Geometry At Work: Maxwell, Escher And Einstein

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 ;)

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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

Raspberry Multiverse

Stephen Hawking’s Grand design is leading to a non symmetric multiverse. However there is reason to suggest a full symmetric multiverse with point symmetry at the centre of a raspberry shaped pulsating bubble multiverse. For an entangled multiverse, each quantum should have its own 12 fold entanglement relation to all 12 opposite (anti) quantums, each located inside one of the 12 universes. The result is called: Poincaré dodecahedral space symmetry.  (see also below: B.F.Roukema) A dodecahedron is also called “Buckyball” An other evidence for space symmetry and the centre of the multiverse is the so called DARK FLOW. Dark flow seems to be found at the edge of the visible universe!! There the “dark flow centre is represented by a so called "Pitch of spacetime”

Six stages of debugging

Neutron Stars Are Even Weirder Than We Thought

Let’s face it, it’s hard for rapidly-spinning, crushed cores of dead stars NOT to be weird. But we’re only beginning to understand how truly bizarre these objects — called neutron stars — are.

Neutron stars are the collapsed remains of massive stars that exploded as supernovae. In each explosion, the outer layers of the star are ejected into their surroundings. At the same time, the core collapses, smooshing more than the mass of our Sun into a sphere about as big as the island of Manhattan.

Our Neutron star Interior Composition Explorer (NICER) telescope on the International Space Station is working to discover the nature of neutron stars by studying a specific type, called pulsars. Some recent results from NICER are showing that we might have to update how we think about pulsars!

Here are some things we think we know about neutron stars:

Pulsars are rapidly spinning neutron stars ✔︎

Pulsars get their name because they emit beams of light that we see as flashes. Those beams sweep in and out of our view as the star rotates, like the rays from a lighthouse.

Pulsars can spin ludicrously fast. The fastest known pulsar spins 43,000 times every minute. That’s as fast as blender blades! Our Sun is a bit of a slowpoke compared to that — it takes about a month to spin around once.

The beams come from the poles of their strong magnetic fields ✔︎

Pulsars also have magnetic fields, like the Earth and Sun. But like everything else with pulsars, theirs are super-strength. The magnetic field on a typical pulsar is billions to trillions of times stronger than Earth’s!

Near the magnetic poles, the pulsar’s powerful magnetic field rips charged particles from its surface. Some of these particles follow the magnetic field. They then return to strike the pulsar, heating the surface and causing some of the sweeping beams we see.

The beams come from two hot spots… ❌❓✔︎ 🤷🏽

Think of the Earth’s magnetic field — there are two poles, the North Pole and the South Pole. That’s standard for a magnetic field.

On a pulsar, the spinning magnetic field attracts charged particles to the two poles. That means there should be two hot spots, one at the pulsar’s north magnetic pole and the other at its south magnetic pole.

This is where things start to get weird. Two groups mapped a pulsar, known as J0030, using NICER data. One group found that there were two hot spots, as we might have expected. The other group, though, found that their model worked a little better with three (3!) hot spots. Not two.

… that are circular … ❌❓✔︎ 🤷🏽

The particles that cause the hot spots follow the magnetic field lines to the surface. This means they are concentrated at each of the magnetic poles. We expect the magnetic field to appear nearly the same in any direction when viewed from one of the poles. Such symmetry would produce circular hot spots.

In mapping J0030, one group found that one of the hot spots was circular, as expected. But the second spot may be a crescent. The second team found its three spots worked best as ovals.

… and lie directly across from each other on the pulsar ❌❓✔︎ 🤷🏽

Think back to Earth’s magnetic field again. The two poles are on opposite sides of the Earth from each other. When astronomers first modeled pulsar magnetic fields, they made them similar to Earth’s. That is, the magnetic poles would lie at opposite sides of the pulsar.

Since the hot spots happen where the magnetic poles cross the surface of the pulsar, we would expect the beams of light to come from opposite sides of the pulsar.

But, when those groups mapped J0030, they found another surprising characteristic of the spots. All of the hot spots appear in the southern half of the pulsar, whether there were two or three of them.

This also means that the pulsar’s magnetic field is more complicated than our initial models!

J0030 is the first pulsar where we’ve mapped details of the heated regions on its surface. Will others have similarly bizarre-looking hotspots? Will they bring even more surprises? We’ll have to stay tuned to NICER find out!

And check out the video below for more about how this measurement was done.

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

What’s encrypting your internet surfing? An algorithm created by a supercomputer? Well, if the site you’re visiting is encrypted by the cyber security firm Cloudflare, your activity may be protected by a wall of lava lamps.

Cloudflare covers websites for Uber, OKCupid, & FitBit, for instance. The wall of  lamps in the San Francisco headquarters generates a random code. Over 100  lamps, in a variety of colors, and their patterns deter hackers from accessing data.  

As the lava lamps bubble and swirl, a video camera on the ceiling monitors their unpredictable changes and connects the footage to a computer, which converts the randomness into a virtually unhackable code.

Codes created by machines have relatively predictable patterns, so it’s possible for hackers to guess their algorithms, posing a security risk. Lava lamps, add to the equation the sheer randomness of the physical world, making it nearly impossible for hackers to break through.

You might think that this would be kept secret, but it’s not. Simply go in and ask to see the lava lamp display. By allowing people to affect the video footage, human movement, static, and changes in lighting from the windows work together to make the random code even harder to predict.

So, by standing in front of the display, you add an additional variable to the code, making it even harder to hack. Isn’t that interesting? 

via atlasobscura.com

That’s the secret of programming.

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.

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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.

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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

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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.

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