How Is Geometry Entangled In The Fabric Of The Universe?

Regarding Fractals and Non-Integral Dimensionality

Alright, I know it’s past midnight (at least it is where I am), but let’s talk about fractal geometry.

Fractals

If you don’t know what fractals are, they’re essentially just any shape that gets rougher (or has more detail) as you zoom in, rather than getting smoother. Non-fractals include easy geometric shapes like squares, circles, and triangles, while fractals include more complex or natural shapes like the coast of Great Britain, Sierpinski’s Triangle, or a Koch Snowflake.

Fractals, in turn, can be broken down further. Some fractals are the product of an iterative process and repeat smaller versions of themselves throughout them. Others are more natural and just happen to be more jagged.

Fractals and Non-Integral Dimensionality

Now that we’ve gotten the actual explanation of what fractals are out of the way, let’s talk about their most interesting property: non-integral dimensionality. The idea that fractals do not actually have an integral dimension was originally thought up by this guy, Benoit Mandelbrot.

He studied fractals a lot, even finding one of his own: the Mandelbrot Set. The important thing about this guy is that he realized that fractals are interesting when it comes to defining their dimension. Most regular shapes can have their dimension found easily: lines with their finite length but no width or height; squares with their finite length and width but no height; and cubes with their finite length, width, and height. Take note that each dimension has its own measure. The deal with many fractals is that they can’t be measured very easily at all using these terms. Take Sierpinski’s triangle as an example.

Is this shape one- or two-dimensional? Many would say two-dimensional from first glance, but the same shape can be created using a line rather than a triangle.

So now it seems a bit more tricky. Is it one-dimensional since it can be made out of a line, or is it two-dimensional since it can be made out of a triangle? The answer is neither. The problem is that, if we were to treat it like a two-dimensional object, the measure of its dimension (area) would be zero. This is because we’ve technically taken away all of its area by taking out smaller and smaller triangles in every available space. On the other hand, if we were to treat it like a one-dimensional object, the measure of its dimension (length) would be infinity. This is because the line keeps getting longer and longer to stretch around each and every hole, of which there are an infinite number. So now we run into a problem: if it’s neither one- nor two-dimensional, then what is its dimensionality? To find out, we can use non-fractals

Measuring Integral Dimensions and Applying to Fractals

Let’s start with a one-dimensional line. The measure for a one-dimensional object is length. If we were to scale the line down by one-half, what is the fraction of the new length compared to the original length?

The new length of each line is one-half the original length.

Now let’s try the same thing for squares. The measure for a two-dimensional object is area. If we were to scale down a square by one-half (that is to say, if we were to divide the square’s length in half and divide its width in half), what is the fraction of the new area compared to the original area?

The new area of each square is one-quarter the original area.

If we were to try the same with cubes, the volume of each new cube would be one-eighth the original volume of a cube. These fractions provide us with a pattern we can work with.

In one dimension, the new length (one-half) is equal to the scaling factor (one-half) put to the first power (given by it being one-dimensional).

In two dimensions, the new area (one-quarter) is equal to the scaling factor (one-half) put to the second power (given by it being two-dimensional).

In three dimensions, the same pattern follows suit, in which the new volume (one-eighth) is equivalent to the scaling factor (one-half) put to the third power.

We can infer from this trend that the dimension of an object could be (not is) defined as the exponent fixed to the scaling factor of an object that determines the new measure of the object. To put it in mathematical terms:

Examples of this equation would include the one-dimensional line, the two-dimensional square, and the three-dimensional cube:

½ = ½^1

¼ = ½^2

1/8 = ½^3

Now this equation can be used to define the dimensionality of a given fractal. Let’s try Sierpinski’s Triangle again.

Here we can see that the triangle as a whole is made from three smaller versions of itself, each of which is scaled down by half of the original (this is proven by each side of the smaller triangles being half the length of the side of the whole triangle). So now we can just plug in the numbers to our equation and leave the dimension slot blank.

1/3 = ½^D

To solve for D, we need to know what power ½ must be put to in order to get 1/3. To do this, we can use logarithms (quick note: in this case, we can replace ½ with 2 and 1/3 with 3).

log_2(3) = roughly 1.585

So we can conclude that Sierpinski’s triangle is 1.585-dimensional. Now we can repeat this process with many other fractals. For example, this Sierpinski-esque square:

It’s made up of eight smaller versions of itself, each of which is scaled down by one-third. Plugging this into the equation, we get

1/8 = 1/3^D

log_3(8) = roughly 1.893

So we can conclude that this square fractal is 1.893-dimensional.

We can do this on this cubic version of it, too:

This cube is made up of 20 smaller versions of itself, each of which is scaled down by 1/3.

1/20 = 1/3^D

log_3(20) = roughly 2.727

So we can conclude that this fractal is 2.727-dimensional.

(Source: Adam Plouff)

Programmer LoGiC

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.

——

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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