also interesting observe that also water presents a structure like a tetrahedron, but in this case isn't a regular polyhedron. In particular:

The most common arrangement of liquid water molecules is tetrahedral with two hydrogen atoms covalently attached to oxygen and two attached by hydrogen bonds. Since the hydrogen bonds vary in length many of these water molecules are not symmetrical and form transient irregular tetrahedra between their four associated hydrogen atoms.^{(2, 3)}

(1) MathWorld: Weisstein, Eric W. Tetrahedron; Jackson, Frank and Weisstein, Eric W. Regular Tetrahedron
(2) Wikipedia: Tetrahedral molecular geometry
(3) P. E. Mason and J. W. Brady (2007). "Tetrahedrality" and the Relationship between Collective Structure and Radial Distribution Functions in Liquid Water. J. Phys. Chem. B 111 (20): 5669–5679

This is water

This comics watercolor is drawned by Davide Osenda, a computer engineer and cartoonists. His first comics is about mathematics, the italian graphic novel L'ultima lezione a Gottinga (The last lesson in Gottinga), about Cantor and the mathematics of transfinite numbers. Now it seems that he's working to a comics inspired by This is water, a speech by David Foster Wallace. So I propose you the audio of that speech:

The science behind tears

From the collaboration between the National Cartoonists' Society and the Center for Cartoon Studies, it is born the first issue of the first volume of the on-line magazine The Cartoon Crier. The tabloid is a collection of the saddest strips and cartoons from a lot of great cartoonists. In particular there is also a science comics, Lacrimal studies 101 by Jon Chad, from the Fizzmont institute of rad science⁽¹⁾.

The structure of the comic is like the series of stories named A Goofy Look At... and Goofy as a famous hystoric persons drawned in particular by Hector Adolfo de Urtiága, one of the cartoonists of the Jaime Diaz's studios.
But stop to write about comics and start with science, in particular about the three type of tears that our eyes can produce: basal, reflex and emotional.

Basal tears are produced by our eyes constantly to keep them moist. These tears contain glucose, mucin, lysozyme, lactoferrin, lipocaln, potassium and sodium

Reflex tears are produced when an irritant either physical (a poke in the eye) or chemical (onion fumes) agitates an eye!!

About the emotional tears⁽²⁾ I find, instead, an interesting paper published last year on Science⁽³⁾. First of all there is the composition:

Tears are drops of liquid produced by the lacrimal, accessory lacrimal, and Meibomian glands, which contain proteins, enzymes, lipids, metabolites, electrolytes, and traces of drugs. In mice, tears contain a chemosignal or pheromone. Because the chemical makeup of human emotional tears differs from that of reflexive eye-protective tears, we hypothesized that human tears may similarly convey a chemosignal.

The research team, in order to test their hypothesis, choses a group of women between 30 and 31 years and has occurred the effect of their tears on various groups of men. For various types of emotions and tears were used different groups of donors (each group had an average age 30 years old) and after the samples are submitted to the attention of different groups of men (mean ages of groups were between 28 and 29 years old). First of all, we must note that the first test has been necessary to understand if the tears had some odor able to distinguish them than, for example, a saline solution. After determining that the tears do not have characteristic odors, they went ahead quietly with the actual experiment that aimed to test one of the two hypotheses under consideration, i.e. either that tears contain chemical signals related to the context of sadness in which they were produced, or that human tears, such as those of the mice, are capable of signaling information related to the behavior sociosexual.
We can summarize the results with the following paragraph from the abstract:

We found that merely sniffing negative-emotion–related odorless tears obtained from women donors induced reductions in sexual appeal attributed by men to pictures of women's faces. Moreover, after sniffing such tears, men experienced reduced self-rated sexual arousal, reduced physiological measures of arousal, and reduced levels of testosterone. Finally, functional magnetic resonance imaging revealed that sniffing women's tears selectively reduced activity in brain substrates of sexual arousal in men.

The results, of course, carry with them a series of questions, such as which are the substances inside the tears responsible for this type of response, or if such signals is restricted to emotional tears, or if we can still find the same effect even in the tears of men than women.
It's also interesting to note what is not said in the paper, namely that this kind of research can provide the best information to more effectively convey a certain kind of commercial messages. Discover something about ourselves, like in this case, presents a downside: it can trivially be used against us. However I think that the beauty of the world around us is equal to similar risks, especially if certain findings are not closed to the rooms of the researchers and donors.

(1) A fake institute where we talk about REAL science!
(2) About emotional tears, I translate you a breaf quote from an italian pdf about tears:

This last type of tears [the emotional tears] contains very high percentages of manganese and some hormones including prolactin

(3) Gelstein, S., Yeshurun, Y., Rozenkrantz, L., Shushan, S., Frumin, I., Roth, Y., & Sobel, N. (2011). Human Tears Contain a Chemosignal Science, 331 (6014), 226-230 DOI: 10.1126/science.1198331 (4) Other links about research: Christine Dell'Amore for National Geographic and Janelle Weaver for Scientific American

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