Remote Space Learning - Mathematics for our World

Lecture 03: Mathematics for our World

Mathematics for Our World highlights how mathematics shapes our understanding of nature, technology, art, and society. It reveals patterns like the Fibonacci sequence in plants and the Golden Ratio in architecture, helping us see structure and beauty in the world around us. Mathematics also drives innovations in technology, from encryption to artificial intelligence, and cultivates critical thinking skills essential for problem-solving. Ultimately, mathematics is a universal language that allows us to explore, interpret, and improve our world.

Mathematics has been an essential companion to humanity throughout history. Perhaps it began with early farmers counting livestock or rulers assessing the strength of their armies. As centuries passed, mathematics became increasingly vital, adapting to humanity’s evolving needs. Today, we rely on mathematics in nearly every field: pilots and seafarers calculate distances and directions to navigate the Earth; chefs ensure correct ingredient proportions in recipes; epidemiologists model viral outbreaks using differential equations; and agronomists fine-tune farming practices to boost agricultural yields. From ancient times to the present, mathematics has remained indispensable, continuously expanding to meet the demands of modern life.

The Ancient Babylonian Tablets are some of the earliest records of mathematical and astronomical knowledge, dating back over 3,000 years to the time of the Mesopotamian civilization. These clay tablets, inscribed with cuneiform script, provide fascinating insights into how the Babylonians approached mathematics, using a base-60 (sexagesimal) numeral system that influenced timekeeping (60 minutes in an hour) and geometry (360 degrees in a circle).

Babylonian tablets often feature calculations related to practical problems, such as land measurement, taxation, and commerce. Some contain multiplication tables, reciprocal tables, and methods for solving linear and quadratic equations. The level of sophistication on these tablets shows that the Babylonians not only performed basic arithmetic but also understood mathematical relationships and patterns, making them one of the earliest civilizations to develop a formalized mathematical system.

These tablets have been instrumental in understanding the history of mathematics, revealing that advanced calculations were in use well before similar concepts were documented in ancient Greece or other civilizations. The ancient Babylonian approach to mathematics has continued to impact modern fields, from geometry to time measurement, underscoring the timeless value of their discoveries.

In the 2017 movie The Wall, Sgt. Allen Isaac (played by Aaron Taylor-Johnson) uses mathematics to accurately locate his enemy. By carefully measuring the difference between when a bullet impacts near him and when he hears the sound of the shot, he calculates the distance to his target. He achieves this by factoring in the average speeds of both the rifle bullet and the sound wave. This scene highlights how mathematical principles can be applied in real-life scenarios requiring precision and timing.

In this lecture, we’ll explore the diverse applications of mathematics in the modern world, focusing on its roles in organization, prediction, and control. Through these examples, we aim to deepen our appreciation of mathematics and understand its essential role in shaping and improving our daily lives.

The world is incredibly complex! Every day, countless changes happen simultaneously. Around 385,000 babies are born worldwide, over 500 million tweets are posted, thousands of airplanes crisscross the skies, stock prices fluctuate, and approximately 463 exabytes (463 billion gigabytes) of new data are generated. To navigate this vast amount of information, we rely on systems to help us organize and make sense of it all. Mathematics is essential in providing these systems.

In this section, we’ll explore a few mathematical tools that help us manage and interpret complex information. While there are many examples, we’ll focus on three: networks, the coordinate plane, and matrices.

A network is a structure consisting of objects, represented as nodes (or vertices), connected by lines called links (or edges). Networks are widely used by computer scientists, mathematical modelers, statisticians, and experts in many other fields to illustrate systems and the relationships between components effectively.

In the illustration on the right, the blue dots represent nodes, while the green lines are the links. Notice how each link connects two nodes, symbolizing a relationship between them.

The network shown is an example of a simple network, where each link connects two different nodes, and each pair of nodes has at most one connecting link. In contrast, a non-simple network may have a link connecting a node to itself or multiple links between the same pair of nodes, representing more complex relationships.

Networks are invaluable across scientific disciplines, helping to map connections in fields ranging from social sciences to biology, communication, and more.

The Human Disease Network illustrates the relationships between different human diseases based on shared genetic defects. In the network, two diseases are connected if they share a common gene defect, meaning a person with one disease may have an increased likelihood of developing related diseases due to genetic overlap. The thickness of each link indicates the likelihood or strength of this relationship—the thicker the link, the higher the chance of complications that could lead to the connected disease. This research was conducted by Dr. Kamyar Hedayat of the Full Spectrum Health Center and Dr. Jean-Claude Lapraz.

The thickness of each link between nodes conveys the strength of the association; thicker lines indicate a stronger genetic or causal relationship, suggesting a higher probability of comorbidity. This detail is especially useful for clinicians and researchers who aim to understand the potential progression of diseases, allowing them to monitor patients for related conditions more closely. For instance, a patient with diabetes might be at higher risk for cardiovascular disease if the two are strongly connected in the network.

The British Royal Family Tree is an example of a network, where each family member is a node. Members with blue borders represent those related by blood, while those with gray borders signify members who joined through marriage. Black links indicate marital relationships, gray links represent parent-child connections, and red dashed links show divorces. This network allows us to visualize the complex relationships within the royal family in an organized and easy-to-understand structure.

This network not only includes blood relatives (in blue borders) who are direct descendants of the monarchy but also marital connections (in gray borders), symbolizing the extended network that brings in new alliances and strengthens ties with other noble families and countries. The use of black links for marriages shows stable marital relationships, while red dashed lines indicate divorces, highlighting shifts in the family dynamic over time. The gray lines represent parent-child relationships, emphasizing lineage and succession within the royal family.

The tree also helps viewers understand the complexities of royal succession, as only certain family members are eligible for the throne based on their direct bloodline. Additionally, marriages within the family tree often had political and diplomatic implications, historically strengthening alliances with other nations or noble houses.

According to a popular story, the French mathematician René Descartes (pronounced re-NEY deh-CART) was known to be somewhat sickly and spent many days resting in bed. One day, as he lay in his bedroom, he noticed a spider crawling across the door, which was decorated with square tiles. Descartes began to wonder how he could determine the spider's exact location on the door by counting squares from the bottom and left edges of the door. This idea sparked what would later become known as the Cartesian coordinate system.

Descartes formalized this system in his work La Géométrie, under his Latin name, Renatus Cartesius. The Cartesian coordinate system, one of mathematics’ most powerful tools, begins with two perpendicular lines: a vertical line called the y-axis and a horizontal line called the x-axis. These lines intersect at the origin (0,0), and each point on the plane is defined by an ordered pair (x,y)(x, y)(x,y), where x is the horizontal distance from the y-axis (called the abscissa) and y is the vertical distance from the x-axis (called the ordinate). For instance, in the coordinate plane shown, the point (5, 4) has an x-coordinate of 5 and a y-coordinate of 4, specifying its precise location. This system is now foundational in mathematics, allowing us to graph functions, model real-world data, and visualize geometric shapes.

The Cartesian coordinate system has evolved to play a crucial role in locating positions on the globe. This concept is foundational in latitude and longitude coordinates, which are essentially a global grid system that enables us to pinpoint exact locations on Earth. Just as points on a plane are defined by x and y coordinates, any position on the globe can be specified using latitude and longitude, helping us navigate and map our world with precision. This system is essential for GPS technology, global mapping, and navigation, allowing us to determine our position anywhere on the planet.

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