Hey there, curious minds! Have you ever noticed how a small part of a tree looks a lot like the entire tree? Or how the roots underground resemble the branches above? It’s pretty fascinating! And it’s not just trees—blood vessels, rivers, lightning, and even broccoli have similar patterns. Once you start seeing it, you can’t stop! So, what’s the connection between all these things?
The answer lies in something called fractals. Fractals are special shapes that repeat themselves at different sizes. If you zoom in or out, you’ll see the same pattern over and over. This is called self-similarity. Imagine a triangle filled with smaller triangles inside it, repeating infinitely. Mathematician Benoit Mandelbrot named these shapes “fractals” because they exist in a space that’s not quite one dimension, not quite two, but somewhere in between.
Let’s talk about dimensions. Normally, we think of dimensions as length, width, and height. But Mandelbrot’s idea of dimension is about how shapes fill space. For example, if you double the length of a line, you get twice as much line. For a square, doubling the sides gives you four times the area. For a cube, it’s eight times the volume. This is where fractals get interesting. When you scale up a fractal, you get more of it, but not in a whole number dimension like 1, 2, or 3. Instead, you might get something like 1.585 dimensions!
Fractals aren’t just cool math concepts—they help us understand nature. Trees, for example, grow in fractal patterns to soak up sunlight and carbon dioxide efficiently. By branching out, they maximize their surface area without using too much energy. Tree roots do the same to absorb water and nutrients.
Inside our bodies, fractals are at work too. Our lungs have a branching pattern that allows them to hold a lot of air in a small space. This helps us breathe efficiently. Our blood vessels also branch out like fractals, ensuring that oxygen and nutrients reach every part of our body.
Fractals aren’t just found in living things. Rivers branch out to drain water efficiently from land, and lightning bolts spread energy through the air in fractal patterns. Even cracks in surfaces and the way snowflakes form can be explained by fractals.
What’s amazing is that there’s no single rule or gene that makes all these things grow in fractal patterns. Each system, whether it’s a tree, a river, or a lung, has evolved to be as efficient as possible. This has led them to develop similar fractal patterns, even though they solve different problems.
Fractals show us that nature is full of surprises and hidden patterns. They let us see the world in a new way, making everything infinitely interesting. So next time you look at a tree or a river, remember the magic of fractals and stay curious!
Grab some paper and colored pencils, and let’s create fractal art! Start by drawing a simple shape, like a triangle or a square. Then, inside that shape, draw smaller versions of the same shape. Keep repeating this process to create a beautiful fractal pattern. Notice how the pattern looks similar at every level!
Head outside and explore your surroundings. Look for natural objects that have fractal patterns, such as leaves, trees, or clouds. Take pictures or sketch what you find. Share your discoveries with the class and discuss how these patterns help the objects function efficiently in nature.
Using a ruler and graph paper, draw a simple line, square, and cube. Measure and calculate how their dimensions change when you double their size. Then, try this with a fractal pattern, like the Sierpinski triangle, and see how its dimension changes. Discuss how fractal dimensions differ from whole number dimensions.
Research how fractals are present in the human body, such as in the lungs or blood vessels. Create a poster or presentation explaining how these fractal patterns help our bodies function efficiently. Share your findings with the class and discuss the importance of fractals in biology.
Use a computer or tablet to explore fractal simulations online. Websites and apps can generate fractal patterns like the Mandelbrot set. Experiment with zooming in and out to see the self-similarity of fractals. Share your favorite fractal discoveries with the class and discuss what you learned from the simulation.
Sure! Here’s a sanitized version of the transcript:
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Hey, smart people! Joe here. Ever notice how if you look at part of a tree, it looks a lot like an entire tree? And why does this underground part of a tree look so much like the rest of the tree? That’s pretty interesting. This isn’t a tree, but it sort of looks like one. And so does this. And these branches sure look an awful lot like these branches, except those are blood vessels, and so are these, which also kind of look like a tree, although this part reminds me of a river. Lightning, lungs, cracks in the ceiling—what’s going on here? Why do all these things look so similar? Once you start seeing it, you see it everywhere. It’s like there’s some connection between rivers, lightning bolts, broccoli, trees, and all sorts of living and non-living things.
Well, all these objects have one thing in common: zoom in or out, and we see the same branching pattern repeat itself over and over at different scales. These are fractals, a special kind of self-similar shape that mathematicians, and the rest of us, find fascinating. And this video is about why we see them everywhere.
I don’t know if you’ve ever looked at a tree as deeply as I have, but that interesting thing where part of the tree also looks like a tree is called self-similarity. It’s like one of those triangles with an infinite number of smaller triangles inside it. Unlike the self-similar shapes we see in nature, these perfectly self-similar shapes are infinite. We could zoom in or out and continue to see those patterns repeat forever! Mathematician Benoit Mandelbrot named these self-repeating shapes fractals because they exist sort of in between dimensions or in fractured dimensions.
What does that mean? Let’s take a quick sidebar to talk about how mathematicians use words differently than we do. You and I think of dimensions as the three that we live in, or the two that exist on paper, or even the one dimension of a line, because that’s what we learned in geometry class. What Mandelbrot meant by “dimension” has to do with how different shapes fill space as they get bigger or smaller, and this is key as we explore fractals in nature.
You can double the length of this line, and you get twice as much line. Another way of saying that is you scale it up by two to the power of one. If we do the same to a square, double its length and width, you get four times as much square, or you scale it up by two to the two. Do it to a cube, double the length, width, and height, and we get eight times as much cube or two to the three. This power is the dimension Mandelbrot was talking about, and for simple shapes, it matches with our usual idea of dimension.
But what’s interesting about a fractal is that when you scale it up by two, you get three times as much fractal. That exponent isn’t one or two; you get 1.585 dimensions. Even though the fractal sits in a two-dimensional plane, just like a regular triangle does, when you scale it up, it doesn’t fill space quite the same as a two-dimensional object. The same thing is true for fractals with volume. To a mathematician, it’s more than two-dimensional, but not quite three-dimensional. Fractals exist in this weird in-between space, and that’s part of what Mandelbrot found fascinating about them.
Mandelbrot pointed out that fractals are not just a toy for mathematicians to create art. They can help us understand nature better because they’re everywhere. To start off, why do trees even look like trees? Biologically speaking, there’s no such thing as a tree. Sure, there are things we call “trees” because of the way they look. But if you look at a tree closely, many of the plants we call “trees” are more closely related to things that aren’t trees and more distantly related to other things that do look like trees. So, “tree” is just a way of describing plants that look kind of tree-like.
It’s almost as if growing fractal-like branches that look similar at different scales was the solution to some problem that all these different plants faced, and that problem is soaking up sunlight and carbon dioxide. Growing tall is one solution to that problem, or maybe growing just a few gigantic leaves on top of a trunk, or even a canopy the size of a city block with all the leaves on the very tip. But all of these options require spending a lot of energy to grow for not much gain. Luckily, there’s a better way to do it, and that’s where being a fractal is really useful.
A perfect fractal lets you put infinite surface area in a finite amount of space. This snowflake isn’t getting any bigger, but you can keep zooming in and keep finding another smaller layer just like the first. You can keep doing this forever, meaning its outer edge, the line you need to draw this shape, is infinitely long. Trees do something similar; by growing out each level as a smaller version of the previous level, a tree can pack a lot of surface area into its volume. It’s not an infinite amount like a mathematically perfect fractal, but it’s a pretty cool way of soaking up more sunlight without wasting energy by getting too bulky.
And it’s no coincidence that tree roots grow in a similar way; they need lots of surface area to soak up water and nutrients, and fractal branching is the most efficient way to maximize the volume that the tree can draw from without wasting energy building plumbing that’s too big. Meanwhile, inside our bodies, we have our own little trees. A lung’s job is to take in oxygen, and an adult body needs around 15 liters of oxygen every hour. If our lungs were just two balloons, they’d never keep up. Fractal branching means our lungs can hold half the area of a tennis court while staying packed nicely inside our chest.
And our lungs aren’t the only trees we have inside us. Our entire circulatory system looks kind of like a bunch of fractal branches too. We have almost 100,000 kilometers of blood vessels in our bodies delivering oxygen and nutrients and removing waste. Fractal branching lets our circulatory system pack in as many blood vessels as we need to connect every point A with every point B while also spending the least possible energy building our body’s plumbing and manufacturing all the blood that runs through it.
In a way, it’s like each of these living systems has a goal. A tree wants to soak up sunlight and carbon dioxide, a lung wants to take in air, and a blood vessel wants to exchange nutrients with every cell in the body. In all these cases, fractal branches are the most efficient way to scale up while staying basically the same size. This secret pattern shows up in non-living things too. All around the world, from their sources to their ends, rivers arrange themselves into branching shapes. At their source, fractal branching is the most efficient way to drain water from a given area of land. And at their mouths, we see fractal branching as sediment piles up and splits a river into smaller and smaller strands.
Cracks and lightning bolts are both ways of dissipating energy, and it shouldn’t surprise you that fractal branches are the most efficient way to do that inside a given space. When scientists model all these ways of growing, it turns out that, like perfect mathematical fractals, these branching shapes are best described as in-between dimensions. At this point, it might be tempting to think there’s one universal rule that underlies every branching fractal pattern that we see around us, but as usual, nature isn’t so predictable.
We also see fractal branches in crystals, the shapes of snowflakes, and even strange mineral deposits that people sometimes mistake for ancient plant fossils. Similar fractals, but for different reasons. Here, things like temperature, humidity, and the concentration of different chemicals act as a set of rules for building the structure. As these structures grow, those rules repeat themselves at multiple scales, giving us self-similar fractal shapes.
What’s amazing is that as much as these fractal shapes pop up in nature, there isn’t a single gene or law of physics making all these things grow fractal branches. But one by one, as each of these systems evolved to be as efficient as possible, they all landed on the same solution to their individual problems, letting us look at things in an interestingly new dimension and making them infinitely interesting. Stay curious!
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This version maintains the essence of the original transcript while removing any informal or potentially confusing elements.
Fractals – Complex geometric shapes that can be split into parts, each of which is a reduced-scale copy of the whole, often found in nature. – The branching pattern of trees and the structure of snowflakes are examples of fractals in nature.
Dimensions – Measurements that define the size and shape of an object, such as length, width, and height. – In mathematics, a cube has three dimensions: length, width, and height.
Patterns – Repeated designs or recurring sequences that can be found in numbers, shapes, or natural phenomena. – The Fibonacci sequence is a famous pattern that appears in various biological settings, such as the arrangement of leaves on a stem.
Nature – The natural world, including plants, animals, and landscapes, often studied to understand biological processes and structures. – Scientists study the patterns in nature to learn more about how ecosystems function.
Self-similarity – A property where a shape or pattern looks the same at different scales, often seen in fractals. – The coastline of a country can exhibit self-similarity, appearing jagged and irregular whether viewed from afar or up close.
Trees – Large plants with a trunk, branches, and leaves, which play a crucial role in ecosystems by producing oxygen and providing habitats. – The branching structure of trees is an example of a fractal pattern in biology.
Lungs – Organs in the respiratory system that allow for gas exchange, taking in oxygen and expelling carbon dioxide. – The branching of airways in the lungs resembles a fractal pattern, maximizing the surface area for gas exchange.
Blood – The fluid that circulates in the bodies of humans and other animals, delivering nutrients and oxygen to cells and removing waste products. – Blood flows through a network of vessels that branch out like a tree to reach every part of the body.
Rivers – Natural flowing watercourses, usually freshwater, that flow towards an ocean, sea, lake, or another river. – The branching patterns of rivers can be modeled using fractal geometry to understand their complex networks.
Energy – The ability to do work or cause change, which can be found in various forms such as kinetic, potential, thermal, and chemical. – Plants convert sunlight into chemical energy through photosynthesis, providing energy for themselves and other organisms in the ecosystem.
