Dec 30, 2023

Mathematics: Key to Understanding Patterns and Relationships in the World

Mathematics stands as the key to understanding patterns and relationships in the world.

Mathematics is the discipline that explores the connections between numbers, quantities, and shapes. It encom passes various branches such as arithmetic, algebra, trigonometry, geometry, statistics, and calculus. In addition to enhancing human traits like creativity, reasoning, critical thinking, and spatial awareness, mathematics offers a means to solve both simple and complex problems in real-world contexts using diverse approaches. It serves as a universal language, allowing us to make sense of concepts and rules through mathematical symbols, signs, proofs, language, and conventions.

Mathematics organizes observed world patterns, directly or indirectly linking most natural patterns to its principles. Moreover, the astonishing extent to which natural phenomena adhere to mathematical series and numbers is remarkable. By predicting and influencing the behavior of nature and occurrences in the world, mathematics becomes indispensable for the betterment of mankind. As a science of patterns, mathematics enables students to identify, apply, and generalize patterns found in numbers, shapes, and their environment. Thus, possessing these skills equips students to become adept problem solvers and fosters a deeper understanding and appreciation of nature and the world.

Students should therefore have the opportunity to analyze, synthesize, and create a range of patterns, using pattern-based thinking to comprehend and represent both mathematical and real-world phenomena. These explorations offer endless possibilities for problem-solving, verifying generalizations, and building competence in mathematics and science.

Objectives:

Upon completing Chapter I: Mathematics in our World, learners should have the ability to:

Firstly: Explain the significance of mathematics in their daily lives.
Secondly: Recognize patterns in nature and regularities present in the world.
Thirdly: Engage in discussions about the nature of mathematics, including its definition, expressions, representation, and applications.
Finally: Demonstrate appreciation for mathematics as a human endeavor.

A. Patterns and Numbers in Nature and the World

Patterns found in nature exhibit visible regularities and encompass various complex formations. These patterns include spirals, symmetries, mosaics, stripes, spots, and more. They endure across various contexts, subject to mathematical modeling. Upon deeper examination, it becomes evident that these natural patterns share similarities and resemblances.

Early Greek philosophers such as Plato, Pythagoras, and Empedocles studied patterns to interpret the inherent order in nature, contributing to our modern understanding of visible patterns.

In the 19th century, physicist Joseph Plateau investigated soap films; as a result, it led him to formulate the concept of minimal surfaces.Biologist and artist Ernst Haeckel painted numerous marine organisms, emphasizing their symmetry. D'Arcy Thompson, a Scottish biologist, pioneered the study of growth patterns in both plants and animals, demonstrating how simple equations could explain spiral growth. In the 20th century, mathematician Alan Turing predicted the mechanisms of morphogenesis that give rise to patterns like spots and stripes. Moreover, Biologist Aristid Lindenmayer and mathematician Benoît Mandelbrot illustrated how fractal mathematics generates plant growth patterns.

Figure 1. Patterns in Nature

Additionally, numbers are prevalent in nature, with mathematicians observing their appearance in various patterns: two wings of a bird, three leaflets of a clover, four hooves of a deer, five petals of a buttercup, six legs of an insect, seven colors of a rainbow, eight arms of an octopus, among others. As scientists delved into the study of numbers, they recognized their importance in everyday life.

Figure 3. Samples of numbers in nature

B. The Fibonacci Sequence

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers. It starts with 0 and 1, and each subsequent number is obtained by adding the two numbers preceding it. So, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

This pattern was first introduced to the Western world by the Italian mathematician Leonardo of Pisa, also known as Fibonacci, in his book Liber Abaci in the early 13th century. However, the sequence was described earlier in Indian mathematics.

The fascinating aspect of the Fibonacci sequence is that it appears in various natural phenomena and structures, for example, the growth patterns of plants, the arrangement of leaves and seeds, the branching of trees, and even the proportions of animal bodies and human anatomy. This occurrence of the Fibonacci sequence in nature has intrigued researchers and has been a subject of study in disciplines such as biology, physics, and art.

The ratio between consecutive Fibonacci numbers approaches the value of the golden ratio (approximately 1.61803398875).

The Golden Ratio or Phi

The Golden Ratio, often represented by the symbol φ (phi), is a mathematical constant that has fascinated artists, architects, and mathematicians for centuries. It is an irrational number, approximately equal to 1.61803398875. The Golden Ratio is derived from the Fibonacci sequence, where the ratio between consecutive Fibonacci numbers approaches phi as the sequence progresses.

The Golden Ratio, tied to aesthetics, embodies harmony and balance, visually pleasing to the human eye. For instance, artists, designers, and architects often employ this ratio for its aesthetic appeal, expressed as follows:

When dividing a line segment, creating a ratio where the whole segment to the longer part equals the ratio of the longer to the shorter part results in the Golden Ratio.

Mathematically, if "a" is the longer segment and "b" is the shorter segment, then (a+b)/a = a/b = φ. This ratio can be observed in various forms, such as rectangles, spirals, and polygons.

Throughout history, architects utilized the Golden Ratio in structures for instance the Parthenon, and artists like Leonardo da Vinci incorporated it in their paintings. Human body proportions also reflect this ratio, with measurements like the height of the navel to the total body height approximating phi.

While opinions on the importance and significance of the Golden Ratio vary, its prevalence in different fields of study and its perceived aesthetic appeal have made it an intriguing concept for scholars and creatives alike.

The Golden Spiral in Shapes

The golden spiral is a specific type of logarithmic spiral that increases in size based on the golden ratio, also known as Phi (ϕ). The golden ratio, approximately equal to 1.618, is a mathematical constant that appears in various aspects of arts, architecture, and nature.

In a golden spiral, the ratio of the length of each quarter-turn to its width is equal to the golden ratio. This creates a spiral shape that continually expands while maintaining the same overall shape. It is formed by drawing a series of quarter circles that grow in radius according to the golden ratio.

The golden spiral can be found in numerous natural phenomena, such as the shape of certain seashells, the arrangement of leaves on plants, the structure of galaxies, and even the proportion and shape of some famous artworks and architectural designs. Its aesthetically pleasing form and prevalence in nature have made it a fascinating subject of study and a source of inspiration for many artists, mathematicians, and scientists alike.

Figure 4. Golden Spiral

The Fibonacci Sequence in Nature

Nature is fascinating! It often uses patterns and sequences that mathematicians and scientists study. The Fibonacci sequence, which we talked about earlier, shows up in many places in nature, especially in flowers:

Flowers with 3 petals: Lilies and irises are examples. This means when you look at these flowers, you'd typically count 3 petals.
Flowers with 5 petals: Wild roses, larkspurs, buttercups, and columbines are examples. So, for these flowers, you'd see 5 petals.
Flowers with 8 petals: An example is delphiniums. If you count the petals, you'd find 8.
Flowers with 13 petals: Ragworts, corn marigolds, and cinerarias fit this category. You'd count 13 petals on these flowers.
Flowers with 21 petals: For example, Asters, chicories, and black-eyed susans are here. They typically have 21 petals.
Flowers with 34 petals: For example, Pyrethrums and plantains belong to this group. These flowers have a whopping 34 petals.

It's surprisingly amazing how this mathematical sequence is evident in nature, right in the flowers we might see in gardens or out in the wild!

The Fibonacci Sequence goes like this:

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1,597 2,584 4,181 6,765 10,946 17,711 28,657 46,368, 75,025 121,393 196,418 317,811...

Every number is the total of the two numbers before it.

Table 1. Fibonacci Table

When you divide one Fibonacci number by the one before it, you get a number close to 1.618, called the Golden Ratio. The bigger the Fibonacci numbers you use, the closer you get to this number.

A	B	B/A = ø
2	3	1.5
3	5	1.6666666667
5	8	1.6
8	13	1.625
...	...	...
144	233	1.6180555556
233	377	1.6180257511
377	610	1.6180371353
610	987	1.6180327869
...	...	...
1,597	2,584	1.6180338134
2584	4181	1.6180340557
4181	6765	1.6180339632
6765	10,946	1.6180339985
...	...	...
75,025	121,393	1.6180339887
121,393	196,418	1.6180339888
196,418	317,811	1.6180339887
317,811	514,229	1.6180339888

To illustrate, here are a few examples of where the Fibonacci sequence can be observed in nature and other phenomena:

Pinecones and Sunflower Seeds: The arrangement of scales on a pinecone and the positioning of seeds in a sunflower both follow spirals that align with Fibonacci numbers. These patterns optimize space efficiency and allow for maximum packing.

Flower Petals: Many flowers, such as daisies and roses, exhibit a specific number of petals that corresponds to Fibonacci numbers. For example, lilies often have 3 petals, buttercups may have 5, daisies can have 34 or 55 petals, and so on.

Tree Branching: The branching patterns of trees, including the arrangement of branches, twigs, and leaves, often adhere to the Fibonacci sequence. This can be observed in various tree species, such as oak trees and beech trees.

Nautilus Shells: The spiral shapes of nautilus shells conform to logarithmic spirals, which are closely related to the Fibonacci sequence and the golden ratio.

Bee Family Tree: The breeding patterns and family tree of honeybees follow a sequence derived from the Fibonacci sequence. Male bees typically have one parent, in contrast to female bees, which have two. Each generation traces back to the previous two generations.

Architecture and Design: Elements of architecture, such as the proportions and dimensions of buildings and structures, often incorporate the golden ratio, which is closely linked to the Fibonacci sequence.

These examples highlight the presence of Fibonacci numbers in various aspects of the natural world and how they assist in optimizing efficiency and aesthetic appeal.

Figure 6. Fibonacci Petals

C. Patterns and Regularities in the World as Organized by Mathematics

Mathematics encompasses the deep connection between patterns, relationships, and functions, serving as a unifying theme in this field of study. Many natural phenomena showcase exquisite beauty and can be accurately described through mathematical concepts. These spectacular patterns, such as rainbows, water waves, cloud formations, tree branching patterns, mud-crack patterns, butterfly markings, leopard spots, and tiger stripes, are underpinned by scientific and mathematical principles.

The behavior of waves on various water surfaces, be it puddles, ponds, lakes, or oceans, is governed by mathematical relationships involving their speed, wavelength, and water depth. By exploring similarities between diverse elements, like the spiral patterns of a snail shell and the swirling stars in a galaxy, or the branching patterns of a tree and a river network, hence, we can appreciate the majestic beauty and richness of natural pattern formation, all orchestrated by mathematics.

Order and symmetry form integral components of our world.

The cyclic patterns of days and nights, changing seasons, and alternating sunrises and sunsets provide order, while various forms of symmetry, like the fractal patterns in a Romanesco broccoli, the spiral shape of a nautilus shell, the bilateral symmetry of a peacock's tail, or the circular structure of spider webs, contribute to similarity, predictability, and regularity in nature.

The concept of symmetry captivates philosophers, astronomers, mathematicians, artists, architects, and physicists alike, as it seems to be present in countless aspects of our surroundings. Motion and behavior, such as the swinging motion of a pendulum, the reflection in a plane mirror, the free-falling motion of objects, and the action-reaction forces between interacting objects, are all guided and organized by mathematical principles. These phenomena display regularities and symmetry in their motion, adhering to mathematical laws.

For instance, the swinging motion of a pendulum exemplifies a complex yet harmonic mathematical relationship. The period, or the time it takes for the pendulum to swing back to its original position, is related to its length, following a non-linear relationship. Simply doubling the length of a pendulum does not result in a period that is twice as long, but the intricacies of this relationship can be elucidated through mathematics.

The formation of an image by an object in a plane mirror presents a fascinating phenomenon.

The image appears to be the same size as the object and is positioned behind the mirror at the same distance as the object is from the mirror. Without a doubt, Mathematics explains this regularity in size and distance through the law of reflection.

A free-falling object is one that falls solely under the influence of gravity. Its motion follows the equations of uniformly accelerated vertical motion, as it experiences no forces other than gravity. In every interaction between two objects, a pair of forces act upon them. The force exerted on the first object equals the force exerted on the second object in both magnitude and opposite direction. Hence, this principle of equal and opposite action-reaction force pairs underscores all interactions.

In summary, mathematics permeates the world around us, providing the tools to understand and describe the patterns, regularities, and symmetries inherent in nature. It unravels the complexities of phenomena such as the motion of a pendulum, image formation in a mirror, free-falling motion, and the action-reaction forces between objects, ultimately showcasing the beauty and order that mathematics brings to our understanding of the world.

Figure 7. Spectacular Patters

In the meantime, our world operates on established sequences, from the daily transitions of day to night to the cyclical change of seasons and the predictable rise and fall of the sun. Natural entities, whether it's the fractal design in a Romanesco broccoli or the symmetry of a spider's web, exude uniformity, predictability, and routine, all rooted in the world's mathematical structure.

Figure 8. Symmetrical Pattern

The idea of symmetry captivates a vast audience, from philosophers and astronomers to artists and physicists. Moreover, this mathematical underpinning of symmetry seems ubiquitous in our surroundings. For instance, consider the rhythmic swing of a pendulum, the reflection we see in a flat mirror, the descent of an object influenced solely by gravity, or the consistent application of action and reaction forces.These phenomena showcase mathematical consistency and symmetric behavior.

To elucidate, the pendulum's motion, though complex, resonates harmoniously. Its cycle duration, the time it takes to return to its starting point, correlates with its length in a non-linear fashion. A pendulum twice the length of another doesn't necessarily take twice as long to complete a swing, but through mathematics, this relationship becomes clear. Likewise, the image created by a plain mirror is intriguing: the reflected image maintains the object's size and appears as far behind the mirror as the object stands in front of it. This consistency can be demystified using the principles of reflection.

An object in free fall, under the exclusive pull of gravity, adheres to mathematical laws denoting uniformly accelerated vertical motion.

Every physical interaction produces a pair of forces acting on the interacting entities. These forces are equal in magnitude but opposite in direction, embodying the principle of action and reaction. This underlines the mathematical order that organizes and defines our universe.

Figure 9. Regularities in the world organized by mathematics

D. Phenomena in the World as Predicted by Mathematics

Nature exhibits various patterns, ranging from numerical patterns like Fibonacci numbers in sunflowers to shape patterns like those found in snowflakes. These patterns can be observed at every scale, from the smallest particles to galaxies. In addition, Mathematics plays a pivotal role in understanding these patterns by describing the underlying processes that break symmetry. For instance, sand dunes and zebra stripes may be caused by different physical processes, but mathematically, they share similarities. Mathematics also tackles enigmas found in nature, such as the orbital movements of planets, employs calculus to model changing quantities, predicts and controls physical systems, and explores logical relationships and proof.

Simplicity emerging from complexity can be witnessed in various phenomena like the formation of water droplets, population dynamics, and also in the arrangement of daisy petals, which often involve Fibonacci numbers.

Figure 10. Physically different but mathematically similar patterns

Through abstracting observations, mathematics allows us to identify commonalities and disparities between different phenomena and organize the natural world into distinct sets that can be studied using similar mathematical principles. Thus, nature serves as a bridge between the tangible everyday environment and the abstract realm of mathematics.

Likewise, Mathematics aids in summarizing, formalizing, interpolating, and extrapolating from recorded observations, enabling a deeper understanding of the natural world.

E. Nature and Occurrences in the World as Controlled by Mathematics for Human

Mathematics combines logic and creativity and is pursued for both practical and intrinsic reasons. While some appreciate the beauty and intellectual challenge of mathematics, others, including scientists and engineers, value its application in their respective fields. Given its central role in modern culture, a basic understanding of mathematics is essential for scientific literacy. In order to achieve this, students should view mathematics as part of scientific inquiry, grasp the nature of mathematical thinking, and acquire key mathematical concepts and skills.

The application of mathematics to medicine is an exciting and emerging field within applied mathematics. Nurses rely on mathematical calculations, such as addition, fractions, ratios, and algebra, to administer medication accurately and monitor patients' health. Nursing schools often assess new students' mathematical abilities and provide remedial courses in medical math when needed. Sharp mathematical skills are also crucial for nurses and healthcare practitioners in managing state-of-the-art medical facilities.

Another area where mathematics significantly contributes to health and medicine is the study of life expectancy. Life expectancy represents the projected remaining years of life for individuals and varies across different populations due to various factors. Census data is used to calculate life expectancy.

In the realm of political science, mathematics and statistics are employed to predict group behavior. This field involves utilizing various mathematical applications, including computer science, database management, statistics, and economics. Economists employ mathematical functions to explain the relationship between different variables, such as the causes of price increases, unemployment, or inflation. Mathematics also serves as a tool to better comprehend and analyze real-life phenomena in economics. As students delve deeper into the study of economics, they come to recognize that mathematical principles are indispensable in explaining concepts like prices, quantity of goods sold, and costs.

F. Applications of Mathematics in the World

Mathematics finds practical applications in everyday life, serving as a universal language across different places, times, settings, and circumstances. While it is often associated with counting, mathematics goes far beyond that. From purchasing products and following recipes to decorating a room, math principles are at work. These principles have been employed for thousands of years, transcending borders and continents.

Farming and gardening offer rich opportunities for mathematical exploration.

Time and money, two crucial elements in agriculture, also rely on mathematical foundations. Mathematics has enhanced the economic efficiency and productivity of farming by serving as an organizational system for effective time management and financial decision-making. Farmers utilize numbers daily for various tasks, such as measuring, weighing, and land marking. Sometimes unnoticed, the calculations and formulas involved require mathematical knowledge, and farmers apply problem-solving and money management skills regularly. Specifically, basic geometry, proportions, multiplication, and measurement skills come into play in agricultural activities. Technological advancements have further streamlined farming processes, allowing larger operations with fewer workers, highlighting the increasing importance of mathematics in the field.

Even something as routine as planning a shopping list and grocery shopping involves mathematical knowledge.

Operations like addition, subtraction, multiplication, and division are essential, along with estimation and percentages. To illustrate, calculating price per unit, determining percentage discounts, comparing unit and bulk prices, and estimating the total cost are all part of the mathematical considerations in grocery shopping. Modern trends like electronic banking and payment methods also rely on mathematical principles. Mathematics can be found throughout the home, from symmetric furniture arrangements to measuring ingredients and calculating cooking times in the kitchen.

Mathematics plays a role in various aspects of travel, such as calculating fuel requirements based on distance, estimating expenses for toll fees, checking tire pressure, and navigating different routes. Construction and building projects heavily rely on mathematics, from accurate measurements of lengths, widths, and angles to detailed material estimation and resource optimization. This practical application of mathematics is known as engineering mathematics, which combines theoretical principles, practical engineering, and scientific computing in order to address contemporary technological challenges. Engineering mathematicians work in diverse fields, ranging from designing high-end cars to inventing robotics and automated devices.

Experts agree that strong math skills are crucial in financial decision-making.

People with deficiencies in math fundamentals tend to make more significant financial mistakes and underestimate the impact of compounding interest. Understanding concepts like exponential growth and compound interest empowers individuals to manage debt and therefore make more informed financial choices.

Time is a valuable asset for everyone, and effective planning is essential to avoid wasted time and accumulating tasks. Managing time well requires more than simply using a clock and calendar; it demands mathematical skills. While abstract mathematical concepts may not always come into play, the fundamental skills developed through basic math lessons resonate throughout a person's lifetime and prove useful in solving real-life problems in various situations, both at work and in the world.

Examples:

Choose the figure that completes the pattern.

Answer: D. The sketch is being gradually constructed, with each stage adding a new line that never intersects with the last line from the previous stage.

Answer: B. In each figure, three shapes interact: the external shape, the middle shape, and the inner shape. Notably, the external shape of one figure becomes the middle shape in the next, and the inner shape transforms into the external shape two steps ahead. For instance, in the 4th figure, a circle becomes a pentagon, and a hexagon emerges as the inner shape, creating a dynamic progression.

Thus, the next number is 28 (23 + 5).

Solutions:

Firstly:

The sequence 1, 4, 9, 16, 25, 36 can be written as 12, 22, 32, 42, 52, 62.

Thus, the next number is 72, that is, 49.

Secondly:

1, 4, 9, 16, 25, 36, 49 (4-1) 3 (9-4) 5 (16-9) 7 (25-16) 9 (36-25) 11 3+2=5; 5+2=7; 7+2=9; 9+2=11; 11+2=13

Thus, the next number is 49.

Examples 1 and 2 are commonly encountered in aptitude tests. In order to identify the subsequent shape or number, we need to carefully observe the objects, analyze their properties, and examine their connections with other objects. By doing so, we can form hypotheses, make predictions, and establish generalizations based on the observed patterns.