Hausdorff dimension: How to measure fractional dimensions

In class, we talked about how dimensions can be non-integer values.  We were given some examples of fractals, shapes with non-integer dimensions, and we were able to calculate the dimensions of some fairly simple fractals.  But what about more complex shapes, that can’t be easily doubled, quadrupled, or so on?  How does one measure, arbitrarily, the dimension of any shape?

Well, one of the ways to do so is to find the Hausdorff Dimension of a set.  This concept of measuring dimensions was developed by (big surprise) Felix Hausdorff back in 1918.

The key idea is this:  a “circle” with dimension 1 (a line) has its length vary proportionally to its “radius”.  A circle with dimension 2 has its area vary proportionally to its radius squared, and so on with spheres and volume.  To extrapolate, a “circle” with any dimension p would have its “p-volume” vary proportionally to its radius to the power of p.  So, if p were 2.5, a 2.5 dimensional circle would, when doubled, increase in its 2.5 dimensional volume by 22.5 = about 5.66.

Now, hold onto your hats (if you don’t have a hat, go get one real quick, then hold onto it) because here’s where things get interesting.  Let’s say you wanted to cover a 3-dimensional object with a bunch of smaller spheres, and measure the “4 dimensional volume” of the spheres.  Well, we know that the 4-dimensional volume of an object is proportional to its radius4, so we can get an idea of how the 4-dimensional volume changes by simply adding together the radii4 of every sphere covering the object.

Now let’s take an arbitrary covering of our object by spheres and measure its 4-dimensional volume as above. We will get some number A. Now, how does that number change as our spheres shrink? Let’s say we, for example, replaced each sphere with 8 spheres, each having half the radius, and still managed to cover the object. Note that while the 3-dimensional volume of our spheres has remained the same (8 spheres of radius 1/2 have the same volume as 1 of radius 1), our 4-dimensional volume has been cut in half!

The main idea is that by shrinking the radii of our spheres, we can arbitrary decrease the 4-dimensional volume of the spheres covering our object. Since we can cover our object with spheres having arbitrary small 4-dimensional volume, it would make sense that the object would have 0 4-dimensional volume, which is consistent with how one would think about 4-dimensional area.

The key here is that this is true for any d-dimensional area if d is greater than 3, because our object is 3-dimensional.  Thus, if we took the infimum of values for d such that this is true, we would get 3.

To reiterate, we had a 3-dimensional object.  We were able to determine it was 3-dimensional because for any dimension d higher than 3, we could cover our object with spheres such that their d-dimensional volume was arbitrarily small.  To be very precise:

The Hausdorff content of dimension d of an object is the infimum of numbers δ ≥ 0 such that there is some cover of the object by balls of radius r1, r2,… such that (r1d+r2d+r3d+…)< δ.

The Hausdorfff dimension of an object is the infimum of numbers d such that the hausdorff content of dimension d of an object is equal to 0.

Now, this approach matches quite nicely with our definition of integer dimensions, and provides a very nice way for us to expand that notion into other, non-integer dimensions.  For example, this approach can actually be used in a surprising way:  to find the dimension of coastlines.

Perhaps you’ve heard of the coastline paradox:  that the more precise you try to measure the coastline, the longer it gets, seemingly without bound.  This should signal to us that the coastline behaves as a fractal, and since the above method gives us a way to measure the dimension of arbitrary objects, we can use it to try and measure the dimension of the coastline.

Measuring the dimension of the coast of Great Britain. Image: Prokofiev, via Wikimedia Commons.

Here’s how it’s done:  first, cover the coastline in large circles and measure the sum of the radii all put to some power p.  Then, shrink the circles and measure again the sum of the radii all put to the power p.  If this number keeps getting smaller, you’ve overestimated the dimension of the coast.  If the number keeps getting bigger, you’ve underestimated it.  If you keep doing this, you can fine-tune your estimate of the dimension.  In fact, Mandelbrot did this very thing, and got that the coast of Great Britain had a fractional dimension of about 1.25, while the coastline of South Africa had a fractional dimension of about 1.02.  While these are just estimates, it’s still cool to see how abstract ideas such as this can be used to measure things in real life.

Sources used:

http://en.wikipedia.org/wiki/Hausdorff_dimension

http://mathworld.wolfram.com/HausdorffDimension.html

http://mathworld.wolfram.com/HausdorffMeasure.html

What is a Dimension?

When you read and watch a lot of bad science fiction, like myself, you get used to a lot of cliches and sci fi tropes.  The heroes will obviously have to travel through time, fight their doubles, and outsmart a supercomputer.  Despite their predictability, and no matter how far-fetched they are, I almost always love them.  The one thing I can not stand, though, is when writers use words incorrectly in order to make things sound “cool” and “sciency”.  Things like cosmic rays to explain the fantastic 4’s powers, any time quantum mechanics is brought up in fiction, and people saying infinite when they simply mean unimaginably large.  The worst one, though, is the misuse of the word dimensions.

Quantum Mechanics image: Randall Munroe.

A dimension is not an alternate reality where everyone is evil and has facial hair.  Dimensions are properties of space that we can use to denote position.  A square and a circle have 2 dimensions, but a cube and a sphere have 3.  This is similar to how most movies are portrayed on a flat screen, but when watching 3-D movies you can put on special glasses and they suddenly have depth along with length and width.  This can go beyond 3 into 4, or 5, or n dimensions.

Movies and pop culture often shows its ignorance in many different dimensions.  You may have seen one of the many Marvel superhero movies featuring The Tesseract as a Macguffin.  A tesseract is the 4-dimensional version of a cube, just like how a cube is the 3-dimensional version of a square.  However, The Tesseract in the movies just appears as a blue glowing cube with magic powers.  It isn’t shown to have any properties unique to a real tesseract, or hypercube.  In the episode “Through the Looking Glass” of the TV show Farscape, the characters are separated into different colored dimensions.   Here dimensions are treated as if they are different universes each with a different color and a different effect on peoples senses.  Countless other tv shows, movies, games, and pulp novels provide just as poor and worse depictions of dimensions.

A rotating tesseract by Jason Hise.

In defense of these creators, thinking about 4-dimensional and higher objects can be difficult.  One way to help our imaginations is to think about going from 0 dimensions to 1 dimension to 2 dimensions and then up to 3 dimensions.  If we start with a point and stretch it some length we get a line, then if we stretch the line the same length we get a square, then if we stretch the square out we get a cube.  We can extend this to higher dimensions.  If we stretch the cube out into the fourth dimension we get a hypercube or tesseract. If we want to think of a 4-dimensional ball coming into our 3-dimensional space, we can similarly think of an analogy in lower dimensions.  You can imagine a ball dropping through the surface of a pool of water.  In this case, the ball is a 3-dimensional sphere going through the 2-dimensional plane of the water’s surface.  As the ball first touches the surface, a small solid circle would suddenly appear.  Then as the ball starts sinking down, the circle would grow and grow until it got half way in the water.  Once it passed the halfway mark, it would start shrinking until it was just a tiny circle and finally disappear.  Something similar would happen with a 4-dimensional sphere appearing in our 3-dimensional world.  It would appear as a small sphere, grow until it got halfway, and then start shrinking until it disappeared.  Basic geometric shapes are the easiest things to imagine in higher dimensions, because of all the symmetries they have.

Image by NerdBoy1392.

When you want to think about more complicated things, you need a way to represent the fourth dimension.  When we thought about the 4-dimensional sphere, we used time as the fourth dimension.  As time advanced the sphere passed through our space so that we could see its facets.  However, if we wanted to play a 4-dimensional version of chess time may not be the most useful method to think of the fourth dimension.  A regular chessboard is already 2-dimensional.  We can then stack the boards on top of each other to get a 3-dimensional box to play in.  If we want to extend the game further we need a way to represent the next dimension.  If we get 8 separate 3-dimensional game boxes each one can be treated as a different level in the the next dimension.  If we wanted to actually play chess we would also need a set of rules that work in higher dimensions, like this version.

Thinking of these higher dimensions may seem like it is only a fun exercise for your imagination, but they are meaningful and have some applications.  Einstein’s theory of special relativity combines the 3 spatial dimensions we live in with time to form Minkowski Space (Devlin 322).  This allows physicists to think geometrically about space and time.  Higher dimensions also comes up when looking at some higher mathematical ideas, like algebraic curves.  Graham’s number, one of the largest numbers ever used in a scientific paper, comes about when looking at cubes of n dimensions.  In 1868, Julius Plucker created a new type of geometry that doesn’t think of space as a bunch of points but rather as an infinite number of lines.  In this new line geometry, our normal 3-dimensional space made up of dots becomes an infinite amount of lines defined by 4 parameters (Boyer 498).  Thinking of higher dimensions can be a useful tool to help us analyze the world around us.

Not every work of fiction fumbles over depictions of different dimensions.  There are several examples of TV shows, games, and books accurately depicting higher and lower dimensions.  Futurama and The Simpsons are known for their many accurate references to mathematical ideas.  In “Treehouse of Horror IV”, Homer stumbles on the third dimension and becomes a computer generated 3-D version of himself.  In the Futurama episode “2-D Blacktop”, the Professor, Leela, Bender, and Fry end up smashed into 2-D versions of themselves with all the restrictions that brings with it.  When they start expanding back into 3-D, the characters are seen travelling amongst fractals, which have fractional dimensions.   The indie video game FEZ is about a creature who has experienced life in 2 dimensions being introduced to the third dimension by a 4-dimensional being.  Many of the game’s puzzles revolve around treating the 3-dimensional world as a 2-dimensional projection.  There is also the book Flatland by Edwin A. Abbott, which describes a square’s adventures to higher and lower dimensions.  All of these provide accurate representations of what dimensions are and what living in higher and lower dimensions would be like.

Higher dimensions are an abstruse but interesting concept.  This idea can be used as a tool to help with mathematics and physics, or as something fun to just spice up an old board game.  There is plenty of depth to the fourth and higher dimensions to write and create interesting sci fi stories.  These stories can be interesting, consistent, and can go beyond just using the word dimension as a way to sound more “sciency”.

Sources

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Devlin, Keith. The Language of Mathematics: Making the Invisible Visible. New York, NY: W.H. Freeman and Company, 1998. Print.