Passionate About Mathematics: My Journey to Research

My journey into the world of mathematics began in a junior high school math class where the definition and origin of pi were introduced. Our teacher demonstrated the Buffon needle experiment using a computer, randomly placing thin needles between parallel lines and showing how the probability of intersection with the lines was related to pi. At that moment, I was deeply shocked. My understanding of pi was limited to the ratio of the circumference to the radius of a circle. Why did pi appear when there was no circle? This sparked a lifelong fascination with the elegant and beautiful 'art' of mathematics.

Studying A-Level Mathematics unveiled the joy of delving into mathematical theory, but it wasn't enough. I yearned to delve deeper, seeking to uncover the essence of each concept presented in textbooks. For instance, while studying FP3, I discovered the intriguing correlation between hyperbolic trigonometric functions and e^x, similar to how trigonometric functions relate to e^ix. I meticulously derived the formulas for hyperbolic trigonometric functions from their geometric definitions using calculus. This process made me reflect on mathematicians' thought processes, further fueling my desire to explore the world of mathematics.

As an advanced learner among A-level mathematics students, I founded a Maths Club at our school. For our school's 'Science Fair' event, our club chose 'Minesweeper' as our subject. Initially, we struggled to find a mathematical connection. However, realizing that Minesweeper followed only two conditions, I drew upon my knowledge of the binomial distribution from S2, and we found a research direction: whether Minesweeper follows a binomial distribution. Opinions varied, but I focused on the impact of revealing a square on the probability of other squares being mines. It deviates from a binomial distribution, but I recalled a similar example in the textbook. We adjusted mine counts and calculated probabilities in different conditions, finally reaching a conclusion: the probability of a square being a mine followed a binomial expansion when the total number of mines was big enough. During this activity, the research process enhanced my teamwork and communication skills, reinforcing my realization that mathematics is omnipresent. The profound sense of fulfillment I derived from this process intensified my aspiration to immerse myself in the pursuit of groundbreaking mathematical research. Additionally, I achieved high scores in the AMC12, AIME, and SMC.

One memorable moment during my mathematical journey was the discovery of a fascinating pattern while studying the binomial theorem. Arranging Pascal's Triangle in a right-angled triangle fashion, I observed that the sum of all terms along the diagonal downwards formed the Fibonacci sequence. Intrigued, I sought to prove this phenomenon by deriving a general formula for the sum of the diagonal terms and utilizing the definition of the Fibonacci sequence in my proof. Through this exploration, I uncovered the underlying pattern by carefully observing the number of terms described by Pascal's Rule. This revelation left me in awe, wondering how the Fibonacci sequence could be concealed within Pascal's Triangle with seemingly no intersection. Countless unanswered questions in the realm of mathematics inspire my determination to explore and unravel its mysteries.

Mathematics has not only provided me with knowledge but also the ability to question and challenge axioms. It has equipped me with rationality to approach everything with a discerning mind. I aspire to devote myself to mathematical research, and gaining admission to a prestigious UK university for undergraduate studies would be the first step in my exploration of the beauty of the mathematical world.