Field of Science

A bit of math behind Ant-Man

cc @sarofsky
I write also for Lo Spazio Bianco (The White Space), a web-zine about comics. Thanks to Carlo Coratelli, that contacted Sarofsky Corp., we can realize an interview with Erin Sarofsky, President, Owner and Executive Creative Director of the company. They realized some titles for Marvel Studios movies, in oparticular Ant-Man, and I proposed two questions about this movie. The questions, with Erin's answers, are the following, but you can read the complete interview here:
We loved the opening animation in Ant-Man. It seems it's inspired by the Power[s] of Ten of the Eames. Is it really like that?
Ab-So-Lutely! We were very inspired by the Eames film. Luckily though, our universe is the Marvel Universe... so being 100% accurate was not necessary.
About Ant-Man, we also loved the work with the mathematical modeling you used to produce the animation and we think it would be quite interesting to elaborate about the actual realization method. How does it work?
We actually worked in reverse. We planned our moves and then did the math after the fact. We used the surface of the grass as our zero; anything above that is positive and anything below is negative.
Andy Zazzera, CG Director on the job, and our wicked smart math guy did all the estimations. (Again, nothing is exact because our world is fake... but the math is legit).
For a while I was tempted to quest something of more detailed about the math developing, but I thought that it could not so interesting for our regular readers. However, I'm convinced that it is not for Field of Science's readers!

One hundred years

Creativity is the residue of time wasted
This interesting quotation by Albert Einstein linked him with Henri Poincaré not only with the contribution of French math to special relativity, but also for the utility of the creative leisure. Indeed Poincaré told that, after some fruitless attempts to sole a particularly difficult mathematical problem, he decided to go away for a geological excursion, in this way stimulating conditions to resolve the problem!
But the most important reason in order to write something about Einstein is the general relativity birthday, that was presented by Einstein on the 25th november 1915 at the Prussian Accademy of Sciences.

A brief history of neutrinos' oscillations

I just write a more detailed post about the model behind neutrino's oscillations. Here I would simply recall that the idea was proposed by Bruno Pontecorvo in 1957 and developed by Ziro Maki, Masami Nakagawa e Shoichi Sakata in 1962. Today I try to summarize the experimental way.

Some geometrical aspects of 3- and 4-spaces

A couple of abstracts about the geomtery of space:
Historically, there have been many attempts to produce the appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions, Gibbs' vector system using the dot and cross products. We illustrate however, that Clifford's geometric algebra (GA) provides the most elegant description of physical space. Supporting this conclusion, we firstly show how geometric algebra subsumes the key elements of the competing formalisms and secondly we show how it provides an intuitive representation and manipulation of the basic concepts of points, lines, areas and volumes. We also provide two examples where GA has been found to provide an improved description of two key physical phenomena, electromagnetism and quantum theory, without using tensors or complex vector spaces. This paper also provides pedagogical tutorial-style coverage of the various basic applications of geometric algebra in physics.
James M. Chappell, Azhar Iqbal & Derek Abbott (2011). Geometric Algebra: A natural representation of three-space, arXiv:
We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms, and other elementary 4-dimensional solids.
J. Scott Carter & David A. Mullens (2015). Some Elementary Aspects of 4-dimensional Geometry, arXiv:
There's also a minimalistic introduction to euclidean planes.

Water on Mars

"Our quest on Mars has been to 'follow the water', in our search for life in the universe, and now we have convincing science that validates what we've long suspected. This is a significant development, as it appears to confirm that water -- albeit briny -- is flowing today on the surface of Mars."
John Grunsfeld from the press release. The abstract of the paper follows:
Determining whether liquid water exists on the Martian surface is central to understanding the hydrologic cycle and potential for extant life on Mars. Recurring slope lineae, narrow streaks of low reflectance compared to the surrounding terrain, appear and grow incrementally in the downslope direction during warm seasons when temperatures reach about 250–300 K, a pattern consistent with the transient flow of a volatile species. Brine flows (or seeps) have been proposed to explain the formation of recurring slope lineae, yet no direct evidence for either liquid water or hydrated salts has been found4. Here we analyse spectral data from the Compact Reconnaissance Imaging Spectrometer for Mars instrument onboard the Mars Reconnaissance Orbiter from four different locations where recurring slope lineae are present. We find evidence for hydrated salts at all four locations in the seasons when recurring slope lineae are most extensive, which suggests that the source of hydration is recurring slope lineae activity. The hydrated salts most consistent with the spectral absorption features we detect are magnesium perchlorate, magnesium chlorate and sodium perchlorate. Our findings strongly support the hypothesis that recurring slope lineae form as a result of contemporary water activity on Mars.

Ojha, L., Wilhelm, M., Murchie, S., McEwen, A., Wray, J., Hanley, J., Massé, M., & Chojnacki, M. (2015). Spectral evidence for hydrated salts in recurring slope lineae on Mars Nature Geoscience DOI: 10.1038/ngeo2546

Freedom and truth in mathematics

The very essence of #mathematics is its freedom. (Georg #Cantor)
The way we deal with today's numbers in schools is essentially the same manner used by our ancestors Pythagoreans, who saw the numbers as concrete objects, of course, but in a way that prevented them from conceiving the infinity. The only ancient mathematician who approached the infinity was Archimedes, but in the history of mathematics can be considered a fairly unique case of lack of development mainly due to the isolation of the mathematicians at that time and of noticeable difference in quality between the Sicilian and colleagues. In order to return to touch the wall of infinity and use them in a profitable way the Earth had to wait the arrival of Georg Cantor.
The German mathematician actually faced numbers, revolutionizing mathematics, using essentially sets and logic, two tools that enabled him not only to approach, but even manipulate the infinite thanks to the transfinite numbers. Leading his steps was probably the following conviction:
The very essence of mathematics is its freedom.
According to Daniel Bonevac, this veritable mantra, written in 1883, is emblematic of the Cantor's libertarian approach to mathematics. With this milestone, Bonevac try to write a theory of mathematical truth, in order to explain some facts more or less established:
1) that the mathematical statements are either necessarily true or necessarily false;
2) that mathematical truth is derived primarily from logical truth;
3) that the existence in mathematics involves a kind of modality, which requires only the consistency or the constructability.

Hyperbolic Pascal triangles and other stories

A new set of mathematical abstracs. We start with the hyperbolic Pascal trianlges:

Fibonacci and Pell sequences in the hyperbolic Pascal triangle
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.
Hacene Belbachir, László Németh & László Szalay (2015). Hyperbolic Pascal triangles, arXiv: