Making Math Model Nature: Soot, Scale, and Special Numbers

Making Math Model Nature: Soot, Scale, and Special Numbers

broccoflower1.jpgTonight at my home university (SIUE) we have a special guest speaker, Dr. Christopher Sorensen of Kansas State University, giving a talk on his research (more on that below). I’m currently sitting in the audience waiting through the 10 million distinguished introductions for the talk to begin. I had the opportunity to meet with Sorensen earlier today and was struck with what a cordial person he is. He was interested in learning a little bit about everyone around him and honestly was interested in talking with students and the most junior of faculty. While there are many extremely nice people in science, I’m still always caught off guard when the most distinguished individuals come off as just the nice guy next door.

Sorensen, in reality, isn’t just the guy next door. He holds 5 patents, has written over 200 research papers, and has won numerous awards for his teaching and research, including the Carnegie Foundation Professor of the Year award and the David Sinclair Award for his work on aerosol fractals.

Tonight, he’s giving a talk titled: “Fire, Fractals and the Divine Proportion” and it’s finally starting.

His primary areas of research are aerosols and colloids – Basically the study of suspended particles. For instance clouds are nothing more than water (and pollutants) suspended in our atmosphere (this makes them an aerosol). Colloids are the liquid equivalent, with particles suspended in fluid. Both these systems are in non-equilibrium, and they rest a breath away from collapse. Ever drop vinegar into milk? The run away curdling of the cream is a collapse of a temporarily balanced colloid.

Of all the different types of aerosols to talk about, tonight Sorensen’s focusing on soot. Coming from Kansas State, he lives in a land of prairie fires and airborne ash. It isn’t the only place that fire exists, however. Researchers working to understand fire set all sorts of cool stuff a lite. In one image he showed a “swimming pool” at Los Alamos (a pool fire) that had been filled with jet fuel and set on fire. The flame shot tens of feet into the air. He also showed a giant tank filled with coal that was experimentally set a blaze. Soot Soot Soot. And if you (as he did) look in your tail pipe, you’ll find still more fire by-products (that would be more soot).

Looked at soot beneath an electron microscope, it becomes networks of little circular particles joined together into a lacy aggregate. This soot has a magical property to it: It is scale invariant. This means you can look at it across many different size scales, from 100 microns to 1000 microns to even larger, and as you zoom out and out and out, the image doesn’t change. Another word for scale invariant is that something is a fractal.

Fractals crop up everywhere. The coastline of England is invariant from island sized scales to foot print sized scales – it is simply broken and craggy to both the ant and the airplane pilot. Mountains (sans vegetation) is the same way, with both pebbles and peaks equally reach for the sky with equal geometric shapes. It keeps going, from twigs to trees, and broccoflower (image above: Sasha Harris-Cronin) clumps to heads all show the same structure. And even crumpled paper from the tinest shred to the biggest piece of new paper fractally folds in similar ways.

Fractals are amazing, and not only are they visibly stunning, they are also mathematically definable and understandable. For instance, we talk about how things scale; if you double the size of a human – make them twice as tall, twice and fat, etc – their mass goes up as the cube (2x2x2 = 2 cubed). Other objects have different ways that they grow. It is possible through mathematical analysis to figure out the fractal number of any scale invariant object.

The question is, do certain fractal dimensionalities crop up over and over?

With soot the dimensionality is 1.8. With polystyrene balls it is 1.8. With milk it is 1.8. With every colloid and aerosol Sorensen and his team measured, they found the number 1.8.

This implies there is a some sort of an underlying physics that is telling all the suspended particles to diffuse around and bond in very specific ways that build networks with the same structure independent of what it is doing the diffusing. Using computer simulations he tried to simulate the 3 dimensional motions and found that he could actually watch (in his code) the particles move in 3 dimensions and build 2 dimensional networks with fractal dimension of 1.8. If you have objects traped in 2d, you get 1.44, both in their computer and in reality, for instance as things coalesce while trapped on the surface of water.

The processes that build these fractals are the same processes that build jello. As the particles of jello powder diffuse through hot water they bond together in progressively larger and larger networks until the network fills your jello mold. What is really neat about jello is you go from a completely chaotic system of random particles to an ordered set of fractal structures jammed together.

Gels differ from scale invariant fractals in many ways, but one of them is easy to understand. Imagine that once the fractals hit a certain size (say the size of England), and you started jamming them together (like a bunch of Enlands jammed into an imaginary ocean). This jamming together of pockets of fractals leads to a new fractal structure with a different fractal dimensionality (Fractals within Fractals within Fractals…)

And, Sorensen has figured out how to make jello out of soot. Or at least, he’s figured out how to make (and patent!) a gel of soot that is even lower density than aerogel! He just showed the most amazing movie (look here for mpg and here for other versions) of him and his graduate student making one of these soot gels in the lab. The soot looks like the most amazing chocolate frosting.

And his team is finding they can make a gel out of just about anything.

So, having figured out how to make it in the computer and in the lab, it is time to figure out how things build together from physical and mathematical rules.

This work is still ongoing, but one fascinating result has fallen out. As things build, they build mathematically in a Fibonacci series.

1, 1, 2, 3, 5, 8, 13, 21…

This series comes from starting with 1, adding 1 to that and getting 2, and then taking that 1 and adding it to the brand new 2 to get 3, and then adding that 2 and 3 to get 5, etc etc. The Fibonacci series can take one to what’s called the Golden Ratio (go read and listen to this) Mathematically, and number n, in the series is equal to the sum of the previous number and the number two numbers back.

The Golden Ratio is a two dimensional feature of our reality that crops up in galaxies, sea shells, flowers, and everywhere else.

And Sorensen has expanded the idea into additional dimensions. For 3-D he comes up with 1, 1, 1, 2, 3, 4, 6, … where each number is the sum of the number before it and the number 3 before it. In 4-D he gets 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14… where each number is the sum of the number before it and the number 4 before it. In limits similar to those used to find the Golden Ratio, he pulls out all the dimensionalities of his fractals: 1.44 and 1.8.

From soot to geometry, and from random ash to ashy gel he has taken a wild ride of discover. I have probably done a terrible job summarizing this. He is a charismatic speaker who has combined good images, excellent humor, and a dynamic presentation that included physical humor. Go explore his website and his other website (And if you need a distinguished speaker, he’d be good for your list).


  1. Heather
    Dec 6, 2007

    Wow! Thank you for this! Fractals were my favorite part of math way back in high school and I never got to learn enough about them. Sounds like an amazing lecture-thank you for pointing to more reading.

  2. Freiddie
    Dec 6, 2007

    Love to know this! I never even though there was an extension of Fibonacci’s series to more dimensions. That explains the 1.8 dimension. Now you’ve described the entire relationship between fractals and Fibonacci’s series. Thanks!

  3. michael cassidy
    Dec 6, 2007

    Your mpg links aren’t working on his site.

    Interesting post; I am jealous he is having more fun than I am.

  4. RJ
    Dec 6, 2007

    Does Dr. Sorenson have another website. There isn’t any information on the site you linked in the article.


  5. pamela
    Dec 6, 2007

    Hi RJ, I just added another link 🙂

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