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.

——

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.

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Everyone who reblogs this will get a pick-me-up in their ask box.

Every. Single. One. Of. You.

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jupyterjones

7 years ago

Interesting submission rk1232! Thanks for the heads up! :D :D

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jupyterjones

5 years ago

Eclipse Across America

August 21, 2017, the United States experienced a solar eclipse!

An eclipse occurs when the Moon temporarily blocks the light from the Sun. Within the narrow, 60- to 70-mile-wide band stretching from Oregon to South Carolina called the path of totality, the Moon completely blocked out the Sun’s face; elsewhere in North America, the Moon covered only a part of the star, leaving a crescent-shaped Sun visible in the sky.

During this exciting event, we were collecting your images and reactions online.

Here are a few images of this celestial event…take a look:

This composite image, made from 4 frames, shows the International Space Station, with a crew of six onboard, as it transits the Sun at roughly five miles per second during a partial solar eclipse from, Northern Cascades National Park in Washington. Onboard as part of Expedition 52 are: NASA astronauts Peggy Whitson, Jack Fischer, and Randy Bresnik; Russian cosmonauts Fyodor Yurchikhin and Sergey Ryazanskiy; and ESA (European Space Agency) astronaut Paolo Nespoli.

Credit: NASA/Bill Ingalls

The Bailey’s Beads effect is seen as the moon makes its final move over the sun during the total solar eclipse on Monday, August 21, 2017 above Madras, Oregon.

Credit: NASA/Aubrey Gemignani

This image from one of our Twitter followers shows the eclipse through tree leaves as crescent shaped shadows from Seattle, WA.

Credit: Logan Johnson

“The eclipse in the palm of my hand”. The eclipse is seen here through an indirect method, known as a pinhole projector, by one of our followers on social media from Arlington, TX.

Credit: Mark Schnyder

Through the lens on a pair of solar filter glasses, a social media follower captures the partial eclipse from Norridgewock, ME.

Credit: Mikayla Chase

While most of us watched the eclipse from Earth, six humans had the opportunity to view the event from 250 miles above on the International Space Station. European Space Agency (ESA) astronaut Paolo Nespoli captured this image of the Moon’s shadow crossing America.

Credit: Paolo Nespoli

This composite image shows the progression of a partial solar eclipse over Ross Lake, in Northern Cascades National Park, Washington. The beautiful series of the partially eclipsed sun shows the full spectrum of the event.

Credit: NASA/Bill Ingalls

In this video captured at 1,500 frames per second with a high-speed camera, the International Space Station, with a crew of six onboard, is seen in silhouette as it transits the sun at roughly five miles per second during a partial solar eclipse, Monday, Aug. 21, 2017 near Banner, Wyoming.

Credit: NASA/Joel Kowsky

To see more images from our NASA photographers, visit: https://www.flickr.com/photos/nasahqphoto/albums/72157685363271303

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

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jupyterjones

5 years ago

I got a migraine and threw up because of this

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jupyterjones

7 years ago

Permutations

Figuring out how to arrange things is pretty important.

Like, if we have the letters {A,B,C}, the six ways to arrange them are: ABC ACB BAC BCA CAB CBA

And we can say more interesting things about them (e.g. Combinatorics) another great extension is when we get dynamic

Like, if we go from ABC to ACB, and back…

We can abstract away from needing to use individual letters, and say these are both “switching the 2nd and 3rd elements,” and it is the same thing both times.

Each of these switches can be more complicated than that, like going from ABCDE to EDACB is really just 1->3->4->2->5->1, and we can do it 5 times and cycle back to the start

We can also have two switches happening at once, like 1->2->3->1 and 4->5->4, and this cycles through 6 times to get to the start.

Then, let’s extend this a bit further.

First, let’s first get a better notation, and use (1 2 3) for what I called 1->2->3->1 before.

Let’s show how we can turn these permutations into a group.

Then, let’s say the identity is just keeping things the same, and call it id.

And, this repeating thing can be extended into making the group combiner: doing one permutation and then the other. For various historical reasons, the combination of permutation A and then permutation B is B·A.

This is closed, because permuting all the things and then permuting them again still keeps 1 of all the elements in an order.

Inverses exist, because you just need to put everything from the new position into the old position to reverse it.

Associativity will be left as an exercise to the reader (read: I don’t want to prove it)

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jupyterjones

7 years ago

How Do Hurricanes Form?

Hurricanes are the most violent storms on Earth. People call these storms by other names, such as typhoons or cyclones, depending on where they occur.

The scientific term for ALL of these storms is tropical cyclone. Only tropical cyclones that form over the Atlantic Ocean or eastern and central Pacific Ocean are called “hurricanes.”

Whatever they are called, tropical cyclones all form the same way.

Tropical cyclones are like giant engines that use warm, moist air as fuel. That is why they form only over warm ocean waters near the equator. This warm, moist air rises and condenses to form clouds and storms.

As this warmer, moister air rises, there’s less air left near the Earth’s surface. Essentially, as this warm air rises, this causes an area of lower air pressure below.

This starts the ‘engine’ of the storm. To fill in the low pressure area, air from surrounding areas with higher air pressure pushes in. That “new” air near the Earth’s surface also gets heated by the warm ocean water so it also gets warmer and moister and then it rises.

As the warm air continues to rise, the surrounding air swirls in to take its place. The whole system of clouds and wind spins and grows, fed by the ocean’s heat and water evaporating from the surface.

As the storm system rotates faster and faster, an eye forms in the center. It is vey calm and clear in the eye, with very low air pressure.

Tropical cyclones usually weaken when they hit land, because they are no longer being “fed” by the energy from the warm ocean waters. However, when they move inland, they can drop many inches of rain causing flooding as well as wind damage before they die out completely.

There are five types, or categories, of hurricanes. The scale of categories is called the Saffir-Simpson Hurricane Scale and they are based on wind speed.

How Does NASA Study Hurricanes?

Our satellites gather information from space that are made into pictures. Some satellite instruments measure cloud and ocean temperatures. Others measure the height of clouds and how fast rain is falling. Still others measure the speed and direction of winds.

We also fly airplanes into and above hurricanes. The instruments aboard planes gather details about the storm. Some parts are too dangerous for people to fly into. To study these parts, we use airplanes that operate without people.

Learn more about this and other questions by exploring NASA Space Place and the NASA/NOAA SciJinks that offer explanations of science topics for school kids.

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

Credits: NASA Space Place & NASA/NOAA SciJinks