We worked with fractal pioneer Benoit Mandelbrot at IBM and Yale University. AMA

A side of Benoit Mandelbrot many people do not know. An anecdote.

Benoit talked about his financial models in the summer workshops we ran for high school teachers. One of the teachers asked if he had made any money on the stock market using these models. (The answer is no. The models aren't predictive. They just show that risk is greater than most people think)

Benoit answered, "I never discuss 4 things in public. My portfolio, religion, politics, or sex."

So, I asked him, "I understand why you don't discuss your portfolio, religion, or politics. But really, Benoit? Has anyone ever asked you about sex?"

He responded, "No, but I keep hoping."

The room was silent for a second, and then everyone burst into laughter when they realized this was an example of Benoit's great sense of humor.

Thank you for this AMA.

My question is: What is one little known fact about fractals you feel goes underappreciated and you'd love the world to know?

Thanks again!

Fractals are a wonderful way to see connections between things we might not think are related.

Example: Why does a tree in the winter look like our lungs look like a network of rivers?

Have you seen this fractal zoom?

Last Lights On - Mandelbrot fractal zoom to 6.066 e228 (2⁷⁶⁰⁾

At the end there appears to be a figure that is very similar (even more than the self-similar nature of fractals themselves), almost identical to the macroscopic, total view of the Mandelbrot set.

My question: do you know how they chose the coordinates to find that shape? Is there any feasible way to search for a particular geometrical pattern within a fractal?

I studied fractals with my physics prof and comp sci prof in response to people claiming to draw meaning out of Jackson Pollock paintings via the change in fractal dimensions over his career. Example, Jackson Pollock: No. 32

Once someone learns about fractals, one sees them everywhere

Thanks for this AMA!

I had not seen that before. Thank you!

Every little Mandelbrot set can be located by solving a particular equation. My guess is they knew the size of the set they wanted, solved that equation, and that's where they magnified.

If you know how the pattern is related to the equation describing the fractal, yes. But mostly, its just look around and find something that looks interesting.

Richard Taylor is the person who computed the dimensions of Pollack's paintings. Some people disagreed with his conclusions, but Benoit thought Taylor had it right.

Other than "The formula is incorrect" - how do you feel about Jonathan Coulton ?

He admits that the formula in the song is inaccurate. And he knows. And you don't have to tell him. ;)

Sorry. I don't know anything about Jonathan Coulton.

Benoit DID know this song. He was charmed by it.

Benoit never expected to have a song written about him.

From my point of view, its a cute song but not music I'm wild about. I prefer Boxer Rebellion or Fleet Foxes.

Professor Frame, you're about halfway through the lecture. Any chance you'll stop and tell a quick joke like you used to in class?

If I tell the joke now, I won't be able to tell it in class today.

What fractal image would you consider to be the most beautiful?

What is the most unexpected application of your research?

What book would you recommended for learning fractal mathematics?

Do you think that mathematics is a fundamental feature of reality or a man-made construct?

*edit wording

Most unexpected application Surely you must mean Benoit's. Mine is trivial in comparison.

Traffic on the Internet is fractally distributed. This has implications for how the Internet infrastructure is built. (# of routers, size of buffers, things like that)

I think reality IS mathematics. I'll just leave it at that.

Book recommendations

This depends on the level. For a high school level recommendation, Kenneth Falconer released a book just 5 days ago titled "Fractals: A Very Short Introduction." Also, you might look at Benoit's and my website http://classes.yale.edu/fractals/index.html

College level: also by Falconer "Fractal Geometry: Mathematical Foundations and Applications

Most beautiful

Mathematical fractal, the Mandelbrot Set. Ive been looking at it for over 20 years, and it still surprises me.

For natural fractals, I'd have to say this cast of a human lung by Dr. Ewald Weibel is the most beautiful. http://classes.yale.edu/fractals/panorama/biology/physiology/Lung1.gif

How can a high school student learn more about fractals? Are there any books or computer programs that someone without a math background could understand?

Kenneth Falconer released a book just 5 days ago titled "Fractals: A Very Short Introduction"

Back in the '90s I saw a great documentary called "The Colors of Infinity" done by Arthur C. Clarke. It featured Benoit and the M-set quite extensively and I found the whole thing fascinating and mind-blowing. The idea that you could just keep zooming in forever and seeing such complexity was amazing! I still have a copy on VHS.

I remember when they were discussing future real-world applications of fractals one thing that came up was image enhancement and that it could be used to extrapolate or "guess" what a pixel should look like based on the rest of the image. Has this come to pass?

Also, what are some good examples of fractal geometry in use in our everyday lives technology-wise?

Edit: a word.

This was called fractal interpolation. I don't know of any successful application of that idea.

Some examples fractal geometry in everyday life:

The structure of the internet.

There are fractal capacitors that can store more electricity than a normal capacitor.

There are fractal chemical mixers that can mix chemicals with less turbulence.

Fractal fuel cells are more efficient than non-fractal fuel cells.

There is an application of fractal antenna technology to build microwave invisibility cloaks. This was created by Nathan Cohen. If you search Google for this, you'll get lots of good pictures.

Another anecdote about Benoit

During World War II, Benoit's family stayed in the town Tulle, and although there were close calls, he stayed safe there.

Not long after we began working together, Benoit planned to spend part of the summer in France. He was invited to be the guest of honor at a celebration in Tulle. He said he'd go, because he wanted to tell the people of the town that he owed them his life. Without their kindness and generosity, he might well not have survived the war.

When Benoit returned at the end of the summer, I asked about his visit to Tulle. "It was fine," he replied. Only "fine" I thought. Why wasn't it wonderful? "Did you tell them?" I asked. Benoit said that he had not yet lost control of his emotions in public, so he just couldn't tell the story then. He'd tell them later.

But he didn't. Other work came up, other travels, other meetings. Then he focused on writing his memoirs. Then he died.

This story has been on my mind a lot after Benoit died. The lesson I take is that while you might think you have time enough to do important things later, you may not. Benoit didn't tell the people of Tulle that they saved his life, and I didn't tell Benoit how impossibly lucky I was that he brought me into his world. I'm sure he knew it, but I never told him, and I regret this every day. Look at your own life. What do you need to do?

First of all thanks so much for taking the time to do this. Fractals are beautiful and so are you!

I'll try to keep this short and clear. I'm currently a math undergrad. A year or so back I was working on calculating the function that would allow a square to revolve smoothly (a bike with square wheels, for instance). Once I had that figured out a few professors I was working with encouraged me to try to generalize the function for other figures. Triangles, hexagons, five pointed stars and so on. At this point I didn't know how to do all of the math by myself so they pointed me in the right direction with materials and theory already done by others. Then they approached me with a new "twist" on the problem (at least, it seemed to be new as we couldn't find any material on it). This is where I get to my question.

Do you think a function could be calculated that would complement a fractal in the way I suggested? Or in other words, could a fractal ever "roll smoothly?"

Thank you for your time!

The answer would be, the function would also have to be fractal because it would have to match the bumps of a fractal that occur on many different sizes.

Are there any convex fractals?

Many convex shapes - for example, triangles, squares, and circles - can be made fractal by filling the inside with what's called an Apollonian packing of circles. The Wikipedia entry "Apollonian gasket" gives some nice pictures for circles. But my understanding of the original question, about extending to fractal wheels the fact that a bicycle with square tires can roll smoothly over an appropriately curved road, would require that the bicycle wheel have a fractal perimeter. I can't see how to do this without a fractal road, matching the cascade of bump sizes of the fractal wheel.

That's what confuses me. I have trouble imagining what the opposite/complement of a fractal would look like. It's why I have so much trouble answering my question. I suppose these matching pairs could exist in theory, as indicated by Professor Frame's response... but not all shapes have them. (an equilateral triangle, for instance, will never roll perfectly smoothly). I think there is more of an answer out there than just a theoretical speculation.

I'm sorry I didn't think about this more carefully earlier.
Our classes have just finished and I'm in the middle of writing final exams. I've got an idea about how you
might find a road over which a fractal wheel will roll
smoothly. But I don't know for sure that this will
work. Still, it's an idea you might try. Please reference these images: http://imgur.com/a/VqdfU

Start with a square wheel. We know square wheels roll
smoothly on a truncated catenary road. We'll build a
fractal one step at a time starting from the square.
If you can find step at a time modifications of the
catenary road on which each successive modified square
rolls smoothly, then the fractal wheel will roll
smoothly on the road that results from all these
modifications.

Start with the square shown in red in (wheel1.jpg).
Replace each straight line segment of the square with
the two equal legs of an isosceles triangle with
altitude 0.1 base.

Step 1 of building the road: find a road over which
this (non-regular) octagon rolls smoothly.

Next, apply the isosceles triangle construction to
each side of the octagon, obtaining the shape in
(wheel2.jpg). Find a road over which this wheel
rolls smoothly.

Repeat with (wheel3.jpg). In (wheel4.jpg) we see
a later iterate, close to the limiting fractal wheel.

Because each step in the construction of the wheel
is a simple modification of the previous step, I
have some hope - though it is only a hope, I've not
tried this - that simple modifications, similar
patterns repeated on ever smaller scales, will take
one stage of the road to the next.

I hope this helps.

So what did his office look like? (And did he have one of those fractals screensavers?)

Benoit Mandelbrot did not use a computer. His office (at Yale) was cluttered with books and papers. There was a table in his office where his wife, Aliette, would proofread for him. She drove him to Yale every day, because he had fallen asleep and skidded off the road on one driving trip to Yale. She had accompanied him and worked with him every day after that incident.

You say he did not use a computer. Is he just 'one of those old people' or does he simply not need one?

Also, how long did the daily commute take for him and his wife? How long did this go on?

Mandelbrot did not know programming languages, but he knew what he wanted programmers to do. An any given moment, he had 4-5 programmers working for him, including me, occasionally.

Benoit split his time between IBM and Yale, Usually he and Aliette came here on Tuesdays and Thursdays. The commute was from Scarsdale, NY, to New Haven, CT. I don't know how much time the drive took them. I believe Aliette drove for at least a decade, and maybe longer.

What would we be most surprised to learn about Dr. Mandelbrot?

The Benoit I knew was almost always exuded self confidence, and he had a self-promoting personality. But I knew a time when he was depressed. Around 1993, he had developed cataracts in his eyes, which interfered with his ability to see colors. For someone who depended on looking at the world and images and "seeing" things that no one else had seen, impaired vision was very depressing.

I believe I was the person who told him to have cataract surgery, which he did in 1994. With his restored visual acuity and full color vision, he was revitalized, and became the old, energetic, enthusiastic, productive Benoit.

He was very good mechanically. He was a good carpenter and he fixed up cars.

He was driving with some colleagues to a conference in NY, and the car broke down. Benoit looked under the hood, identified the problem, fixed it, and when they arrived to the conference, Benoit was covered in grease. He had to explain what had happened to conference attendees.

What are your thoughts on the application of Mandelbrot's theory to financial markets?

This was a part of Benoit's work I never understood. My wife balances our checkbooks.

Have you guys any promising approaches to how fractals can be applied to architecture?

As an architecture student, I'm interested in applying the logical organisation and aesthetic beauty of fractal geometries in creating habitable spaces, but physicalizing these idealised mathematical constructs is often unfeasible due to real-world issues such as materials and construction techniques.

I've done research on using Additive Manufacturing technologies such as 3D printing, and while promise is there, it's far from a perfect solution due to limitations in print resolution, speed, etc.

example:fractal beam

Thanks for a cool AMA!

Benoit thought that the Paris Opera House was a good example of Fractal architecture because it has decorations on many scales.

Another place to look is the book "African Fractals" by Ron Eglash. There are people currently working on fractal design in architecture to construct spaces with ambiguous scales.

Some current applications are to produce spaces that can be used on many different scales.

For example, two people talking together, or a smaller group, or a larger group, or a lecture group, all done in the same space.

Good bye. I've got to go teach my fractals class. Thank you for the questions. Benoit would have loved this.

A quote, after he'd given a talk to a bunch of students. The students had said how much they appreciated his talk. His comment, "Truly and deeply each marked a very sweet day. Let me put it more strongly. Occasions like that make my life."

Benoit did not teach much, but he really appreciated teachers.

Hey guys -- I got my PhD at Yale -- great place!

My question is this -- have there been any successful attempts to use fractals to explain the spread of culture?

Or, at the very least, has it been use to any advantage in linguistics?

Thank you! I was talking about Mandelbrot in my marketing class today! Fucking ace.

I'm unfamiliar with fractal analysis of the spread of culture.
In fact, I'm not entirely sure what you mean by that. But if culture spreads through contacts - personal or digital - then the branching structure of acquaintance networks and of the internet suggests the possibility of the fractal spread of ideas.

Linguistics has seen some applications of fractals, but I've not followed these closely and so can't make informed comments on them. But I will mention that one of the first projects that started Benoit thinking about scaling was Zip's law of word frequencies. Benoit's uncle fished from his wastebasket a review of Zipf's book when Benoit asked for something to read on the metro ride home. So in this sense, it's correct to say that the beginning of fractals was rescued from a wastebasket.

I've heard a lot of applications for fractals and fractal research. The worst had to be t-shirt designs with fractal imagery. What would you say is the best application?

Benoit once told me that the best tasting bagels are those with a fractal distribution of holes. If the bagels are cooked at the right temperature, the carbon dioxide generated by the yeast forms bubbles with a fractal distribution. So before you buy a bunch of bagels, cut one open, look at the different hole sizes, and they range from very small to very large, they were cooked properly.

Of course, you can always taste one bagel before buying a bunch from the same batch. But that's not the "fractal way" to judge your food.

Benoit never mentioned anything about bagels to me, but I do have a funny food story.

We would often go to a little local restaurant for lunch. Benoit always had pea soup and rye toast and for desert he asked for a bowl of butter pecan ice cream with a funny hat.

The waitress looked at me, lost. I could understand Benoit quite well. What he actually meant was a cone. He wanted the cone on top of the ice cream, to look like a hat.

One application that was thought to be very important was to design fractal-based antennas for cell phones and other mobile devices. The idea was that a fractal antenna could pick up a larger range of frequency bands on which signals were transmitted. But the downside was that there was lots of noise in this broad range of frequencies, and the fractal antenna picked up this undesirable noise, obscuring the desired signal in the narrower frequency band where it was being transmitted.

So fractal antennas were abandoned. Not everything works as planned. But you have to keep experimenting, and sometimes you find "gold."

Pretty sure all cellphones use fractal antennas these days.

I've checked with Nathan Cohen, whose company,
FracTenna, builds and sells fractal mobile
antennas. Here are some recent real-world
applications of fractal antennas.

Safety backup controls in the Boeing 787,

Nextivity's active cellphone booster to fix
dropped calls, and

pickup antennas in subways, airports, and
other places, to extend cell and mobile
coverage.

Fractal antennas are alive and well.

So, the best isn't actually something that is built. It is using fractals to understand subtle patterns in nature, especially biology.

Heartbeat patterns are multi-fractal, for example.

Another is that the metabolic rate of animals scales in a fractal fashion. We need to understand this to scale drug trials from mice up to humans.

[deleted]

Primarily, modelling nature, but then there's another way to think about it. Evolution has discovered fractals are efficient for a lot of biological processes. This is starting to inform manufacturing. We are just now starting to discover what nature has known for 100 Million years... that fractals are efficient.

Hey, that's really cool.

I don't have a proper question, just wanted to ask if you ever told him this joke: "What does the B in Benoit B. Mandelbrot stand for?" "I don't know" "Benoit B. Mandelbrot"

Yes. Several hundred people have told me that joke. Benoit had heard it as well.
Benoit's answer was, "the B stands for nothing. If the S in Harry S. Truman was good enough for Truman, a B is good enough for me."

I suppose it would wear a little thin. Sorry to have bored you with it again, and thanks for the answer!

Your mentioning this wasn't a bore at all. For one thing, how would you know I'd heard it before? For another, I am a bit amused tracking the way this joke spreads. Thanks for the additional datapoint.

This is absolutely crazy, but I was just reading Chaos by James Gleick. Thank you for this AMA.

Did Madelbrot ever have a 'Eureka' moment while he was working with you on a difficult problem? Can you please describe your experience?

And I'm not a technical person, but anyhow - can we use fractal mathematics to solve computational problems instead of regularly shaped grids (tetrahederons/quad)? Especially in meteorology. Is this even possible, or is it already in use?

I never saw Benoit have a Eureka moment. Most of Benoit's work was in his head. He usually wouldn't speak about something until he thought it through pretty carefully. There were certainly moments where we left a problem unresolved in the evening and when we returned in the morning, he had a solution that flabbergasted me. So, I guess the AHA moments happened at home.

My condolences on the loss of Dr. Mandelbrot. Was he aware of the Jonathan Coulton song, and if so what did he think about it?

Please see the answer above. Benoit knew of this song and was flattered.

I have a little bit of experience in developing CFD code, and it always amazed me how all the dimensionless numbers in fluid mechanics allow for problems to become, in my mind anyways, self similar at many different combinations of whatever equivalent dimensionless number is being targeted. Is this an example of fractal type self similarity or am I way off track?

Benoit's studies of fluid turbulence in the early 1970s were the first steps in developing multifractals.
One of his papers is "Intermittent turbulence in self-similar cascades." Also, Leonardo Da Vinci's "The Deluge" and Hokusai's "The Great Wave off Kanagawa" show that these artists were familiar with self-similarity in fluid turbulence. Benoit loved these examples.

I've heard that physicists have hypothesized that fractals are possibly the most accurate representation of the structure of the universe. Do you believe this to be true? And how do you think this would relate to the possibilty of alternate dimensions, if it were to be proven scientifically?

Well, I'll pass on the alternate dimension part, but for the other there is good evidence that the distribution of galaxies is fractal on a wide range of scales.

This may represent bumpiness in the original big bang.

Fascinating. Thank you for your response. Would you say fractals are intrinsic to everything in nature, or just most things?

Just most things.

If you drop a pebble into a lake, the ripples are circles. Circles are not fractal.

That makes sense, thanks for clearing that up for me. In your research, do you ever come across data that suggests a connection to phi, the golden mean or the fibonacci sequence? Also, do you have any thoughts as to why many users of psychedelic drugs see fractals as an aspect of their experience? I know this is straying a bit from your area of expertise, but I'd really like to hear your perspective on the matter.

The only applications I know about the golden mean are when people say, incorrectly in my opinion, that logarithmic spirals are fractal. It's true that if you magnify this spiral about its center, you get another spiral. But if you magnify about any other point of the spiral, you don't get another spiral. This is much too limited a form of self-similarity to be called fractal.

About psychedelic drug users seeing fractals, from being around some of my undergrad classmates when they used drugs, as far as I could tell they were too incoherent to have any idea what they saw. Maybe they saw swirly colors, and afterward when they looked at a Mandelbrot set poster, thought that's what they saw. If more dorm rooms had Jackson Pollock posters, maybe the common story would be that people saw drip paintings when they use drugs.

For the lay person, what are the practical uses of fractals?

We talked about a few in previous questions. Please refer to earlier questions about applications.

Michael, music helps me stay focused. What is your favorite genre/band to listen to?

Older stuff... Bach and Mozart of course.

Then, Fleet Foxes, DCFC, Florence + The Machine, Tori Amos, Civil Wars.

I'm old, but not dead.

If you're still here, I am wondering why fractals look like things in nature and why else they're useful.

Many inanimate fractals - coastlines and mountain ranges, for example - are fractal because the same forces sculpt the shapes over a wide range of scales. This produces similar features of many sizes, hence fractals.

Many living fractals arise because the same growth instructions are applied on many different scales. Think of the lungs again. About 23 successive levels of branching, about half a billion alveoli. Why would the genetic instructions to grow the lungs specify every twist and turn?
Rather, just say grow some multiple of the circumference and branch. Keep doing this until the circumference gets to a certain size and then start adding alveoli. We believe the genetic code for the lungs, and many other parts of us, are growing instructions, not descriptions of the finished product.

There's another kind of fractal that's invisible: fractal structures in metabolic pathways. But by now I hope the idea is clear. Replication of the same pattern across many length scales is a consequence of applying the same forces or growth rules across many scales. Fractals are dynamical geometry. Focus on the process, not on the form.

Why do you think that metabolic scaling follows a 3/4 power law and why is the 3/4 power scaling so prevelant in biology?

The answer West proposes has to do with the fractal structure of the lungs in that a lot of the heat we disperse actually comes through our breath, not across our skin surface. The fractal dimensions of the lungs may be responsible for the 3/4 metabolic scaling. There's a nice comparison. The volume of the lungs is 5 or 6 liters and the area is 130 square meters. Folding the area into that volume is like folding an envelope up to fit into a thimble. The fractal structure of the lungs makes that possible.

I've begun to notice academics in the humanities applying the principles of fractal geometry to literature and media theory. Are there any other unexpected applications of the concepts that surprise you?

The one that surprised Benoit and me the most is Dali's painting Visage of War. It is supposed to represent the infinite horror of war, and it is a face with the eye sockets and mouth filled with faces who's eye sockets and mouth are filled with faces and so on... http://bit.ly/1cZLxc4

Benoit pointed this out to me. I did some looking around at books about Dali and found a study of that painting before he did the painting. In the study, the mouths were filled with faces, but the eyes were filled with rings of a tree and a honey comb. Something scary, but not the same pattern repeated over many levels. Going from the study to the painting, Dali discovered the notion of self similarity. This was his method of representing infinity.

When I showed Benoit the study of the painting, he was silent fort almost a minute. Then he looked at me, grinned, and said "Well Michael, today you have earned your pay."

For math grad school admissions, do you think it's better for me to take a graduate course this summer or participate in an REU?

REU for sure. You get experience in what doing math research is like, and that will tell you whether or not you should go to grad school.

What were some of Benoit's favorite hobbies (apart from Fractals)?

Two come to mind. He loved Opera. And he loved storytelling. Specifically, him telling stories to other people.

One of his visitors years ago asked if Benoit was a mathematician, a physicist, or an economist. Without hesitation, Benoit replied "I'm a storyteller." That's what he mostly did. He told us lots of stories.

Do you know if Benoit knows that he has popularised Mathematical thinking among people who would previously of left that behind at the end of their educations? Specifically modern "counter cultures".

To the extent that he often got letters and emails from older people who had been away from math for a long time but had seen photos of the Mandelbrot set and had been inspired.

was it difficult to stop yourselves from becoming lost in the images?

I didn't stop myself. I'm still in the images.

What was he working on towards the end of his life? I know that he had a professional working relationship with Taleb and he was promoting the use of multifractal models in finance and economics.

The last thing he worked on was his memoirs. He died before they were finished. His wife and Merry Morse from IBM finished them up. It has been published and is called "The Fractalist" It came out in 2012. The cover photo is especially interesting, if you look at Benoit's hair. His normally fuzzy hair has been turned into the boundary of the Mandelbrot Set. Its just brilliant. I wish I had thought of it. http://bit.ly/IunKry

Prof. Frame, hello from a Union alum!

My wife and I were in a bookstore yesterday, and commented that there used to be graphics books on fractals - both algorithmic as well as photo books, but that seems to no longer be the case. Being a fan of The Beauty of Fractals and the Science of Fractal Images from years ago, are there any current books that you would recommend that provide source code for generating fractal images, above and beyond the "basic" fractals of 20 years ago? Thanks!

Edit: P.S. My wife is a descendant of Dr. Thomas Wynne, so may be a distant relation to your AMA compatriot. :)

Hi Charles. Great to hear from you. The best book i can think of is "Indra's Pearls" by Mumford Wright and Series.

Email me some time so we can catch up!

I'm a math teacher at a K-8, dual-language (half spanish / half english) schoo. What are your suggestions for any math teachers out there for how to teach math?

Try to involve your students in open ended problems. Give them puzzles to work on. They need to see that math isn't just memorization, but thinking and trying to see answers that are not obvious.