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by Terence Tao [1]“The beauty of mathematics,” someone once wrote to me, “is ...

by Terence Tao [1]“The beauty of mathematics,” someone once wrote to me, “is that it’s matching delights your soul, whereas the beauty of the humanities delights your eyes.” I wrote back, “But doesn’t the beauty of language—whether it be English or Chinese—delight your soul as well? Aren’t all forms of beautiful writing gateways to truth, love, and wisdom?” Mathematics and the humanities are not as far apart as one might think. Both are gateways to beauty and wisdom, and both can captivate one’s thoughts and imaginations.“Mathematics” can be a bit of a dirty word in some circles. People oftenImages think of it as being boring, too cumbersome, or simply uninteresting. But if you strip away the mystical and forbidding language used in mathematics texts and replace it with language that is more accessible and understandable, you might find that what was once thought to be a dry and meaningless collection of numbers and symbols can actually be a source of beauty, insight, and even amusement. Mathematics is not a collection of arcane facts and formulas that is divorced from reality; it is a body of knowledge that grows organically from within human civilization and experience, with profound connections to our lives and the natural world.If you zoom out a bit, mathematics—like any other field—will not seem as daunting. In fact, there are some very simple “ah ha” moments that can be deeply satisfying—moments that are full of the beauty of mathematics.I believe that everyone has their own personal sense of the beauty of mathematics—a personal sense of its charm and magic. I hope that this book will help you find your own “ah ha” moments in mathematics and fall in love with this subject—or reignite your passion for it.—Terence TaoPrefaceMathematics is a subject that is often considered dry, boring, or too difficult. However, when viewed with a fresh perspective, the subject can reveal a wealth of beauty and simplicity that is both captivating and enlightening. In this book, we will explore some of the hidden treasures in the world of mathematics and discover the beauty that is often overlooked.Chapters1. The Beauty of PatternsMathematics is full of patterns that are both beautiful and profound. One example is the pattern of prime numbers. Prime numbers—numbers that can only be divided by themselves and one—provide a wealth of natural patterns in the number series. Another example is the patterns found in geometry, such as the repeating patterns of regular polyhedrons or the recursive patterns of fractals. These patterns have been studied for centuries by mathematicians and scientists and have applications in fields such as art, architecture, and science.2. The Beauty of SimplicityMathematical proofs can often seem complex and technical, but behind these technical details are principles of simplicity and clarity. Simplicity in math means better understanding, easier communication, and a deeper understanding of the subject matter. This chapter explores how mathematical proofs can lead to an appreciation of simplicity, economy, efficiency, and above all else—beauty.3. The Beauty of InsightMathematics is a subject that rewards insight. Sometimes a difficult problem can be solved with a simple idea or trick. This chapter explores some examples of mathematical insights, including Pythagoras’ theorem, Fermat’s Last Theorem, and Nash’s embedding theorem. Each example illustrates how a mathematical insight can transform a problem and provide unexpected answers.4. The Beauty of ApplicationsMathematics may seem disconnected from daily life, but it plays an integral role in numerous applications. This chapter explores some unexpected applications of math, including the use of statistics in criminology, the use of geometry in robotics, and the use of calculus in biology and physics. These examples demonstrate how math can translate into real-world solutions and enhance our quality of life.ConclusionThe beauty of mathematics is not solely for mathematicians but for everyone. This book has hopefully conveyed the idea that mathematics—far from being dry and boring—is a subject full of captivating beauty and wonder. The patterns, simplicity, insights, and applications explored here are just a small sampling of the richness and diversity of mathematical concepts that can captivate our imaginations and enrich our lives. So next time you encounter a math problem or see a formula in a textbook or article, try zooming out a bit to see the beauty within. You might find yourself captivated by a wealth of mathematical delights.[1]^