Patterns and Numbers in Nature and The World

Nature is interesting. It's full of patterns. It's full of numbers. And they're linked.

The following video called "Nature by Numbers" by Cristobal Vila shows us how fascinating nature's patterns are. Watch it.

Before we proceed, let us reflect on the video by answering these guide questions.

• What natural objects were shown in the video?

• What patterns are exhibited in each of these natural objects?

• Is there some mathematical concept that explains these patterns? If yes, what is that?

The goal of this lecture is to identify patterns and numbers in nature. It will be culminated with an appreciation towards mathematics because we will realize that mathematics, through patterns and numbers, is a way to understand nature.

All throughout this lecture, we will see equations or mathematical concepts that may be advanced or are higher than our current level of mathematics. They are only presented here to provide some visual material. There is no need to study these equations in detail. As long as you see that there is mathematics behind this natural phenomena, that would be enough.

A symmetric pattern is present when one part of the object retains its form or shape after some form of transformation or change [3]. In other words, symmetry is present when two or more parts of an object are "balanced" or "identical".

There are two types of symmetries in nature: mirror symmetry (sometimes called bilateral symmetry) and radial symmetry (sometimes called rotational symmetry).

• mirror symmetry - a type of symmetry wherein two sides are identical with respect to an imaginary line (called the line of symmetry) that divides them.

• radial symmetry - a type of symmetry wherein more than two parts of the object are identical with respect to a point (called the point of symmetry) located at the center.

Browse the pictures for examples:

Fractals are patterns of self repeating iterations having fractal dimensions [2]. In other words, a fractal is a pattern wherein a larger object is being repeated many times into smaller and smaller copies of itself. Since the smaller copies are branching out of the larger image, fractals are sometimes called trees.

The most common example of a natural fractal are snowflake crystals.

Remember the line from the Frozen song "...frozen fractals of our time..."? She meant "snowflakes".

Browse the pictures for examples:

A spiral is a pattern wherein a curve revolves around a center focus [4]. Spirals are one of the most common patterns found in nature. There are spiral galaxies, seashells, spiral-shaped plants, and even some animals tend to display spiral patterns.

As seen in the video earlier, some spirals are closely related to the Fibonacci sequence. More on the Fibonacci sequence will be discussed in the next lecture.

See photos below for some examples.

There are different types of spirals. In nature, the most common type of spiral are called logarithmic spirals which are explained using the Fibonacci sequence as we have seen from the video above. This is the type of spiral seen in the configuration of sunflower seeds, spiral galaxies, seashells, and spiraling of fern leaves. More on the Fibonacci sequence will be discussed in the next lecture.

1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , . . .

So how does this sequence generate the spiral?

First, take a square with side of length 1. Then add another square with side of length 1, then another with side of length 2, then another with side of length 3, then of length 5, 8, 13, and so on.

Meanders are patterns of regular winding curves of bends and turns. They are usually described as "sinuous" because they resemble the graph of the sine function from trigonometry.

Meanders are common in riverways, snake movements and other natural phenomena described by the sine wave.

See photos below for more examples:

Wave patterns are described as disturbances as energy is transferred from one place to another, without the actual transfer of matter.

Do not confuse meanders with waves. Generally, meanders occur on a stretch of a curve while waves occur on surfaces. For example, ocean waves occur on the boundary between the surfaces of the ocean and the atmosphere.

Bubbles, such as soap bubbles are spherical. Soap bubbles are made of a mixture of soap and water enclosing some volume of air. Because of cohesive forces, soap and water molecules tend to minimize the surface area of the enclosure (hence, making the coating as thick as possible). And since a sphere is the three-dimensional shape with the smallest surface area with a fixed volume (see explanations in Physics or Calculus), bubbles become spherical in shape.

Foams are a mass of bubbles. Interestingly, when bubbles form a foam, they still obey some mathematical rule, specifically the Plateau's laws. You can read more about this here.

Tessellations are patterns on surfaces covered by regularly repeating two-dimensional shapes. Each individual cell of a shape in a tessellation is called a tile.

Some tessellations are made of self-repeating shapes. Some are made of two or more shapes such as the ones below.

In nature, tessellations can be found in honeycombs of honey bees, paper nests of wasps, fish scales, and others.

Cracks are linear openings on surfaces that form in process of relieving stress. "When an elastic material stretches or shrinks uniformly, it eventually reaches its breaking strength and then fails suddenly in all directions, creating cracks with 120 degree joints, so three cracks meet at a node." [2]

Spots and stripes are patterns found on skins of animals or plants and other natural forms. For animals, biologists have explained that this is a result of some evolutionary process. These skin patterns were developed for different purposes such as camouflage, temperature control, protection from parasites and predators, signaling and communication, and others.

Animal skin patterns are explained mathematically using a model called reaction-diffusion model by the English mathematician Alan Turing. The model uses partial differential equations which is quite an advanced-level mathematics. We can read an explanation not containing much numbers here.

In the previous section of this lecture, we have seen the abundance of patterns. Now, we will understand that nature is also rich with numbers. Just like patterns, numerical rules are also some of the governing principles present in many natural phenomena.

There are many instances wherein a nature is explained through numbers. In this lecture we will name two: population growth dynamics and probability.

The following chart shows the population of the Philippines from the year 1800 to 2020. [5]

Even processes in nature that are usually regarded as "random" such as natural life span of an animal, or height of a person, or the number of puppies a mother dog bears follow some "regular" rule explained in terms of probability.

Take for example the Galton Board below. A Galton board is simply a board with pins forming a triangular grid. Balls are then dropped from the top of the triangle and would randomly fall to a certain slot at the bottom of the triangle. Observe the behavior of the number of balls dropping at each slot.

You can also "play" your own Galton Board (also known as Quincunx) here. What can we observe? There is a certain behavior in the distribution of the number of balls per slot. Hence, the "random" destination of balls as they fall still follow some mathematical rule.

For another example, we obtained a data of the heights and weights of 10 000 randomly selected individuals (5 000 males and 5 000 females) from kaggle.com. We graph these datasets using a histogram and we can obtain the following:

Obviously, the distribution is similar to that of the results of the Galton Board earlier. This behavior is called the normal distribution. The phenomena of normally distributed data are very common in nature from the number of eggs an insect would lay, or the number of fruits a tree would bear in one season, or the age of people when they die. This concept is extensively utilized specifically in statistics and generally in science to analyze data and draw inferences.

[1] Stevens, Peter S. (1974). Patterns in Nature. Little, Brown & Co.

[2] ECStep. Natural Patterns. from https://ecstep.com/natural-patterns/

[3] Patterns in Nature. Patterns in Nature Contain Symmetry. from http://www.patternsinnature.org/Book/PatternsContainSymmetry.html

[4] The Franklin Institute. Math Patterns in Nature. from https://www.fi.edu/math-patterns-nature

[5] Aaron O'Neil (2021). Population of the Philippines 1800-2020 in Statista. from https://www.statista.com/statistics/1067059/population-philippines-historical/

1. What are the nine (9) natural patterns?

2. For each natural pattern, give an example.

3. Give two (2) things in nature that follow some numerical rule.

4. In your own opinion, does nature and mathematics relate to each other? How?