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.

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.

Bilateral Symmetry has significant applications in various fields today, including biology, medicine, engineering, and design.

Such as:

1. Human Anatomy and Surgery: 

The bilateral structure of human body helps in medical diagnostics, surgeries, and prosthetic designs. For example, doctors use the symmetrical arrangement of organs to identify abnormalities. 

2. Computer Vision:

Facial Recognition technology relies on the bilateral symmetry of human faces to improve accuracy. 

3. Building Structure:

Many Buildings and bridges use bilateral symmetry for aesthetic appeal and structural balance

4. Animal Locomotion:

Bilateral Symmetry allows for efficient movement in animals, improving speed and coordination.

5. Genetic Research:

Studying symmetrical development in organisms helps scientists understand genetic disorders and evolutionary adaptations.

6. Aerodynamics:

Cars, airplanes, and ships are designed with bilateral symmetry to ensure balance, stability, and efficiency in motion.

• 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.

Radial symmetry is widely applied in various fields, including biology, engineering, architecture, medicine and design.

Such as:

1. Medicine Imaging and Microscopy:

Many imaging devices (e.g. MRI Scanners, CT Scanners, use redial symmetry in their design for efficient rotation and scanning.

2. Mechanical Designs:

Radial symmetry is crucial in machines and devices, such as wheels, gears, turbines, and propellers, ensuring smooth motion, balance, stability, and performance. 

3. City Planning: 

Many cities use radial symmetry for efficient traffic flow, such as circular road systems and centralized transport hubs.

4. 360 Degrees Censors: 

Radial Symmetry is used in robotic sensors and cameras for full-range environmental detection. 

Fractals are patterns of self repeating iterations having fractal dimensions. 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.

Snowflakes are considered fractals because their shapes show self-similarity and repetition of patterns at different scales—two key features of fractals. 

Here’s why:

1. Branching symmetry – A snowflake grows outward in six symmetrical directions. Each arm branches and sub-branches, and the smaller branches look like miniature versions of the larger ones.
   
   

2. Self-similarity – If you zoom in on one branch of a snowflake, its structure resembles the whole snowflake pattern. This "repetition of shape within shape" is what makes it fractal-like.
   
   

3. Iterative growth – Snowflakes form through a repeating process: as water vapor freezes, molecules attach in a way that reinforces the existing pattern. This step-by-step, rule-based growth is the same process used to generate mathematical fractals.

In short, Snowflakes are fractals because they grow through recursive, symmetrical branching that creates repeating patterns at different scales. 

WATCH: HOW SNOWFLAKES ARE FORMED?

Browse the pictures for examples:

Real Application of Fractal patterns: 

1. In Nature & Science

Fractals appear everywhere in the natural world, and they play a role in how living and nonliving things are structured.

• Growth & Form – Trees branch into smaller twigs and leaves, rivers divide into tributaries, and our blood vessels and lungs branch into smaller pathways. This fractal branching allows organisms to maximize space, efficiency, and distribution—for example, our lungs’ fractal structure lets us absorb more oxygen in a limited space.
  
  

• Weather & Geography – Natural landscapes often follow fractal geometry. Coastlines appear jagged from afar, but when you zoom in, the small portions are equally irregular, just like the whole. Mountain ranges, clouds, and even lightning bolts have repeating, self-similar patterns that scientists can measure and use to model natural processes and predict environmental changes.
  
  

2. In Mathematics & Technology

Fractals are not just natural phenomena—they’re also powerful mathematical tools.

• Modeling Complexity – Traditional geometry describes smooth, regular shapes (like circles or squares), but fractals allow us to represent irregular and chaotic shapes such as coastlines or turbulence. This makes them essential for studying complex systems.
  
  

• Computer Graphics – Fractals are widely used in animation, films, and video games to create realistic landscapes, textures, and natural effects such as mountains, clouds, or terrain in virtual worlds. They provide detail without manually designing every part.
  
  

• Signal Processing – Fractals are applied in data compression (like image or sound files) and in antenna design. Their repeating patterns make signals more efficient, powerful, and reliable, especially in telecommunications.
  
  

3. In Everyday Life & Art

Beyond science, fractals inspire creativity and practical applications in human life.

• Architecture & Design – Many architects and artists use fractal geometry to create aesthetic and functional patterns in buildings, textiles, and artworks. Traditional cultures, like African and Islamic designs, often use fractal-based motifs.
  
  

• Medicine – Human biology itself is fractal-like. Doctors study fractal patterns in heartbeats, brain waves, and lung structures to detect diseases such as arrhythmias or cancer. Fractal analysis provides a noninvasive way to understand health and spot irregularities.

A spiral is a curve that winds around a central point while gradually moving away (or towards) it. Unlike circles, spirals grow or shrink as they rotate. 

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.

TYPES OF SPIRAL PATTERNS: 

1. Archimedean Spiral: The Archimedean spiral is a type of spiral where the distance between each turn is always the same. It was first described by the Greek mathematician Archimedes (287–212 BC). 

Notice how each loop is evenly spaced from the previous one — that’s the defining feature of this spiral. 

Natural Examples of Archimedean Spirals:

1. Logarithmic Spiral: A logarithmic spiral is a self-similar curve that winds around a central point but keeps expanding outward as it does so. Unlike the Archimedean spiral (where the distance between turns is constant), the logarithmic spiral grows exponentially—the spacing between its arms increases in proportion to its distance from the center. 

Natural Examples of Logarithmic Spirals:

Meanders is a winding, curving bend in a river or stream that develops as water flows across relatively flat landscapes. Instead of moving in a straight line, rivers often form snake-like curves due to erosion and deposition processes. 

WHY MEANDERS ARE IMPORTANT?

Meandering rivers showcase the dynamic energy of flowing water as it carves, reshapes, and redirects the land it traverses. These bends are not random; they result from the river’s interaction with terrain, water volume, and flow speed. Large, fast-flowing rivers tend to develop wider and more dramatic bends, while smaller, slower rivers form gentler curves.

Meanders are especially common in flat landscapes, where there are fewer natural barriers to confine the river’s path. Over varied terrain, the bends may become more complex, producing intricate meander formations.

Beyond their visual beauty, meanders play a crucial ecological and geomorphological role. By slowing down water flow, they reduce soil erosion, encourage sediment deposition, and help sustain balance within river ecosystems. Thus, meanders are not only aesthetic features of rivers but also essential natural processes that shape landscapes, enrich floodplains, and regulate river dynamics.

MEANDER PATTERNS IN NATURE

FUN FACTS: 

The Gyrification Index (GI) measures how folded the brain surface is.

• A small GI means a smoother brain with fewer and shallower folds → less surface area.
  
  

• A large GI means a highly folded brain → more surface area, allowing more neurons to fit in.
  
  

What’s interesting is that the pattern of folds and lobes is fairly consistent across people, meaning the brain doesn’t fold randomly. Instead, gyrification is guided by biological and developmental factors—like genetics, cell growth, and connectivity demands—not just by mechanical forces of brain tissue (Ronan & Fletcher, 2015).

A smoother brain has a lower GI, a folded brain has a higher GI, and the fact that folds are consistent across humans shows that brain folding is purposeful, not random. 

Animals with Low GI (Smoother Brains)

• Rodents (rats, mice, hamsters) → very smooth brains with few folds.

• Rabbits → mostly smooth brains.

• Reptiles (lizards, snakes, turtles) → little to no folding.
  
  

Animals with Moderate GI (Some Folding)

• Cats → some folds, allowing moderate surface area.

• Dogs → more folds than cats, related to their social behavior and learning ability.

• Sheep & Goats → moderately folded brains.
  
  

Animals with High GI (Highly Folded Brains)

• Primates (chimpanzees, gorillas, orangutans) → highly folded, closer to humans.

• Dolphins & Whales → extremely folded brains, often even more gyrified than human brains.

• Elephants → very high GI, complex folds supporting memory and social intelligence.

• Humans → among the highest GI, with consistent and highly organized folding patterns.

Wave patterns is a repeating disturbance or vibration that transfers energy from one point to another without the permanent movement of matter. Waves can travel through different mediums such as water, air, earth, or even in empty space (as in electromagnetic waves).

WHY WAVE PATTERNS ARE IMPORTANT?

• They are essential for energy transfer (e.g., ocean waves moving energy across the sea).

• Enable communication (sound waves let humans and animals interact, light waves allow vision).

• Maintain balance in ecosystems (waves mix ocean water, spread nutrients, and shape coastlines).

APPLICATION OF WAVE PATTERNS IN THE MODERN WORLD: 

Examples of Natural Wave Patterns

1. Ocean Waves – caused by wind; move energy across the sea and shape coastlines.

2. Sound Waves – vibrations in air or water; let humans and animals communicate.

3. Light Waves – electromagnetic waves that allow vision and photosynthesis in plants.

4. Seismic Waves – produced by earthquakes; help scientists study Earth’s interior.

5. Brain Waves – rhythmic patterns of electrical activity that control thought and movement.

6. Heat Waves (Infrared Radiation) – transfer warmth from the sun to Earth.

Bubble is a thin film of liquid that traps gas, usually air, inside it. This film is made of two layers of liquid molecules with a very thin layer of air in between. Because of surface tension, which acts like the liquid’s “skin,” the bubble takes on a spherical shape since a sphere has the least surface area for the amount of gas inside. 

Foam is formed when many bubbles are packed closely together, separated only by thin liquid films. It occurs when gas is dispersed within a liquid or solid, creating a cluster of bubbles. Compared to single bubbles, foams are less stable because the thin films between the bubbles can easily break. 

WHY BUBBLES AND FOAMS ARE IMPORTANT?

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. 

Types of Tessellations

• Regular Tessellation – Uses only one type of regular polygon (e.g., equilateral triangles, squares, hexagons). Only three shapes can make a regular tessellation.

• Semi-Regular Tessellation – Uses two or more types of regular polygons arranged in a repeating pattern.

• Irregular Tessellation – Uses non-regular polygons or freeform shapes

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

WHY TESSELLATIONS ARE IMPORTANT?

Crack patterns are natural or man-made fracture lines that form when a material breaks apart due to stress, drying, or cooling. Instead of breaking randomly, cracks often arrange themselves into predictable geometric patterns, showing how nature distributes forces efficiently. 

How Crack Patterns Form?

1. Drying or Shrinking – When mud, paint, or clay dries, it contracts. The stress causes cracks to form, usually making polygonal patterns like hexagons or squares.

2. Cooling – When lava or rock cools, it shrinks and cracks into columnar jointing patterns, often hexagonal.

3. Stress and Pressure – Cracks in glass, walls, or roads spread in branching patterns to release stored energy.

Types of Crack Patterns

• Polygonal cracks – Found in dried mud, salt flats, or cooling lava.

• Branching cracks – Seen in broken glass, ice, or concrete under sudden impact.

• Radial cracks – Spread outward from a single point, like in shattered windshields or dried fruits.

WHY CRACK PATTERNS ARE IMPORTANT?

Spots and stripes are common natural patterns found in animals, plants, and even some geological formations, and they play important roles in survival and adaptation. 

Spots and stripes are not just decorative — they are biological adaptations that help organisms survive (camouflage, temperature control, protection from parasites and predators), reproduce (signaling and communication), and interact with their environments. 

a. LOGICAL PATTERNS

Logic reasoning and pattern observing are the first two math standards, which are the most important measurement of IQ and the core component of many careers.

Logical patterns are usually the first to be observed since making categories or classification comes before numeration. For children, logical patterns include studying shapes and colors. For older ones, logic tests can be seen on aptitude tests wherein takers are shown a sequence of pictures and asked to select which figure comes next among several choices.

To identify logic patterns, you have to look out four (4) things, namely:  

• (1) rotating shapes

• (2) increase and decrease in numbers of shapes or patterns  

• (3) alternating patterns, colors, and shapes  

• (4) mirror images or reflections 

Try to answer the following: 

b. GEOMETRIC PATTERNS

A geometric pattern consists of shapes like polygons and circles that are repeated to create a design. Geometric patterns can be seen in nature and in different artworks. Examples of these patterns are also found in textiles, floor tiles, paintings, and wallpapers.

c. NUMBER PATTERNS

A number pattern is a list of numbers that follow a particular sequence or order.   

Consider the given sequence of numbers: 23, 30, 37, 44, 51, 58, 65 and 72. 

The illustration below will help us see the relationship of the numbers in the sequence for this number pattern, the next number is determined by adding seven to the previous number. This also shows that the difference between two consecutive numbers is seven.  In determining the number pattern, it is important to examine the interval or the difference between the consecutive numbers in the sequence. This will show the rule, the trend, or the pattern in the sequence.

d. WORD PATTERNS

Word patterns can be found in giving the plural of nouns, in forming the past tense of verbs, and in word analogy. They can also be found in the meters of poetry and in the rhythm of the words.

These are examples of word patterns:

Patterns are highly significant to us humans for several reasons:

1. Patterns help us make sense of the world around us by organizing information and providing predictability. They allow us to recognize regularities and understand relationships between different elements..

2. Our brains are wired to recognize and process patterns efficiently. This cognitive ability enables us to process information quickly, make decisions, and adapt to changing environments.

3. Patterns play a crucial role in learning and memory. We learn by recognizing patterns in data, language, and experiences, and we use these patterns to recall information when needed.

4. Identifying patterns can aid in problem-solving by allowing us to apply previous solutions to similar situations. Recognizing patterns can also help us anticipate outcomes and devise effective strategies.

5. Patterns inspire creativity by providing a framework for innovation and artistic expression. Artists, designers, and innovators often draw inspiration from patterns found in nature, mathematics, and culture.

Overall, patterns are fundamental to human cognition, perception, and creativity, shaping how we understand the world and interact with it.

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] 

One of the earliest people to discover this was the English scholar Robert Thomas Malthus. His theories are now widely popular in economics and are now known as the Malthusian Theory.

It was observed that population growth at any given time is always directly proportional to the current population. This means that the number of people who are added (e.g. births, in-migration) and subtracted (e.g. deaths, out-migration) depends on the population of the place. For example, the higher the population, the more women who would be giving birth. This principle is now known as the Malthusian Growth Mode. Mathematicians developed this concept into an equation (specifically, a differential equation) 

And then, using some techniques, solving this equation would yield the exponential function given above. Thus, we are able to mathematically describe the behavior of how population grows over time. 

Knowing the concept of population growth is important because it helps us understand how living things (including humans) interact with resources, the environment, and society. Here are the key reasons: 

1. Environmental Management

• Too much population growth can lead to deforestation, pollution, and climate change.

• Helps governments and scientists predict environmental impacts and design sustainable practices.

2. Food and Resource Planning

• Population growth determines demand for food, water, housing, and energy.

• Helps in planning agriculture, industry, and resource distribution.

3. Public Health and Services

• Understanding growth trends guides healthcare, education, and social services.

• Helps prepare for disease outbreaks, aging populations, or high birth rates.

4. Economic Development

• A growing workforce can boost the economy (more workers and consumers).

• But too rapid growth may cause unemployment, poverty, and inequality.

5. Policy and Decision Making

• Governments use population data to make laws, urban planning, and family planning programs.

• Helps balance development with sustainability.
  
  

In summary:
Knowing about population growth is important because it allows us to plan ahead, use resources wisely, protect the environment, and ensure a better quality of life for present and future generations.

Yes- studying the population growth of bacteria in a dead body helps forensic experts estimate time of death, detect cause, and trace crime scenes. This is a real-world use of the population growth concept in solving crimes. 

Bacteria Growth in a Dead Body

• After death, the body becomes a perfect environment for bacteria.

• Gut bacteria (normally harmless in life) begin multiplying rapidly because the immune system no longer controls them.

• This follows an exponential growth pattern, meaning they can double in number every 20–30 minutes under the right conditions.
  
  

How It Helps in Solving Crimes?

1. Time of Death (Postmortem Interval, PMI)

   • Forensic scientists study the rate of bacterial growth and spread to estimate how long a person has been dead.

   • Example: Specific bacteria appear at predictable stages of decomposition.
     
     

2. Cause of Death

   • If unusual bacteria (like pathogens) are found, it may indicate poisoning, infection, or foul play.
     
     

3. Location of Death

   • Different environments (soil, water, urban setting) have different bacteria.

   • By analyzing bacterial populations, investigators can tell if a body was moved after death.

Connection to Population Growth Concept

Just like human or animal populations, bacteria follow these growth models:

• • Exponential growth (J-curve): Immediately after death, bacteria multiply rapidly.

  • Logistic growth (S-curve): Growth slows as nutrients run out and decomposition stabilizes.

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. 

Another Real-life Example:

Normal Distribution in Nature

The normal distribution (bell curve) naturally appears in many biological, environmental, and physical processes because most natural outcomes are shaped by many small random factors. 

This is explained by the Central Limit Theorem: 

"when many independent random effects combine, the result tends to follow a normal distribution."

Examples of Normal Distribution in Nature

1. Human Height and Weight

• Most people are close to the average height and weight.

• Very short and very tall individuals are rare.

• If you graph a population’s height, it forms a bell-shaped curve.

2. Flower Petal Counts and Plant Growth 🌸

• While Fibonacci numbers appear in patterns, slight variations in growth (like petal size or leaf length) distribute normally around an average.

3. Animal Characteristics 🐦

• Wing lengths in birds, antler sizes in deer, or shell diameters in snails usually cluster near the mean, with fewer individuals at the extremes.

4. Measurement of Traits in Populations 🧬

• Blood pressure, reaction times, enzyme activity, and metabolic rates all show bell-curve patterns.

5. Environmental Phenomena 🌡️

• Daily temperature variations, rainfall over many years, and even measurement errors in climate data approximate normal distributions.

Summary

• Probability exists in nature because many processes are uncertain and random.

• The normal distribution is one of the most common probability distributions in nature.

• It describes patterns in biology (traits), ecology (populations), and environment (climate).

• In short: Nature’s randomness often organizes itself into a bell curve.

Numbers are significant to humans for several reasons:

1. Numbers allow us to quantify and measure things. They give us a way to describe and compare quantities, sizes, distances, durations, and more. This is essential for understanding the world around us and for making informed decisions.

2.  Numbers provide a universal language for communication. They enable us to convey precise information across cultures, languages, and contexts. Whether it's expressing quantities, giving measurements, or describing data, numbers facilitate clear communication.

3. Numbers are fundamental to problem-solving. From simple arithmetic to complex mathematical equations, numbers are used to analyze, predict, and solve problems in various domains, including science, engineering, economics, and everyday life.

4. Numbers help us organize and categorize information. They allow us to arrange data in meaningful ways, such as sorting in ascending or descending order, grouping similar items, or creating numerical hierarchies.

5. Numbers play a crucial role in memory and cognition. They help us remember information more efficiently by providing a structured framework for encoding, storing, and retrieving data. For example, phone numbers, addresses, and dates are easier to recall when represented numerically.

6. Numbers inform decision-making processes. Whether it's evaluating risks, assessing probabilities, or weighing costs and benefits, numbers provide a basis for making rational choices and judgments.

7. Numbers have symbolic significance in various cultures and belief systems. They can represent abstract concepts such as luck, fortune, or spirituality. In addition, numbers are used in symbolism and rituals in fields like astrology, numerology, and religious practices.

8. Numbers are indispensable in technology and innovation. They form the basis of mathematical theories, algorithms, and computations that drive advancements in fields like computer science, telecommunications, and artificial intelligence.

Overall, numbers are deeply ingrained in human cognition, communication, and culture, serving as a foundational tool for understanding the world and making sense of our experiences.

References

[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/

[4] https://www.studocu.com/ph/document/la-consolacion-university-philippines/mathematics-in-the-modern-world/lesson-1-patterns-and-numbers-in-nature-and-the-world/40231664