The Mathematics of Our World: A Synthesis

 When we closely observe the nature we can see that there is a lot of hidden order in nature. Leaves, shells, clouds, rivers, the stars, above all, have some patterns. These are not accidental patters, the mathematical way of describing and understanding them exists.

Types of Patterns

• Symmetry – Butterflies and human faces are almost mirror images. Snowflakes also show perfect six-fold symmetry.
• Fractals – A fern leaf looks like a small fern itself. This "self-similarity" repeats at different scales.
• Spirals – Seashells, hurricanes, and galaxies often form spirals, which can be described mathematically.
• Branching – Trees, rivers, and even blood vessels spread out in branching systems, optimizing space and flow.

Why Mathematics Matters

Mathematics provides the language to describe these patterns. For example:

• Geometry explains symmetry.
• Algebra and ratios explain spirals.
• Graph theory explains networks like rivers and veins.

Nature is not random chaos—it follows rules, and mathematics helps us see the hidden order.

Have you ever noticed how sunflowers, pinecones, or seashells follow a unique spiral? That’s the Fibonacci sequence in action.

The Fibonacci Sequence

The sequence begins with 0, 1 and each number is the sum of the two before it:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34…

This simple rule leads to amazing patterns in nature.

Fibonacci in Nature

• Flowers – Many flowers have petals in Fibonacci numbers (3, 5, 8, 13…).
• Pinecones and pineapples – Their scales are arranged in spirals that match Fibonacci numbers.
• Sunflowers – Seeds form spirals both clockwise and counterclockwise, often following Fibonacci counts.

The Golden Ratio (ϕ ≈ 1.618)

The Fibonacci sequence is linked to the Golden Ratio. As the numbers grow larger, dividing one Fibonacci number by the one before it gets closer to 1.618.

This ratio shows up in:

• Art – Leonardo da Vinci’s Vitruvian Man is drawn with Golden Ratio proportions.
• Architecture – The Parthenon in Greece reflects the ratio in its design.
• Design – Even modern logos (like Apple or Twitter) use Golden Ratio proportions for aesthetics.

Math shows us that beauty often follows a pattern—it’s not just “art,” it’s also numbers.

We often think of math as numbers and formulas, but at its core, mathematics is a language.

Why Mathematics is a Language

• Precision – Unlike spoken words, math avoids ambiguity.
• Conciseness – One equation can summarize what would take pages of words.
• Universality – Whether in the Philippines, Japan, or Europe, “2 + 2 = 4” means the same thing.

Symbols as Communication

Symbols like +, −, =, and variables like x, y allow us to express complex ideas simply.

For example:

• In words: The area of a rectangle is equal to its length times its width.
• In math language:

A=l×w

See how shorter and clearer it is? That’s the power of math as a language.

Mathematics isn’t just about memorizing formulas—it’s about thinking and solving problems.

Two Types of Reasoning

• Inductive Reasoning – Drawing a general rule from specific examples.
  • Example: If the first 5 odd numbers (1, 3, 5, 7, 9) add up to 25, you may guess that the sum of the first n odd numbers is n2n^2n2.
• Deductive Reasoning – Using general rules to make conclusions.
  • Example: All even numbers are divisible by 2. Since 14 is even, 14 must be divisible by 2.

Polya’s Four Steps to Problem Solving

• Understand the problem. What is being asked?
• Devise a plan. Think of possible strategies.
• Carry out the plan. Solve step by step.
• Look back. Check if the answer makes sense.

Example Problem:

"You have 24 apples, and you want to divide them equally among 6 friends. How many apples will each friend get?"

• Step 1: Understand – Divide 24 by 6.
• Step 2: Plan – Use division.
• Step 3: Carry out – 24 ÷ 6 = 4.
• Step 4: Look back – Each friend gets 4 apples, and 4 × 6 = 24. Correct!

Problem-solving is not just for math class—it’s a skill we use in daily life.

Mathematics is not numbers on a piece of paper, mathematics is the design of the universe. It describes the circles of galaxies, the evolution of vegetation, the forms of artists and the logic we apply to problem solving.

We learn to see the real purpose of math as a language, tool to solve problems and describe the patterns around us, thus it helps us understand and associate with the world.

Mathematics is everywhere. When we look at it, we cannot take it back.