Interesting Submission Rk1232! Thanks For The Heads Up! :D :D

New Research Heading to Earth’s Orbiting Laboratory

It’s a bird! It’s a plane! It’s a…dragon? A SpaceX Dragon spacecraft is set to launch into orbit atop the Falcon 9 rocket toward the International Space Station for its 12th commercial resupply (CRS-12) mission August 14 from our Kennedy Space Center in Florida.

It won’t breathe fire, but it will carry science that studies cosmic rays, protein crystal growth, bioengineered lung tissue.

Here are some highlights of research that will be delivered:

I scream, you scream, we all scream for ISS-CREAM! 

Cosmic Rays, Energetics and Mass, that is! Cosmic rays reach Earth from far outside the solar system with energies well beyond what man-made accelerators can achieve. The Cosmic Ray Energetics and Mass (ISS-CREAM) instrument measures the charges of cosmic rays ranging from hydrogen to iron nuclei. Cosmic rays are pieces of atoms that move through space at nearly the speed of light

The data collected from the instrument will help address fundamental science questions such as:

Do supernovae supply the bulk of cosmic rays?

What is the history of cosmic rays in the galaxy?

Can the energy spectra of cosmic rays result from a single mechanism?

ISS-CREAM’s three-year mission will help the scientific community to build a stronger understanding of the fundamental structure of the universe.

Space-grown crystals aid in understanding of Parkinson’s disease

The microgravity environment of the space station allows protein crystals to grow larger and in more perfect shapes than earth-grown crystals, allowing them to be better analyzed on Earth. 

Developed by the Michael J. Fox Foundation, Anatrace and Com-Pac International, the Crystallization of Leucine-rich repeat kinase 2 (LRRK2) under Microgravity Conditions (CASIS PCG 7) investigation will utilize the orbiting laboratory’s microgravity environment to grow larger versions of this important protein, implicated in Parkinson’s disease.

Defining the exact shape and morphology of LRRK2 would help scientists to better understand the pathology of Parkinson’s and could aid in the development of therapies against this target.

Mice Help Us Keep an Eye on Long-term Health Impacts of Spaceflight

Our eyes have a whole network of blood vessels, like the ones in the image below, in the retina—the back part of the eye that transforms light into information for your brain. We are sending mice to the space station (RR-9) to study how the fluids that move through these vessels shift their flow in microgravity, which can lead to impaired vision in astronauts.

By looking at how spaceflight affects not only the eyes, but other parts of the body such as joints, like hips and knees, in mice over a short period of time, we can develop countermeasures to protect astronauts over longer periods of space exploration, and help humans with visual impairments or arthritis on Earth.

Telescope-hosting nanosatellite tests new concept

The Kestrel Eye (NanoRacks-KE IIM) investigation is a microsatellite carrying an optical imaging system payload, including an off-the-shelf telescope. This investigation validates the concept of using microsatellites in low-Earth orbit to support critical operations, such as providing lower-cost Earth imagery in time-sensitive situations, such as tracking severe weather and detecting natural disasters.

Sponsored by the ISS National Laboratory, the overall mission goal for this investigation is to demonstrate that small satellites are viable platforms for providing critical path support to operations and hosting advanced payloads.

Growth of lung tissue in space could provide information about diseases

The Effect of Microgravity on Stem Cell Mediated Recellularization (Lung Tissue) uses the microgravity environment of space to test strategies for growing new lung tissue. The cells are grown in a specialized framework that supplies them with critical growth factors so that scientists can observe how gravity affects growth and specialization as cells become new lung tissue.

The goal of this investigation is to produce bioengineered human lung tissue that can be used as a predictive model of human responses allowing for the study of lung development, lung physiology or disease pathology.

These crazy-cool investigations and others launching aboard the next SpaceX #Dragon cargo spacecraft on August 14. They will join many other investigations currently happening aboard the space station. Follow @ISS_Research on Twitter for more information about the science happening on 250 miles above Earth on the space station.  

Watch the launch live HERE starting at 12:20 p.m. EDT on Monday, Aug. 14!

Make sure to follow us on Tumblr for your regular dose of space: http://nasa.tumblr.com

Speaking of pretty flowers, may I present to you the “Eighteen Scholars”, the flower of my heart-a variation of Camellia japonica L. Its uniqueness lies in the layers and layers of petals-one flower can hold as much as 130 petals.

Named “Eighteen Scholars” in Chinese because at the most, one bush can have up to eighteen of these pretty darlings :3

Geometry at work: Maxwell, Escher and Einstein

Maxwell’s diagram

from the 1821 “A Philosophical Magazine”, showing the rotative vortexes of electromagnetic forces, represented by hexagons and the inactive spaces between them

The impossible cube

invented in 1958, as an inspiration for his Belvedere litography.

Geometry of space-time

The three dimensions of space and the dimension of time give shape to the fourth, that of space-time.

There are 27 straight lines on a smooth cubic surface (always; for real!)

This talk was given by Theodosios Douvropoulos at our junior colloquium.

I always enjoy myself at Theo’s talks, but he has picked up Vic’s annoying habit of giving talks that are nearly impossible to take good notes on. This talk was at least somewhat elementary, which means that I could at least follow it while being completely unsure of what to write down ;)

——

A cubic surface is a two-dimensional surface in three dimensions which is defined by a cubic polynomial. This statement has to be qualified somewhat if you want to do work with these objects, but for the purpose of listening to a talk, this is all you really need.

The amazing theorem about smooth cubic surfaces was proven by Arthur Cayley in 1849, which is that they contain 27 lines. To be clear, “line” in this context means an actual honest-to-god straight line, and by “contain” we mean that the entire line sits inside the surface, like yes all of it, infinitely far in both directions, without distorting it at all. 

(source)

[ Okay, fine, you have to make some concession here: the field has to be algebraically closed and the line is supposed to be a line over that field. And $\Bbb R$ is not algebraically closed, so a ‘line’ really means a complex line, but that’s not any less amazing because it’s still an honest, straight, line. ]

This theorem is completely unreasonable for three reasons. First of all, the fact that any cubic surface contains any (entire) lines at all is kind of stunning. Second, the fact that the number of lines that it contains is finite is it’s own kind of cray. And finally, every single cubic surface has the SAME NUMBER of lines?? Yes! always; for real!

All of these miracles have justifications, and most of them are kind of technical. Theo spent a considerable amount of time talking about the second one, but after scribbling on my notes for the better part of an hour, I can’t make heads or tails of them. So instead I’m going to talk about blowups.

I mentioned blowups in the fifth post of the sequence on Schubert varieties, and I dealt with it fairly informally there, but Theo suggested a more formal but still fairly intuitive way of understanding blowups at a point. The idea is that we are trying to replace the point with a collection of points, one for each unit tangent vector at the original point. In particular, a point on any smooth surface has a blowup that looks like a line, and hence the blowup in a neighborhood of the point looks like this:

(source)

Here is another amazing fact about cubic surfaces: all of them can be realized as a plane— just an ordinary, flat (complex) 2D plane— which has been blown up at exactly six points. These points have to be “sufficiently generic”; much like in the crescent configuration situation, you need that no two points lie on the same line, and the six points do not all lie on a conic curve (a polynomial of degree 2).

In fact, it’s possible, using this description to very easily recover 21 of the 27 lines. Six of the lines come from the blowups themselves, since points blow up into lines. Another fifteen of them come from the lines between any two locations of blowup. This requires a little bit of work: you can see in the picture that the “horizontal directions” of the blowup are locally honest lines. Although most of these will become distorted near the other blowups, precisely one will not: the height corresponding to the tangent vector pointing directly at the other blowup point.

The remaining six points are can also be understood from this picture: they come from the image of the conic passing through five of the blowup points. I have not seen a convincing elementary reason why this should be true; the standard proof is via a Chow ring computation. If you know anything about Chow rings, you know that I am not about to repeat that computation right here.

This description is nice because it not only tells us how many lines there are, but also it roughly tells us how the lines intersect each other. I say “roughly” because you do have to know a little more about what’s going on with those conics a little more precisely. In particular, it is possible for three lines on a cubic surface to intersect at a single point, but this does not always happen.

I’ll conclude in the same way that Theo did, with a rushed comment about the fact that “27 lines on a cubic” is one part of a collection of relations and conjectured relations that Arnold called the trinities. Some of these trinities are more… shall we say… substantiated than others… but in any case, the whole mess is Laglandsian in scope and unlikely even to be stated rigorously, much less settled, in our lifetimes. But it makes for interesting reading and good fodder for idle speculation :)

When 25,000 little dice are agitated in a cylinder, they form into neat concentric circles

Behold the magic of compaction dynamics. Scientists from Mexico and Spain dumped 25,000 tiny dice (0.2 inches) into a large clear plastic cylinder and rotated the cylinder back and forth once a second. The dice arranged themselves into rows of concentric circles. See the paper and the videos here.

https://boingboing.net/2017/12/05/when-25000-little-dice-are-ag.html

gotta catch em all

The Kakeya Problem

Some time ago I (briefly) mentioned that, along with two other students, I was taking a reading course this semester with Dmitriy Bilyk. It hasn’t quite gone in the direction we were initially expecting, but one of our long detours has been an extended sequence of readings around the the Kakeya conjecture. As far as I know, the Kakeya problem (different from the Kakeya conjecture; more on that later) was the first question that fell respectably under the purview of geometric measure theory. So if nothing else, it is interesting from a historical perspective as a question that kickstarted a whole new field of mathematics.

Okay, but what is the Kakeya problem?

As originally posed, it goes like this: given a disk with diameter 1, it is possible to place down a line segment of length 1 into the set, and rotate it (continuously) an entire 180 degrees. But this is not the smallest-area set for which this kind of rotation is possible:

(all pictures in the post are from the Wikipedia page, which is really good)

This set has area $\tau/16$, half that of the circle. So the question naturally becomes: what is the area of the smallest set that allows this kind of rotation?

The answer is known, and it is…

…

zero, basically.

It’s remarkable, but it’s true: you can construct arbitrarily small sets in which you can perform a 180-degree rotation of a line segment! One way to do it goes like this: in the picture above, we reduced the area of the circle by squeezing it until it developed three points. If we keep squeezing to get more points, then the solid “middle” becomes very small, and the tendrils get very thin, so the area keeps decreasing. However, we can still take a line segment and slowly, methodically, shift it back and forth between the set’s pointy bits, parallel-parking style, to eventually get the entire 180 degrees of rotation.

——

You can’t actually get a set with zero area to work. But the reason for that seemed more like a technicality than something actually substantial. So people changed the problem slightly to get rid of those concerns. Now, instead of trying to get a set where you can rotate a line segment through all 180 degrees, you just have to have a set where you can find a line segment in every direction. The difference being that you don’t need to guarantee any smooth way to “move between” these line segments.

Sets that work for the modified problem are often called Kakeya sets (although some people reserve that for the rotation problem and use Besicovitch sets for the modified one). And indeed, there are Kakeya sets which actually have zero area.

The details involved with going from “arbitrarily small” to “actually zero” are considerable, and we won’t get into them here. The following is a simplification (due to Perron) of Besicovitch’s original construction for the “arbitrarily small” case. We take a triangle which clearly has ½ of the directions the needle might possibly take, and split it up into several pieces in such a way that no directions are lost. Then we start to overlap those pieces to get a set (that still has segments in all the same directions) with much smaller area:

We can do this type of construction twice (one triangle “downward-facing” and the other “left-facing”) and then put those two sets together, guarantee that we get all of 180 degrees of directions.

This Besicovitch-Perron construction, itself, only produces sets which are “arbitrarily small”, but was later refined to go all the way to zero. Again, the technicalities involved in closing that gap are (much) more than I want to talk about now. But the fact that these technicalities can be carried out with the Besicovitch-Perron construction is what makes it “better” than the usual constructions for the original Kakeya problem.

——

I should conclude with a few words about the Kakeya conjecture, since I promised them earlier :)

Despite the essentially-solved status of these two classical Kakeya problems, there is at least one big question still left open. It is rather more technical than the original ones, and so doesn’t get a lot of same attention, but I’d like to take a stab at explaining what’s still current research in this sphere of ideas.

Despite the fact that Kakeya sets can be made to be “small” in the sense of measure, we still intuitively want to believe these sets are “big”. There are many ways we can formalize largeness of sets (in $\Bbb R^n$, in particular) but the one that seems to be most interesting for Kakeya things is the notion of Hausdorff dimension. I won’t define the term here, but if you’ve ever heard someone spouting off about fractals, you’ve probably heard the phrase “Fractals have non-integer dimension!”. This is the notion of dimension they’re talking about.

It is known that Kakeya sets in the plane have Hausdorff dimension 2, and that in general a Kakeya set in $n>2$ dimensions has Hausdorff dimension at least $\frac{n}{2}+1$. The proofs of these statements are… difficult, and the general case remains elusive.

One thing more: you can also formulate the Kakeya conjecture in finite fields: in this setting having “dimension $n$” in a vector space over the field $\Bbb F_q$ means that you have a constant times $q^n$ number of points in your set. Wolff proposed this “technicality-free” version in 1999 as a way to study the conjecture for $\Bbb R$. And indeed, a lot of the best ideas for the problem in $\Bbb R$ have come from doing some harmonic analysis on the ideas originally generated for the finite field case. 

But then in 2008 Zeev Dvir went and solved the finite field case completely. Which on one hand is great! But on the other hand, Dvir’s method definitely can’t be finessed to work in $\Bbb R$ so we still have work to do :P

——

Partially I wanted to write about this because it’s cool in its own right, but I must admit that my main motivation is a little more pragmatic. There was a talk at SEICCGTC 2017 which showed a surprising connection between the Kakeya problem and a certain combinatorial game. So if you think these ideas are at all interesting, you may enjoy reading the next two posts in this sequence about that talk.

[ Post 1 ] [ Next ]

Jupiter’s Great Red Spot

Jupiter’s Great Red Spot (GRS) is an atmospheric storm that has been raging in Jupiter’s southern Hemisphere for at least 400 years.

About 100 years ago, the storm covered over 40,000 km of the surface. It is currently about one half of that size and seems to be shrinking. 

At the present rate that it is shrinking it could become circular by 2040. The GRS rotates counter-clockwise(anti-cyclonic) and makes a full rotation every six Earth days. 

It is not known exactly what causes the Great Red Spot’s reddish color. The most popular theory, which is supported by laboratory experiments, holds that the color may be caused by complex organic molecules, red phosphorus, or other sulfur compounds. 

The GRS is about two to three times larger than Earth. Winds at its oval edges can reach up to 425 mph (680 km/h) 

Infrared data has indicated that the Great Red Spot is colder (and thus, higher in altitude) than most of the other clouds on the planet

Sources: http://www.universetoday.com/15163/jupiters-great-red-spot/ http://www.space.com/23708-jupiter-great-red-spot-longevity.html

Calculating the surface area of a sphere. Found on Imgur.