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Do Plants Know Math?
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Mathematical Patterns in Plants: Fibonacci, Golden Ratio & Nature's Hidden Math with Christophe Gole & Nancy Pick
In this episode of Breaking Math, host Autumn interviews authors Christophe Gole and Nancy Pick about the captivating world of mathematical patterns in plants, inspired by their book Do Plants Know Math?. Explore the intersection of mathematics and biology as they discuss the Fibonacci sequence, the golden ratio, and spiral formations that reveal nature's mathematical beauty. Learn about the optimization of plant structures, the role of women in mathematics, and get recommendations for further reading. Topics include phyllotaxis, fractals, and their connections to AI, physics, and topology.
Keywords: mathematics, biology, plant math, Fibonacci, phylotaxis, spirals, golden ratio, fractals, nature, science, women in math,topology, ai, physics, math, plants, gardening
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Transcript
Introduction to Plant Mathematics
00:00:00
Speaker
Welcome to Breaking Math, the podcast that explores the fascinating ways mathematics shapes our world. I'm your host, Autumn Feneff, and today we're diving into the intriguing question and book, Do Plants Know Math? Joining me are two incredible guests who are here to shed light on the remarkable mathematical patterns found in nature.
Christophe Collet: Geometry in Nature
00:00:19
Speaker
Christophe Collet is a mathematician and professor at Smith College, whose work focuses on the geometry and structures in the natural world from spirals and seashells to the arrangement of leaves. He's passionate about revealing how math alongside biology intersect in fascinating ways. Alongside him is Nancy Peck, a talented science writer and former reporter for the Baltimore Evening Sun.
00:00:49
Speaker
Her first book, The Rarest of Rare, was named one of the best science books of the year by Discover magazine. Nancy has a gift for making complex scientific ideas accessible and engaging.
Fibonacci and Plant Geometry
00:01:01
Speaker
Together with Christophe and two other authors, she brings the hidden mathematics of plants to life in their thought provoking book. In today's episode, we'll explore the mathematical wonders that govern the plant kingdom from Fibonacci sequence and sunflower seeds to the intricate geometry of broccoli.
00:01:24
Speaker
Stick with us as we uncover the deep rooted math in the natural world and ask the question, can plants really separately from here? So I took your bios and I just made that.
00:01:50
Speaker
It's a pleasure to have both of you on the show. Now, tell us a little bit about yourselves. You have very colorful backgrounds. And how did you two meet as co
The Path to Plant Math
00:02:00
Speaker
-authors? So I'll start. I fell into math a little bit by accident. um As a way to not do engineering, I was um ah finishing high school in Algeria, where my father was a geologist, and um came back to Paris and did an engineering preparation school. And I was bad at computing integrals, but I was good at topology. And so that that felt like, okay, and and I like literature. All my classmates were, you know, engineering, very tech-oriented and
00:02:42
Speaker
So I thought, oh, math is a little bit more poetic. you know So I switched to math and I ended up coming to the States to do my PhD. My mother was American and and so um I needed to come to the this country to keep my citizenship.
00:03:01
Speaker
and ended up studying dynamical systems, chaos theory, and this kind of stuff, symplectic pathology. And I, yeah you know, I was teaching at some point at UC Santa Cruz and a, um,
00:03:20
Speaker
um A graduate student of mine um did a project on the Mandelbrot set and phyllo taxes, and I thought this was totally wild. And, um. But it was.
00:03:37
Speaker
actually not so wild. And I ended up working with him and switched to bio math and plant math and the rest is history.
Collaborative Journey
00:03:50
Speaker
And how we met Nancy and I, we met at a party at a common France who was a pretty famous artist. and um And we started talking about what we were doing. And Nancy told me about her writing books, and in particular a book about ah the Museum of Natural History at Harvard. and And I told her about my work. And I don't know who said what, when, but within half an hour of knowing one another, we said, oh, we have to write a book together. And so so there we go. And then we brought in ah friends and colleagues that had been working with a physicist and a biologist.
00:04:36
Speaker
So this is a short story about how we got there. Yeah. And on my end, I started out as a newspaper journalist, but eventually morphed into writing books.
Nancy Peck's Science Communication
00:04:49
Speaker
And there were always about history and science, but with all kinds of different angles. So my first book, as Chris mentioned, was called The Rarest of the Rare. And that was about the incredible natural history collections at Harvard.
00:05:04
Speaker
And then over the years, I've also written a book about dinosaur tracks and another related to my family history, which led to me co-producing an off-off Broadway play about a mother murdered by her own son. And um most recently, I translated a French nonfiction book that was written by a woman obsessed with her ancestor, a man who survived a 19th century shipwreck by turning to cannibalism.
00:05:32
Speaker
So you can see I'm drawn to um all kinds of good stories, but of course, plant spirals are really the most compelling story of all. And um yeah, as Chris said, we were at a party given ah by a mutual friend. And um I was really drawn into the botanical world by the Glass Flowers Collection at Harvard, and I wanted to write a book about plants. But my agent didn't like any of my pitches.
00:06:01
Speaker
And then when I was talking to Chris, he had this incredible idea based on his research into plant spirals and the Fibonacci sequence. And I remembered that from um junior high school when we were shown like a film strip and it always stuck in my head how cool it was to um look at the Fibonacci spiral in plants. And I also remembered the movie Fantasia of all things. And I thought, oh yes, this is going to make a great book. and Plus, I have to say, I was a French major in college. So I'm always looking to use my French whenever I can. And we were off and running. And that was 14 years ago. It was a complicated project. And it did take us a little while to get the book written.
00:06:48
Speaker
I mean, that happens a lot. So it's been a while since I've used my f French. so Don't feel bad. Don't feel bad. All of the years of French private Catholic school. So I i completely understand.
00:07:08
Speaker
Now, tell me a little more about who's the audience. I noticed that there are some really cool lessons in the book. Was it difficult to communicate the world of plant spirals to it more of a general audience?
00:07:22
Speaker
So I would say, do plants know math? The title of our book. ah It belongs to a genre known as crossover books, which is to say they're trying to appeal both to a general audience and to a more specialized one like you, Autumn, at the same time.
00:07:39
Speaker
So given my background in journalism, I was determined to make this book readable by non-scientists like me. And the rule was I had to be able to understand everything we put in the book.
Making Science Accessible
00:07:51
Speaker
And we stuck to that, except maybe for the part about dynamical stability, given that physics is really, truly difficult to put into words. And I might be a little shaky on hyperbolic geometry, but that material appears in the appendix at the back of the book.
00:08:07
Speaker
which is where we put the more technical stuff. And the Nancy has to understand rule did not apply to what went into the appendix.
00:08:16
Speaker
she won She was a great enforcer. She was very gracious about it, but very firm at the same time. you know We got a lot of rejects, but but in retrospect, you know we felt, yes, she was right. so So Nancy was very instrumental in being our guide of how you can be understandable by a general audience.
00:08:40
Speaker
I think you folks have a level up, especially having Nancy on the team. You know, dynamical systems can be difficult at times. I know that. It's one of my favorite topics outside of here and topology, but it's really well written for the general audience.
00:08:59
Speaker
Now, how did you really become interested in the intersection of mathematics and plants?
Inspiration from Nature
00:09:07
Speaker
Each of you have your own story, but I know that you brought a lot of these topics together. Within maybe 30 minutes of chatting, what was the real inspiration for that? Well, I think I've already touched on the ah story of my graduate student, Scott Hutton, who did this project. and in Santa Cruz in my graduate seminar. And, um you know, ah again, it was for those in mathematics, um you know, the Mandelbrot set is related to what is called complex um quadratic maps of the form Z goes to Z squared plus C.
00:09:50
Speaker
And so if you iterate this map and on the complex number C, you get spirals. And those spirals, you can count them. And and so he classified those spirals within the Mandelbrot set. And this was totally parallel to work that people had done about plant spirals and in in biology.
00:10:14
Speaker
So there was an interesting parallel there. And then he started looking at models of how those spirals are formed in plants. And then it became his thesis and became his unofficial thesis advisor. And then I moved to Smith and we got him a postdoc at Harvard with our co-author Jacques Dumé.
00:10:36
Speaker
So this is this is how I got into it. But at the very beginning, it was very mathematical. And the more I go, the more I put i i put my hands into dirt. I'd still kill a lot of plants. But but i you know I worked with students. We analyzed data and all that. And so it's been fun to to go from a ah a theoretical mathematician to an applied mathematician in that way.
00:11:04
Speaker
Don't they tell you not to mix theory and application usually? i I love to do both, you know, and and um I still write papers with theorems and I write papers without theorems and simulations, you know, computer simulations and data. And and and I love the the possibility of going between all of these, right? Yes. And, and um you know, ah people and what's fun also is that people like Alan Turing ah spend the two years of
00:11:38
Speaker
the last two years of his life working on this subject. So it's it's fun to also connect with different people in history in that fashion. Now tell us a little bit about the history
Historical Foundations
00:11:51
Speaker
of plant math. When was it discovered?
00:11:54
Speaker
Yeah, so, you know, like any ah field, it's very hard to say, okay, it was that day, right? But ah we could go all the way back to Leonardo da Vinci, who in one of his notebooks, you know, saying, you know, here are four ways that plants arrange their leaves around their stems, you know, and then he goes in the first way, you know, the leaves go opposed to one another in the second and the third and, and on the fourth, he scratches it out. So my daughter and I tried to decipher that and we had a lot of fun ah decoding da Vinci in ah and so we have a hypothesis.
00:12:43
Speaker
But his um his classification is is very you know simple, and and but it's remarkable because it came you know two or three hundred years before anybody else noticed that, because of course nobody read his is notebooks. you know Let me just add that in the book we have a facsimile of Da Vinci's notebook so that you too can try to decipher the mysterious fourth mode.
00:13:13
Speaker
and um so but what ah the the one One time that really marks to me the beginning of the interaction between mathematicians and biologists is in 1754, at least that's the date of publication, of two Swiss scientists, a naturalist, Bonnet and a mathematician, Calendrini. So for instance, they did a classification of plant patterns that were much more solid that than Da Vinci.
00:13:48
Speaker
And their Bonaise study on on the the plant, the leaves, ah led to discovering that the leaves breathe, and that in itself also led to photosynthesis, the discovery of photo photosynthesis by other people. So it's ah it it is really an interesting ah a point of departure for the collaboration between mathematicians and and and biologists. But the really heavily mathematical ah biology collaboration really happened in the 1830s by two teams of
00:14:32
Speaker
a team of young German researchers, two young researchers, Schimper and Brown, that created the that came up with the word phytotaxis, for one, and systematically studied spirals and plants, systematically discovered that plants um displayed the Fibonacci sequence in their number of spirals. and And at the same time, independently, weirdly, a team of two French brothers, the Bravais brothers, who did a beautiful paper that was quite mathematical with proofs hidden in there. It was really written in a proof language, but it was for botanist. So it was ah a beautifully written ah paper that that proved they
00:15:21
Speaker
The relationship between the number of spirals and the angle. Between the successive leaves. In the in a plant and. um So this is really the beginning, I would say, of ah plant mass and and mathematical yeah biology, by the way, right? There's there's no other that I know. There's no other ah genesis story of mathematical biology other than this, right? And and then, of course, there was many, many other um ah collaborations between the two sciences that that occurred later in that century.
00:16:02
Speaker
Wonderful. Now, can you tell us a little bit more about some of the common mathematical patterns that we see in plants? Maybe we can dive into that. Sure. So um yeah, botanists look at patterns a little bit differently than mathematicians would would do. ah For a mathematician, it would be just spirals.
00:16:28
Speaker
and um And spiral lattices and, um, but for botanists, they will make a difference between, uh, say. A grass where you would have a leaf on 1 side and another 1. Going at 180 degrees. Opposite to it, but.
00:16:51
Speaker
upper level and then again and again. um So that's one possibility. But you could see that as a spiral or a helix going at a 180 degrees ah um angle while it goes up. um you have something that's called world phytotaxis, where leaves grow all at the same level in groups of two, three, four. And then the next level, you have again, three, four leaves at an angle that is right in between the former leaves, the the node that's right below, and so on and so forth.
00:17:39
Speaker
But, and then there's the famous spiral phytotaxis, which is really the one that interests us. And that's, um when you take a leaf and then you turn a certain angle ah called the divergence angle and you you go up the stem and then again, the same angle, the same pitch. So in in other words, you you you follow a helix and you put a leaf at a given angle, right? So that's the, ah how should I say, that's that's the theoretical model.
00:18:14
Speaker
Right, but this is not how the plant thinks, right? This is not the way plants thinks. They don't say, oh, I'm going to make an a helix and I'm going to follow that angle precisely.
00:18:26
Speaker
um But those once you do ah a helix like this, and if you compress that helix, ah the points that are closer to one another form other helices, and those are the spirals that we see in plants. So if you think about a A pineapple, for instance, its scales form rows and the rows go as helices go in two directions. And usually you have 13 rows going in one direction and eight going in the other.
00:19:00
Speaker
And um so this is the spiral phyllotaxis, right? You can see ah also versions of that flat in flowers and in fluorescence like daisies and and um or even the the bottom of a pine cone. You can see the the the you know a flat version of those spirals and sometimes those spirals are ah logarithmic.
00:19:25
Speaker
And sometimes they' are ah there they they go more as a square root of the angle. So these are the patterns that we see in nature. And again, what's remarkable is the ah the fact that um the number of spirals is very, very often ah successive numbers of in the Fibonacci sequence.
00:19:54
Speaker
So for those who don't remember the Fibonacci sequence, you start with one, one, you add them, you get two, you add one and two, you get three, you get you add the two last one, two and three, you get five and so on. So it's a recurring pattern. um So I can go on a little bit more about why this pattern, but I'll let you ask you next question.
00:20:20
Speaker
I suppose that there's a lot of stuff in literature and the internet about the Golden Ratio, whether it's in plants or art. So what's true and what's unfounded? Yeah, so I mentioned about the fact that, you know, ah one way to build a um spiral structure is to put points on the helix at a given angle. so ah The French mathematicians ah and and botanists that I mentioned of 1837 posited that the this angle would indeed be the golden angle, which is related to the golden ratio. right The golden angle cuts the circle in a golden ratio proportion.
00:21:16
Speaker
and um So, they said that because because of their understanding of the relationship between the angle and the number of spirals that are visible.
00:21:30
Speaker
ah they They proved that, and that that was really a proof, and it it became known eventually as the fundamental theorem of final taxes. They proved that if you do i ah ah you use the the the golden angle, then you will get a Fibonacci number of spirals.
00:21:53
Speaker
So they say, this is this is it. you know This is a beautiful a mathematical answer to the mystery of why Fibonacci. Just do the golden angle and you'll get Fibonacci spirals. However, ah you know in simulations and models that we've been using, simple models, we see that this rule is it's broken again and again, and you can you can get Fibonacci phyllo taxes without sticking hard to the golden angle rule. um As for the occurrence of the golden ratio in the art, this is a very controversial subject and we tend
00:22:44
Speaker
um The authors in this book tend to think that there's no such thing until the 20th century. And there was a kind of a a ah media coup by a couple of authors in the 19th century that pushed that theory that everything was understandable.
00:23:05
Speaker
by the golden ratio. So they they started analyzing all kinds of art, the Parthenon, the Da Vinci ah paintings and all that, and the the imposed models of the golden ratio proportions.
00:23:21
Speaker
um The problem with that is it was never precise and it was never documented by the artist, including Leonardo da Vinci, who illustrated a book about the golden ratio, but never talks about it in his art, and there's absolutely no indication that he used the golden ratio in his art.
00:23:43
Speaker
same thing with the Parthenon, you know, the the way that people measure those proportions was usually, oh, how many steps in the stairway are you going to use to make your golden rectangle? So so it's very dubious that indeed those artists until the 20th century used the golden ratio in their art, you know, so Le Corbusier, the corbu yeah the the architect did explicitly use the Fibonacci ah members in his art, in his architecture, but um that's documented. I find that to be very interesting because we look at something for the golden ratio and we think whether it's a face or a structure to be perfect, but having that not really brought up until more recent times and applying that into
00:24:39
Speaker
whether it's plants or art or any sort of concept, makes you really reflect on what people think is beautiful. Or is the question of more is beauty symmetry? Or is it going to be something some people find, you know, an imperfection or something completely asymmetric to be beautiful?
00:25:08
Speaker
So is it practicality? Is it application? It's kind of left a little bit open-ended with that. Now it makes you really think about why people are also drawn to spirals and it And when you're having something that's spiraling, ah you know, is that going to be perfectly symmetric? Is that going to ah have a lot of different applications?
Beauty and Math in Nature
00:25:42
Speaker
But for plants, why are people actually drawn to plant spirals? How does this topic pull such a wide range of people into the world of science?
00:25:53
Speaker
and that Well, to me, i mean personally, I find that plant spirals are just really beautiful. Like if you look at the cover of our book, which has a spiraling fern, it's not a Fibonacci spiraling fern, it's a little bit tricky, but um it is a very beautifully spiraling ah fern.
00:26:17
Speaker
It's a little bit hypnotic. I mean, looking at numbers, maybe that brings a kind of joy, but have you ever really stared at the center of a succulent or at the seeds of a sunflower? ah I personally think that plant spirals are a kind of gateway drug to math and science.
00:26:36
Speaker
I mean, in a way, that's how it worked for me. And that's also really one of the goals of our book is to use these spirals to pull in readers who wouldn't normally pick up a science or a math book at the bookstore. Maybe they're gardeners, you know, maybe they have a very casual interest in patterns and design, but they might not pick up a book ah that introduces math and hard science without the hypnotic fern on the cover. um And there are many beautiful photos in the book of plant patterns, but I think also sort of draw people into our world. um the The spirals have also pulled in scientists from many different disciplines, which is kind of interesting. And that's why our book has three scientific co-authors, a physicist, a biologist, and Chris, our mathematician.
00:27:33
Speaker
And um also, there's just something about this enduring mystery. I think it was nicely stated in one of the epigraphs at the beginning of our book by our biologist co-author Jacques Dumé, who is a professor in Valparaiso, Chile, and he writes,
00:27:49
Speaker
Phylotaxis is so appealing. It seems accessible, like pie. So many people have tried to explain pie. And it's the same with phylotaxis. Everybody goes out and says, I'm going to solve this riddle. And every 10 years, someone puts out a theory. And often, it's a step backwards.
00:28:11
Speaker
Or it's a spiral. Or a spiral, exactly. like we we We toyed with this you know model of historical model of of the evolution of science as a spiral. yeah you you You kind of come back to the same place, but maybe you moved ah out a little bit. You moved up the the the understanding a little bit, but but a lot of it is reinventing the wheel. And I plead i plead totally guilty. you know what What I think is was one of my um more important
00:28:48
Speaker
um thought about phytotaxis, somebody already had the germ of that idea more than a hundred years ago. right and so um But it's exciting to to place yourself that way and ah in the history of um of science. ah one One example that might illustrate your question of why the ah found the fascination for plant spiral is the story of why Alan Turing got interested in it, right? He essentially, he wanted to understand life. Yeah, pretty ambitious. But is my and but if you think about life is, you know, you you you start with an end differentiated body of cells, you know, a few cells, and then suddenly they become differentiated.
00:29:46
Speaker
And they create patterns. And how? Why? So he came up with this immensely um ah popular now um paper on morphogenesis and, you know, positing the existence of two um two chemical components ah reacting with one another and forming out of uniformity, ah out of uniform ah distribution of the chemical forming ah patterns, right? And he said, well, the the uniform equilibrium is unstable. So it's called ah the Turing instability.
00:30:31
Speaker
But at the same time, he was trying to think about how to apply this model to phyllo taxes because he thought that phyllo taxes was the simplest pattern formation you could think.
00:30:45
Speaker
right It seems so repetitive and so simple that certainly there was a simple model that could explain it. So one of the first computer program he ever wrote on of one of the ever first computers in the world was a program to model phyllo taxes. And that's because it was a simple model of life as pattern formation.
00:31:11
Speaker
Now talk to us about phyllotaxis.
Challenging Traditional Patterns
00:31:14
Speaker
What is it on a higher level? There's two types, correct? Fibonacci type and quasi-symmetric. How can we tell the difference?
00:31:25
Speaker
Yeah, so the quasi-symmetric, we talked about final um ah Fibonacci phyto-taxis launch here. So what what about this quasi-symmetric thing? So this is ah actually a, um I should not say invention, but but um observation from our co-author, Stephane Doherty.
00:31:50
Speaker
um Well, it came out with our collaboration and ah in the modeling that we were using. And I was saying, oh, you know, if you up the parameter of this disk stacking model, and if you make the radius of the disk decrease too fast, then you don't get Fibonacci phyllo taxes anymore. And he said, so what do you get?
00:32:18
Speaker
And I said, well, you know, the the number of spirals is not Fibonacci, but it's, they they tend to have the same number of spirals on on the same, ah on the on either side. So it would be N and N, or N and N plus one, or N minus one and N, or N and N plus two, but they were, the number of spirals were very close to one another.
00:32:44
Speaker
And he said, oh, this is this is what I've seen all these years and in the corn. And ah and so we we started looking more at plants and indeed court corn doesn't do Fibonacci phyto-taxis. It does this pattern that ah we saw in those simulations. And so he called them quasi-symmetric because those in a Fibonacci phyto-taxis the The spirals in one direction in the the directions with more of them is going to be more vertical are going to be more vertical and the spirals in the other direction are going to be more ah flat. And so they're not symmetric, but if the number of spirals are about the same.
00:33:34
Speaker
then the slope of those spirals is going to be about the same. So that's that's where the word quasi-symmetric came from. And the more we go, the more we find these quasi-symmetric patterns, usually in inflorescence and fruits. so You know, I did study with students this summer where we picked strawberries and on the Smith campus, wild strawberries, and they're all quasi symmetric. And we looked at Jack in the pulpit. We looked at skunk cabbage and and ah peace lilies. And all these are quasi symmetric, the inflorescence.
00:34:15
Speaker
And so, so yes, we want to bring the attention to these other types of patterns that people essentially didn't recognize until recently. Oh, exciting. Now that brings the question of is Fibonacci phyllotaxis optimizing something? Does that have an evolutionary advantage at all?
00:34:41
Speaker
Yes, so that's that's ah a a controversial subject as well. So there are studies that show that, you know, ah there's a packing ah optimization that occurs if you do Fibonacci phyllo taxes.
00:34:58
Speaker
but other and and with the golden angle, but other angles, other noble angles. ah So we're talking about never theory here. ah Noble angles would be angles whose continued fraction expansion would be a long sequence of ones at the end.
00:35:21
Speaker
um would, those noble angles would give the same result. So why Fibonacci and not another angle? um But nonetheless, there is some kind of packing advantage of the Fibonacci phytotaxis and, um but Contrary to the common thought, it's probably not to optimize light absorption because that depends greatly on the shape of the leaf. And leaves are very good at, you know plants are very good at turning their leaves towards the sun if need be. So why would they spend their energy in ah optimizing the ah leaf placement? Something that I believe is more
00:36:13
Speaker
um more promising as a hypothesis is the fact that all these structures are born at, went inside a bud of a plant.
00:36:26
Speaker
ah when the pattern is very, very small. And if you want to optimize your energy in material ah production, ah you better do your pattern in the smallest ah area possible so as to optimize ah material production. So that's that's something that speaks to me at least. I'm not i'm a mathematician, right? I'm not a biologist.
00:36:52
Speaker
And also some people have ah argued that maybe there's a mechanical advantage in having organs that are spread out around the periphery. There are a lot of self-repeating patterns in plants. The thing that I think about quite naturally is Romanesco broccoli. Everybody's familiar with it and how it repeats. so Tell us a little bit about that. Is it infinite? You talk about materials packing. It reminds me of the coastline paradox as well when you see that example of how it's multiplying.
00:37:34
Speaker
Yes, so as in the coastline, right, um ah it is the a ah model of a fractal, which is not um completely mathematical, right? Because in the model of fractals, the mathematical of fractals, you have um you know infinite jaggedness at any scale whatsoever, right? Of course, you cannot have that in nature because you know you you you you pretty soon go into subatomic scale and then it doesn't make sense to to talk about the jaggedness of that. But ah when people think about fractals, they often think about self-similarity.
00:38:24
Speaker
And we had this nice um discussion in the book. and And outside, as we were writing the book with with our co-author, Stephane Duadie, the physicist, um about what a fractal is, and and is the broccoli romanesco a real fractal?
Fractals and Self-Similarity
00:38:43
Speaker
And so,
00:38:44
Speaker
um So I would argue no, because it does not have this infinity jack infinite jaggedness in it. and um And more technically, it doesn't have the Hausdorff dimension. that's That's not an integer.
00:39:05
Speaker
um But ah nonetheless, the concept of fractal is striking in a broccoli romanesco and that's in fact how Stefan got into this field because he saw this broccoli romanesco in a market in Paris and he started looking at the fractal ah properties of it. And then he relies later on, oh, there are spirals too. um But There is this self similarity in broccoli romanesco is really ah captivating because it has seven generations so of self similarity. So you take the the whole.
00:39:50
Speaker
ah flower, the whole broccoli, and then it's composed of many florets. So you look at those florets, and within those florets, there are baby florets. And then within those florets, there's baby, other baby florets, and so on and so forth, all arranged in spirals. And ah we show, and Stefan dissected one of those ah Romanesco broccolis. and and he got to 7th generation of self-similarity, which is pretty amazing.
00:40:21
Speaker
um yeah so um So there we go. They don't go to infinity, but they're strikingly fractal in a friendly sense of the term of that self-similarity. That's exciting to see. Now, i when I was going through the book, there was a little bit of a tale from cell division. And you parallel that to soap bubbles. Tell us the history of that here.
00:40:55
Speaker
Yeah, so ah so that that was worked by our um colleague Jacques Dumé, the co-author, biologist. And so one year he had accepted to take on a a high school student, actually a local student from Northampton and was the son of one of the botanical garden people.
00:41:20
Speaker
And he took him and in his lab, and he was you know looking for work for him to do. And ah he had had the idea that there might be a connection. Well, there there's somebody who had came up come up with the idea that cell divide ah according to very special rules. So they should divide in such a way that the ah The new membrane that separate the the the too so the cell in two should be perpendicular to the walls of the existing cell and in such a way that the area that it divides the cell in.
00:42:04
Speaker
ah Are equal so that that guy was is called Herrera from the 19th century if I recall and so um so Jack started to have this this high school students work on manipulating some bubbles and and ah you know, pushing the the the little wall in between the that separate the bubble ah with little instruments very delicately and pushing them around. And it discovered that, yes, there's an optimal solution for a given
00:42:41
Speaker
um shape of a bubble, but there could be also, you know, local minima or local optimum that also were semi-stable for the the bubbles.
00:43:00
Speaker
And so he started then creating a a mathematical model. Jacques and some postdocs ah created mathematical models of of how to model cell division. and And they found out a number of ah possible equilibria.
00:43:22
Speaker
and And some of those equilibria are indeed ah ah seen in plants. So some are realized in nature and some are not, but some are definitely realized. And and so it explained how ah how nature actually does follow some kind of optimization rule for its cell divisions. And it creates a dynamical system that has stable equilibria. And so so they were able to classify those stable equilibria, which is a really beautiful biomass kind of endeavor.
00:44:00
Speaker
Wonderful. Out of adam curiosity, when I was flipping through at the end, a little Easter egg for anybody who hasn't read the book, ah there was a chapter on recipes.
Culinary Inspirations
00:44:15
Speaker
What inspired that?
00:44:18
Speaker
Well, first of all, as you've noticed, Chris and Stéphane, the physicists, are French, and of course food is a really important part of French culture, and and I had ah spent two years living in France in my youth, and of course that infected me for the rest of my life, and I think yeah All four of us co-authors love to cook. Well, actually, I'm not sure about Jacques. Does Jacques like cooking? but it's why now he's a He's a good cook. He's a good cook, see? So um the recipes were really another strategy for making the book accessible and fun ah for people who might not otherwise pick up a hard science book. And of course, they play off the content of the book.
00:45:01
Speaker
ah There's a recipe for sauteed Romanesco broccoli, which as Chris mentioned is the very plant that first inspired Stiff on the Physicist to begin studying plant spirals. There's a recipe for red cabbage.
00:45:16
Speaker
I don't, maybe this is a spoiler, but if you take a red cabbage and you slice it through the equator, you get these absolutely gorgeous spirals that you can see in both halves. And just the thought that a common cabbage, which is sort of like ah a lowly, not very exciting vegetable would have these beautiful spirals at the core of it for me was very thrilling. um And then there's a recipe for strawberry strudel,
00:45:43
Speaker
which is based on one from my great aunt Thelma who lived to be 100. She was Hungarian and she was an incredible baker. um I modified the recipe from apples to strawberries because strawberry seeds show those quasi-symmetric properties that Chris talked about before.
00:46:03
Speaker
So it's a way to celebrate your um journey through the book. If you get to the end, you know, we thought, ah, this will be this will be fun for people. And we actually hope that people will have a little Philo Texas feast and send us pictures from from their dinners.
00:46:24
Speaker
Now, with all of these recipes that have been in the book, um I'm just thinking about Julia Child always cooking at Smith College, right? um Are there any notable women in the field of plant math that may or may not be in the book?
Women in Plant Mathematics
00:46:43
Speaker
I have to say, as a woman in STEM, Chris, you work at Smith. It's one of the biggest institutions for women.
00:46:52
Speaker
ah Are there any notable women and that do this sort of work that you're aware of? Well, ah in the past, not enough, right? And um in the present, a little bit more. And that' that's hopeful. um But yes, there is a barrier to entry for women in math in particular. And Smith College you know fights that mightily. We created a Center for Women in Math with a post-bac program
00:47:24
Speaker
that has had a pretty big influence on how many women pursue their PhD and pursue mathematical careers and in ah henceforth. So ah in terms of, ah you know, bio-math or plant math, we could not find too many examples, unfortunately. One beautiful example is um ah the sister of this German naturalist,
00:47:54
Speaker
Brown, Cecilia Brown, made all his illustrations and they are absolutely gorgeous and very, very precise ah rendition of pine cones. And, ah you know, with all their angle structures that had to be right and and she did a ah beautifully at the same time, precise and artistic work there. ah There is also the example of a pair of a husband and wife, the snows, that have come up with some important hypotheses on how phyllotaxis happens.
00:48:32
Speaker
um And in the the more recent ah ah times, um I personally worked with this woman called Chauvin Braybrook, and she's a developmental and mechanobiologist who is now at UCLA. And, ah you know, she is great at integrating any science together. And so she's a firebrand.
00:49:03
Speaker
um she's She's a ah great scientist and and open to math, mechanics, and physics, and of course, biology. ah There's ah also a rising star in Finland. Her name is Paula Eloma, who did a great amount of work on the phytotaxis of Gerber daisies with a ah very integrated um team of scientists from all sides.
00:49:28
Speaker
computer science, math, and and you know physics. and ah So she's very good, too. Wonderful. Now, we've talked a little bit about highlighting stories about people who were less well known. Could you particularly pick one out, especially one that surprised you?
00:49:53
Speaker
so Here's where I was going to talk about bonnet that Chris stole my bonnet story.
00:50:03
Speaker
Okay. Okay. It happens. Okay. So maybe I'll just do my spiel and you can choose one. Go for it. Go for it. Was there anything in general that surprised you? How about that?
Bonnet and Plant Studies
00:50:20
Speaker
Well, my personal favorite character in the book is the Swiss scientist Charles Bonnet, who lived from 1720 to 1793. He faced a number of physical challenges, being deaf from an early age. ah He started out studying insects, and he discovered that butterflies breathed through tiny pores, which he named stigmata.
00:50:45
Speaker
ah spiracles, I think we call them now. But all those years of close observation under a microscope weakened his eyesight, and so he turned to plants instead as they were larger and easier to observe. ah He became the first person to organize leaf arrangements into five neat categories, which was quite important. And he was also the first to mention the word spiral in relation to phyllotaxis.
00:51:11
Speaker
So he performed these early science experiments before the scientific method had even really been established. And from his observations, he believed that he had proved that plant leaves are different on the top than they are on the bottom because the leaf bottoms absorbed dew. The thing is, he thought that dew came up from the ground and knocked down from kind condensation in the air. And even if some of his conclusions seem a little silly today,
00:51:38
Speaker
He did truly groundbreaking research on plant respiration, which led to our modern understanding of how plants breathe in carbon dioxide and breathe out oxygen. ah Bonet was also the first to describe what is today known as Bonet syndrome, the visual hallucinations that are caused by the brain when it's struggling to adjust to vision loss. He observed this phenomenon in his grandfather who had cataracts. These are visual hallucinations caused by the brain struggling to adjust to serious vision loss. He observed this phenomenon in his grandfather who had cataracts.
00:52:16
Speaker
a man who was perfectly level-headed, but who saw strange images of birds, carriages, and buildings rising up before his eyes in all their details, seemingly very real. And Oliver Sacks wrote about Boney syndrome in his book, Hallucinations.
00:52:36
Speaker
Wonderful. Now, out of curiosity, do you have any recommendations for further ratings or resources? for listeners who want to delve deeper into plant math and history.
Further Reading Suggestions
00:52:51
Speaker
So, of course, our book. Yeah, so there's really nothing like it out there. um There is, for the for the more mathematical-minded,
00:53:03
Speaker
um there is somebody at in England who studied ah very much Turing and in his influence on phytotaxis. He rediscovered the papers on phytotaxis, interpreted them, and made them known, and he had this wonderful um but citizen ah science experiment where he had people counting spirals on sunflowers.
00:53:31
Speaker
um He's come up, he teaches a course on phytotaxis at Oxford, I think, or Cambridge, I can't remember Oxford, I believe. and um And he came up with a book, and I can't remember if it's out yet, but the book exists, and I i you know i would recommend it very much. so His name is Jonathan Swinton. ah He wrote a beautiful book about ah Turing in Manchester at any rate, which I would recommend greatly. Oh, in Turing's Manchester. yeah Yes.
00:54:05
Speaker
ah And then for me, um there are two very well-known popular science writers ah who both had books that were helpful for me when I was first getting into this topic. um Ian Stewart wrote The Beauty of Numbers in Nature, a classic and another classic.
00:54:28
Speaker
is Philip Ball's patterns in nature. There's a more academic version, and then there's a more popular version, but they both have beautiful pictures as well, and very beautiful science writing, just really a model and an inspiration. And I'm a currently reading a new book about botany. It's not plain spirals, but it's a lot of fun. It's called The Light Eaters, and it's by ah an author named Zoe Schlanger. Wonderful.
00:54:55
Speaker
Um, to wrap things up for today, is there anything that you want folks to take away or that we haven't covered?
Finding Joy in Nature's Patterns
00:55:06
Speaker
I would say, like, um, she benet himself.
00:55:11
Speaker
does in the the beginning of his book is look around, walk in nature, look at nature. you know She will teach you everything you need to know. Well, I would just say that it has been a real privilege for me to be with people who do exactly that on a kind of extraordinary level. You know, it's just a tremendous amount of fun to be with scientists as they are observing in the field. um And I now have um a microscope at home. I'm constantly
00:55:47
Speaker
taking things out of my garden and looking at them in detail in a way that I never would have before. I find it really very thrilling to see the patterns in nature. I think um i think they're a real source of joy. Wonderful.
00:56:01
Speaker
I thank you both for coming on Breaking Math podcast. And for listeners, if you're interested in getting the book, Do Plants Know Math? It is on sale in the United States on September 24th, 2024.