As a parent, we should be following up on our children’s educational progress, especially their understanding of the fundamental scientific principles taught in Basic Science subjects at the junior levels or in Physics, Chemistry, and Further Mathematics at the senior levels. My first child, Muhammad, would ask me tons of questions related to the above subjects since when he was in the junior level, some of which I had to research before answering. Today, their Further Mathematics teacher started treating differential calculus. He asked me when he came back home from school, ‘What is this differential calculus all about and what are its applications?’ The popular way to refer to calculus is the notation dy by dx pronounced by reading the letters separately as dydx. I thought about sharing my responses with the public with the hope that someone with a similar question as Muhammad’s could benefit from my simple introduction to calculus. Happy reading!
How It All Began – The Falling Apple Observation
Most of us are aware of Newton’s falling apple story. Sir Isaac Newton (1642 – 1727), the greatest scientific genius the world has ever seen, was in a garden when he saw an apple fruit falling off from its tree. This simple observation sent the young Newton – who was not even up to the age of 26 at the time – to begin to wonder about the nature of gravity. He thought to himself, ‘Does the moon also fall toward the Earth in a similar way as the apple fruit fell toward the Earth?’. He answered in the affirmative after performing thought experiments in which he projected an object on the Earth’s surface with increasing velocities thereby picturing, for the first time, how an object can be placed in orbits around the Earth like satellites and how an object can escape the Earth’s gravitational influence like spaceship. In fact, Newton calculated for the first time the minimum Earth’s orbital velocity as 5 miles per second and escape velocity as 7 miles per second. Back to the falling moon problem, Newton did not have the mathematics to analyze it. So, he invented calculus, almost on a dare, as Neil dyGrasse Tyson (the most famous astrophysicist of our time) would put it.
Newton did not stop from just analyzing the falling moon problem, he went on further to predict the motion of the numerous moons of Jupiter with great accuracy. Newton became so fond of modeling the positions of celestial bodies that he would say, tell me the present state of a system and I will tell you how it will behave tomorrow and the day after tomorrow. Hence, calculus is the first mathematical tool invented for modeling physical systems and that is why it has applications in almost all engineering disciplines.
Gravity
From this apple fall story, Newton understood that the Earth pulled the apple toward it with a force of gravity. He came up with Newton’s Law of Universal Gravitation after formulating the 3 Newton’s Laws of Motion. When these laws were applied to predict the path of Mercury, the closest planet to the Sun, there were small deviations.
Fast forward to the 1900s when Albert Einstein (the greatest genius of the 20th century) came up with his groundbreaking theories of relativity. The first one was in 1905, called the Special Theory of Relativity which modified Newton’s laws of motion to incorporate objects traveling at very high speeds, close to the speed of light, and introduced time as the 4th dimension in what he described as spacetime. Einstein again formulated the 2nd theory in 1915 called the General Theory of Relativity which modified Newton’s Law of Universal Gravitation to describe the motion of objects near a high-gravity source like the Sun and corrected Newton’s understanding of gravity as a pull by heavier objects on smaller ones with a simple space push on the smaller objects whose space is warped by the heavier objects. With Einstein’s theory of relativity, the path of Mercury was more accurately predicted without any deviations than with Newton’s laws because Mercury was near a high-gravity source, which was the Sun. In essence, when you plug in low gravity and low speeds into Einstein’s equations, they will reduce to Newton’s equations. This means that Einstein’s equations are special cases of Newton’s equations for a high-gravity source such as the stars and black holes and high speeds that approach the speed of light.
Conclusion
The foregoing introduction and applications of calculus with specific cosmic examples and historical perspectives could spark an excellent interest in any fresh student of calculus. Science and mathematics teachers should devote so much time and effort to introducing new topics to their students with a view to sparking a passion in the students and making them understand some of these complex principles better. One of my physics teachers at the secondary level, Mal. Ibrahim Physics would introduce all Physics topics to us like what I did above and I can still remember vividly, over 29 years now, his specific practical examples and explanations of physics principles to this day. His explanations created a strong bond between Physics and me to this day.
Salihu Lukman is an Assistant Professor of Civil Engineering at the University of Hafr Al Batin, Saudi Arabia
Is God a Mathematician? Unveiling the Divine Patterns in Nature
By: Dr. Salihu Lukman
Highlights:
The Mathematical Tapestry of Nature
Physicists as Unravelers of the Mathematical Universe
The Power of Prediction and Modeling
Studying Science
The Importance of Scientific Literacy
Most Profound and Impactful Mathematical Principles of Nature
Introduction
In the pursuit of understanding the mysteries of the universe, humanity has often contemplated the connection between God, mathematics, and the natural world. As we delve into the depths of scientific exploration, we discover that the elegance, precision, and exactness of mathematics are intricately woven into the very fabric of nature. From the graceful dance of celestial bodies to the orchestrated symphony of natural processes, mathematics emerges as a universal language, revealing profound insights into the mysteries of creation.
The Mathematical Tapestry of Nature
Nature’s affinity for mathematics becomes evident when we observe the intricate patterns that govern it. From the spiraling patterns of seashells to the fractal structures of fern leaves, mathematical concepts such as Fibonacci sequences and fractals manifest everywhere. These patterns not only showcase the aesthetic beauty of nature but also reveal the underlying mathematical principles that govern its behavior.
Here are a few examples of the mathematical tapestry woven into the natural world and the cosmos:
Kepler’s Laws of Planetary Motion: Johannes Kepler, a renowned astronomer, formulated three laws of planetary motion based on observations and mathematical analysis. These laws describe the elliptical paths of planets around the sun, the equal-area rule (planets sweep out equal areas at equal times), and the mathematical relationship between a planet’s orbital period and its average distance from the sun.
Newton’s Law of Universal Gravitation: Sir Isaac Newton’s law of universal gravitation mathematically describes the force of attraction between two objects based on their masses and the distance between them. This law helps us understand the gravitational interactions between celestial bodies, such as planets, moons, and stars.
Einstein’s Field Equations: Albert Einstein’s theory of general relativity provides a mathematical framework for understanding gravity as the curvature of spacetime caused by mass and energy. His field equations describe the relationship between the distribution of matter and the geometry of spacetime, offering insights into the behavior of massive objects and the bending of light.
The Fibonacci Sequence in Nature: The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, etc.) and its related Golden Ratio have fascinating connections to nature. They can be observed in the spiral patterns of seashells, the branching of trees, the arrangement of leaves on a stem, and even in the proportions of the human body. These mathematical patterns exhibit a remarkable aesthetic beauty found throughout the natural world.
Fractals in Nature: Fractals are intricate mathematical patterns that exhibit self-similarity at different scales. They can be found in various natural phenomena, such as the branching patterns of trees, the intricate shapes of coastlines, the structure of clouds, and even in the cosmic web of galaxy clusters. Fractals highlight the inherent mathematical complexity and intricacy present in the natural world.
These examples illustrate how mathematics unveils the underlying order and patterns in the cosmos, enabling us to describe and understand the behavior of celestial bodies, the growth of natural forms, and the fundamental forces shaping the universe. The mathematical tapestry of the cosmos reflects a profound interconnectedness between mathematics and the natural world.
Physicists as Unravelers of the Mathematical Universe
Physicists, relentless in their pursuit of knowledge, have played a pivotal role in uncovering the mathematical underpinnings of the natural world. Through rigorous experimentation and mathematical modeling, they have revealed profound equations that describe the fundamental laws of nature. From Isaac Newton’s laws of motion to Albert Einstein’s theory of relativity, these mathematical frameworks provide us with a deeper understanding of the intricate workings of the cosmos.
The Power of Prediction and Modeling
One of the most remarkable applications of the mathematics of nature lies in its ability to make reliable predictions. Engineers and scientists utilize mathematical models to forecast natural disasters, predict weather patterns, and comprehend the behavior of complex systems. By harnessing the power of mathematics, we gain valuable insights that aid in disaster preparedness, resource management, and mitigation strategies, ultimately saving lives and minimizing the impact of natural calamities.
Studying Science
A Pathway to Appreciating God’s Wonders: The exploration of the mathematical tapestry of nature not only enriches our scientific knowledge but also deepens our appreciation for the wonders of creation. As we unlock the secrets of the universe, we are granted glimpses into the divine intelligence behind it all. Science becomes a means to marvel at the precision and orderliness that permeates every aspect of existence.
The Importance of Scientific Literacy
In an age dominated by technology and scientific advancements, understanding science is more important than ever. Scientific literacy empowers us to critically analyze information, make informed decisions, and contribute positively to society. By studying and comprehending the mathematics embedded in nature, we gain a profound appreciation for God’s creative genius, fostering a sense of awe and reverence for the natural world.
I am not saying that everybody must have a science background or must study a science or applied science course. However, possessing a fair understanding of some basic scientific concepts and principles cannot be over-emphasized. For someone without a science background, understanding the basic scientific concepts and principles can provide a foundation for comprehending the processes of nature and the universe. Here are some fundamental scientific concepts that can help:
Scientific Method: The scientific method is a systematic approach used by scientists to investigate and understand the natural world. It involves making observations, formulating hypotheses, designing experiments or gathering data, analyzing the results, and drawing conclusions. Understanding the scientific method allows for critical thinking and evaluating scientific claims.
Laws and Theories: In science, laws and theories are different. A scientific law describes a phenomenon or relationship observed in nature, expressed as a mathematical equation or principle (e.g., Newton’s Laws of Motion). A scientific theory, on the other hand, is a well-substantiated explanation of a natural phenomenon, supported by substantial evidence and capable of making accurate predictions (e.g., Theory of Evolution or Theory of General Relativity).
Matter and Energy: Matter refers to anything that occupies space and has mass. It exists in various states (solid, liquid, gas) and is composed of atoms and molecules. Energy, on the other hand, is the ability to do work or bring about change. Understanding the concepts of matter and energy helps in comprehending the behavior and interactions of objects and systems.
Forces and Motion: Forces are pushes or pulls that can cause objects to accelerate, decelerate, or change direction. Newton’s Laws of Motion describe the relationship between forces, mass, and motion. Understanding these principles helps explain phenomena such as gravity, planetary motion, and the behavior of objects in everyday life.
Cells and DNA: Cells are the basic building blocks of life. They make up all living organisms and have various structures and functions. DNA (deoxyribonucleic acid) is the genetic material found within cells that carries the instructions for an organism’s development and functioning. Familiarity with these concepts aids in understanding life processes, genetics, and the diversity of living organisms.
Ecosystems and the Environment: Ecosystems are communities of living organisms interacting with their physical and chemical environments. Understanding the interconnectedness of organisms and their environment helps comprehend topics such as biodiversity, ecological relationships, and the impact of human activities on ecosystems.
Earth Systems: Earth is a complex system comprising various interconnected components, such as the geosphere (land), hydrosphere (water), atmosphere (air), and biosphere (living organisms). Understanding Earth’s systems helps grasp phenomena like weather patterns, climate change, and natural resources.
By familiarizing yourself with these basic scientific concepts, you can develop a better understanding of the natural world and the processes that shape it. They provide a foundation for further exploration and appreciation of scientific knowledge.
Most Profound and Impactful Mathematical Principles of Nature
The world as we know it today has been shaped by numerous profound and impactful mathematical principles and theories inherent in nature. The exploration of the cosmos has led to the development of several profound and established theories that have significantly shaped our understanding of the universe. Here are a few notable examples:
Calculus: The development of calculus, pioneered by Sir Isaac Newton and Gottfried Wilhelm Leibniz, revolutionized our understanding of change and motion. Calculus has been instrumental in fields such as physics, engineering, economics, and computer science. Its applications range from modeling the trajectory of celestial bodies to designing efficient transportation systems.
Probability Theory: Probability theory, developed by mathematicians like Pierre-Simon Laplace and Andrei Kolmogorov, has had a profound impact on fields such as statistics, finance, and risk assessment. It enables us to understand uncertainty, make informed decisions, and calculate the likelihood of events occurring. Probability theory plays a crucial role in fields like insurance, weather forecasting, and epidemiology.
Number Theory: Number theory, the study of integers and their properties, has both practical and theoretical implications. It forms the foundation for modern cryptography, which secures our digital communications and transactions. Number theory also plays a key role in coding theory, which ensures reliable data transmission and storage.
Chaos Theory: Chaos theory, pioneered by mathematicians such as Edward Lorenz, studies the behavior of dynamic systems that are highly sensitive to initial conditions. It has revolutionized our understanding of complex systems, including weather patterns, population dynamics, and the stock market. Chaos theory has highlighted the importance of seemingly insignificant factors in shaping outcomes and has influenced fields like meteorology, economics, and biology.
Fractal Geometry: Fractal geometry, introduced by mathematician Benoit Mandelbrot, explores complex and self-repeating geometric patterns. Fractals can be found in natural phenomena such as coastlines, clouds, and snowflakes. The applications of fractal geometry range from computer graphics and digital imaging to understanding the structure of galaxies and modeling the growth of biological systems.
Big Bang Theory: Proposed by Georges Lemaître, a Belgian cosmologist and Catholic priest. The Big Bang Theory is the prevailing cosmological model that explains the origin and evolution of the universe. It posits that the universe began as an incredibly hot and dense singularity around 13.8 billion years ago and has been expanding ever since. The Big Bang Theory has revolutionized our understanding of the cosmos, providing crucial insights into the formation of galaxies, stars, and planets.
Theory of General Relativity: Proposed by Albert Einstein, the Theory of General Relativity revolutionized our understanding of gravity. It describes gravity as the curvature of spacetime caused by massive objects. General Relativity has been crucial in explaining the structure and behavior of celestial bodies, from the motion of planets to the bending of light around massive objects.
Quantum Mechanics: Quantum Mechanics is a fundamental theory that describes the behavior of particles at the subatomic level. It has transformed our understanding of matter, energy, and the fundamental forces of nature. Quantum Mechanics has practical applications in fields such as electronics, telecommunications, and computing, leading to advancements like transistors, lasers, and quantum computing. Quantum Mechanics was developed by several scientists, including Max Planck, who introduced the concept of quantization of energy in 1900. Other key contributors include Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, and Paul Dirac, among others.
Cosmic Inflation Theory: Proposed by physicist Alan Guth in 1980. Cosmic Inflation Theory proposes that the universe underwent an exponential expansion in its early stages, shortly after the Big Bang. This theory helps explain the uniformity of the cosmic microwave background radiation and the large-scale distribution of galaxies. Cosmic Inflation Theory has had a significant impact on our understanding of the early universe and sets the stage for subsequent structure formation.
Dark Matter and Dark Energy: The existence of dark matter and dark energy, though not fully understood, has shaped our understanding of the universe. Dark matter is a hypothetical form of matter that does not interact with light but exerts gravitational influence, explaining the observed discrepancies in the motion of galaxies. Dark energy, on the other hand, is an unknown form of energy that is believed to be responsible for the accelerated expansion of the universe. The concept of dark matter was first proposed by Swiss astronomer Fritz Zwicky in the 1930s, while dark energy was proposed in the late 1990s when observations indicated that the expansion of the universe was accelerating. Dark matter and dark energy are fascinating concepts that continue to baffle scientists. Let me break it down for you. Dark matter is estimated to make up about 27% of the universe’s mass and energy. It is an invisible and mysterious form of matter that does not interact with light or other electromagnetic radiation. Instead, it exerts gravitational forces, helping to explain the movement of galaxies and the formation of structures in the universe. Despite its significant presence, scientists have yet to directly detect or fully understand what dark matter is composed of. Dark energy, on the other hand, is an even more perplexing phenomenon. It is believed to constitute about 68% of the universe’s energy density. Dark energy is associated with the acceleration of the expansion of the universe. Rather than being a substance or matter, it is thought to be a property of space itself. The exact nature of dark energy remains elusive, and scientists are actively researching and formulating theories to better comprehend its origins and behavior. The primary reason why dark matter and dark energy are still not fully understood is their elusive and intangible nature. They do not interact with ordinary matter and energy in ways that we can easily detect or observe. Scientists rely on indirect observations, mathematical models, and experiments to study and infer their existence. Ongoing efforts in particle physics, cosmology, and astrophysics aim to shed more light on these mysteries and unravel their secrets. It means that we only have some understanding of a mere 5 % of total matter and energy in the universe, while the remaining 95 % is still not fully understood.
These principles and theories represent just a fraction of the mathematical concepts that have shaped our world. Mathematics has been instrumental in advancing scientific knowledge, engineering innovations, and technological breakthroughs. By uncovering the mathematical principles embedded in nature, we have gained a deeper understanding of the universe and harnessed its potential for the betterment of society. The cosmic theories have transformed our understanding of the universe, influenced technological advancements, and inspired further exploration and scientific inquiry. They have paved the way for groundbreaking discoveries and continue to shape our world by providing a framework for understanding the cosmos and our place within it.
Conclusion
As we ponder the question, “Is God a mathematician?” we uncover a mesmerizing tapestry of mathematical intricacies woven throughout the cosmos and natural processes. Physicists, engineers, and scientists tirelessly work to unravel these mathematical mysteries, providing us with insights that enable us to predict and understand the world around us. By embracing scientific literacy and marveling at the mathematical wonders of nature, we embark on a journey of discovery that deepens our understanding of the universe and enhances our appreciation for the divine intelligence behind it all.
Salihu Lukman is an Assistant Professor of Civil Engineering at the University of Hafr Al Batin, Saudi Arabia
