Want to get a taste of the laws of the universe? Then you have to take a bite of this vegetable

The Chinese film "Cosmic Exploration Editorial Department" released some time ago has received a lot of praise from the audience. In the film, the spiral shape appears many times, for example, Tang Zhijun saw two spiral lights in the video recording; Sun Yitong used bubble gum to pinch the scene he saw into a double spiral; Tang Zhijun also saw the spiral shape on the mural in the cave; until the special effects at the end of the film, it was also made into a spiral.

Movie poster of "Space Exploration Editorial Department"

The film's point of view is that the spiral is "the law of operation of all things in the universe." From galaxies in the universe to the centers of various plants to the genetic material DNA of living organisms, they are all spiral structures.

Back to reality, many things also grow and operate according to the law of "spiral".

01

Spirals in plants

1. Romanesco Broccoli

Romanesco, a hybrid of cauliflower and broccoli, is an example of fractal chemistry.

The surface of its flower head is composed of many spiral-shaped florets. The whole Roman broccoli has nearly 300 such florets. Each floret is a small cone and is also tightly arranged in a spiral shape.

Roman broccoli, copyrighted image, unauthorized reproduction

The florets are arranged in a spiral with the center of the flower ball as the axis of symmetry. This pattern is a natural representation of the Fibonacci sequence (also known as the golden sequence).

The first two terms of the Fibonacci sequence are both "1", and starting from the third term, each term is equal to the sum of the previous two terms, for example: 1, 1, 2, 3, 5, 8, 13, 21, 34...

When the number of items is large enough, the ratio of the previous item to the next item gets closer and closer to the golden ratio - 0.618. People will consider applying the Fibonacci sequence in many pattern designs.

2. Pine cone

You may often see pine cones fallen on the ground in the park. They are a kind of woody, scaly "fruit".

If you look closely, you will definitely notice the spiral of their scales , which can close tightly when it is wet or cold to protect their seeds, and then open when the temperature and humidity are right, allowing the wind to help the seeds spread.

The threads at the bottom of a pine cone. Image source: Wikipedia

The spiral pattern at the base of a pine cone, the numbers formed by these threads when counted from both directions are all Fibonacci numbers.

For example, in the pine cone above, the number of lines rotating counterclockwise is 8, while the number of lines rotating clockwise is 13;

In addition to 8 and 13, the number of lines that rotate could also be 5 or 21, which are numbers that exist in the Fibonacci sequence;

However, numbers that are not in the Fibonacci sequence, such as 7, 9, 10, 11, and 12, will not appear. Isn’t it amazing?

3. Succulents

In botany, fractals are called "spiral phyllotaxy," where the leaves on a plant are arranged in a spiral.

The leaves of many succulents grow in a tightly curled spiral structure, forming a self-similar spiral with the golden ratio - the golden spiral .

This arrangement helps carry rainwater to the core of the plant and prevents the top leaves from shading the bottom leaves .

Spiral leaf arrangement of aloe vera, copyrighted image, unauthorized reproduction

Of course there are some different opinions.

A mathematician once hypothesized that the spiral patterns of all plants, as well as our fingerprints, appear to relieve the stress of growth. For example, in plants, as cells divide and grow in different directions, they may experience mechanical stress due to the constraints of neighboring cells or tissues, which can cause the tissue to bend and fold.

02

What is a fractal?

The word "fractal" is mentioned so many times above. Is fractal spiral growth?

Fractal is a concept in geometry, which refers to the repeated occurrence of simple similar patterns in different dimensions to form complex figures or shapes . This configuration is ubiquitous in nature.

As early as 1967, American mathematician Benoit B. Mandelbrot published a paper titled "How long is the coastline of Britain?" in the journal Science. The paper pointed out that for irregular shapes like coastlines, if different measurement scales are selected, the measurement results will be very different.

As shown in the figure below, when a large measurement scale is selected, small areas are not measured and the obtained values ​​will be relatively small.

If you use a small scale to measure, the result will be much larger. The smaller the scale you use, the larger the value you get.

Different measuring scales measuring the UK coastline, Image source: Wikipedia

That is to say, in reality, such complex irregular-bordered figures do not have an accurate perimeter , and as the measurement scale decreases, its perimeter will tend to infinity. This is the beginning of fractal theory.

Later, people believed that if this irregular boundary presents similar characteristics at small scales and large scales, and this self-similarity exists even when it is infinitely subdivided, then this geometric shape can be called a "fractal."

03

Fractals are not all “spiral”

In fact, there are many examples of fractals in our daily lives. In addition to the Romanesco broccoli and pine cones mentioned at the beginning, the growth of pineapples and the formation of ice crystals also follow fractal laws.

1. Snowflakes

No two snowflakes are exactly alike, but they all have their own unique fractal shape.

Copyright image, no permission to reprint

The branches of a snowflake will give rise to their own side branches, and if the snowflakes continue to accumulate water and merge, then the snowflakes may continue like this forever.

The snowflakes we see in our lives are actually formed through countless iterations.

Schematic diagram of Koch snowflake, image source: Wikipedia

The most famous fractal pattern is called the Koch snowflake , which is formed by one equilateral triangle superimposed on another equilateral triangle, which then forms the next equilateral triangle.

After countless iterations of the Koch snowflake, its corners become so rugged that a snowflake of finite area would have an infinite circumference.

This is similar to the question of "how long is the coastline?"

2. Tree branches

For plants, growing according to fractal rules can allow them to be exposed to sunlight to the maximum extent , allowing them to perform good photosynthesis. At the same time, it can also efficiently transport nutrients to various parts of the body.

The growth of trees is one of the most typical fractals in nature.

As the trunk grows, branches sprout from it, and these branches themselves, like the trunk, develop their own branches.

If you look closely, you'll notice that the entire tree can be seen as a repeating "Y" shape.

Tree fractals. Image credit: Robert Fathauer

This fractal design, like the spirals of succulents, helps the trees optimize their exposure to sunlight and prevents top branches from shading lower branches.

3. Copper crystal

Fractal geometry is also common in chemistry, and copper crystals are a typical example.

Copper crystals branch out in all directions like tree branches, and each branch is a new growth point that continues downward.

Copper crystal, photo taken by: Lao Mao

It is this continuous branching that forms solid metallic copper.

Due to their psychedelic tree-like structure and unique reddish-brown color, copper crystals are often used as works of art in real life.

4. Rivers

When we look at a map, we see that rivers always have a winding "S" shape. Although streams can sometimes form a straight line, they quickly become curved as they encounter different obstacles.

Just one disturbance can disrupt the flow of a river, causing it to bend all the way.

Copyright image, no permission to reprint

A closer look at these rivers reveals that their widths are very formulaic, with the general curve length always being six times the width of the channel.

This self-similarity is the hallmark of fractals and is why rivers around the world look similar.

5. Bubbles

In nature, bubbles that emerge when ocean waves crash or raindrops fall create a self-similar pattern, with thin films of liquid separating air cavities of various sizes.

There are small bubbles scattered next to the big bubbles, and there are even smaller bubbles scattered next to the small bubbles.

Copyright image, no permission to reprint

The foam we see on our coffee, in the sink, and in the bathtub is a fractal phenomenon.

Every fractal in nature has its own logic. What other fractal phenomena have you seen in nature?

References:

[1] "UA Mathematicians Predict Patterns in Fingerprints, Cacti." University of Arizona. 2004.

[2] Peng, Sheng-Lung, Rong-Xia Hao, and Souvik Pal. "Proceedings of First International Conference on Mathematical Modeling and Computational Science." Springer Nature. 2021.

Author: Denovo Popular Science Writer

Reviewer: Zhang Lei, Researcher, Beijing International Center for Mathematical Research, Peking University

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