Comparison of mathematics and Japanese Haiku poetry

From: mk_thisisit

Professor Dawid Kielak from Oxford University explores the concept of elegance and beauty in mathematics, drawing an analogy with Japanese Haiku poetry [00:00:08]. He aims to explain what mathematical elegance means to mathematicians [00:00:27].

Understanding Haiku

Haiku is a form of Japanese poetry characterized by three main properties:

• Conciseness Haiku is very concise, typically consisting of only three lines [00:01:13].
• Deep Content It is designed to inspire a deeper understanding of reality, encouraging reflection and meditation [00:01:30]. This property is linked to Japanese culture and Zen Buddhism [00:01:33].
• Aesthetic Beauty The connection between the concise form and the deep content must be beautiful and aesthetically pleasing [00:02:00]. This aesthetic quality is what Professor Kielak refers to as elegance [00:02:09].

Haiku is also associated with graphic forms like miniatures and calligraphy, often made economically yet beautifully [00:02:18].

Haiku in Science and Mathematics

Professor Kielak suggests that scientific formulas can also be seen as Haiku:

• Einstein’s E=mc² This formula is concise, has deep content, and exhibits a certain elegance [00:02:47].
• Schrödinger’s equation Describes how a particle evolves in quantum physics [00:02:55].
• Boltzmann entropy States that entropy does not decrease (the second law of thermodynamics) [00:03:07].
• Einstein’s field equations Describe the geometry of space-time [00:03:15].

These scientific expressions, like Japanese poetry, can appear indecipherable to those unfamiliar with their “language” [00:04:44].

Euler’s Formula: The Most Elegant Haiku in Mathematics

Euler’s formula, e^(iπ) + 1 = 0, is presented as perhaps the most elegant and beautiful formula in mathematics [00:03:42]. Leonhard Euler, an 18th-century Swiss mathematician, was exceptionally prolific [00:03:46].

Euler’s formula fulfills the three properties of Haiku:

• Concise Form The formula is undoubtedly concise [00:04:19].
• Deep Content It embodies profound mathematical relationships [00:04:25].
• Elegance The relationship between its concise form and deep content is considered very elegant [00:04:33].

Deconstructing the “Deep Content”

Euler’s formula is considered beautiful because it combines five fundamental mathematical constants: 0, 1, Pi (π), the imaginary unit (i), and Euler’s number (e) [00:05:15]. These numbers originate from three distinct areas of mathematics: trigonometry, complex numbers, and mathematical analysis (differential calculus) [00:05:32]. The formula serves as a meeting point for these different fields [00:05:42].

1. Trigonometry and the Unit Circle

The formula sin²x + cos²x = 1 can be understood by relating it to the Pythagorean theorem and the introduction of a coordinate system [00:06:24].

• In a right-angled triangle with legs of length cos(x) and sin(x), the hypotenuse has a length of 1 [00:06:38].
• By placing the origin of a coordinate system at one end of this hypotenuse, the other end lies on a circle with a radius of 1 [00:07:42].
• The coordinates of any point on this unit circle are (cos(x), sin(x)) [00:08:02].
• The variable x represents the length of the arc along the circle from the positive x-axis to the point (cos(x), sin(x)) [00:08:43].
• For example, traveling a quarter of the circumference (length π/2) leads to the point (0,1), meaning cos(π/2) = 0 and sin(π/2) = 1 [00:09:03]. Traveling half the circumference (length π) leads to the point (-1,0), meaning cos(π) = -1 and sin(π) = 0 [00:09:49].

2. Complex Numbers

The introduction of a coordinate system allows for operations like adding points on a plane [00:10:20].

• Points (a,b) and (c,d) can be added as (a+c, b+d) [00:10:46]. This can be visualized as vector addition [00:11:05].
• To enable multiplication of points on a plane, a new notation a + ib (replacing the comma with a plus and an ‘i’) is introduced [00:11:50].
• The multiplication (a + ib)(c + id) would produce a term i² [00:13:14]. To make multiplication meaningful for points on a plane, the imaginary unit i is defined such that i² = -1 [00:13:22]. This seemingly nonsensical number allows for the definition of multiplication of complex numbers [00:13:43].
• Using this notation, the point (-1,0) (corresponding to cos(π) = -1, sin(π) = 0) can be written as cos(π) + i sin(π) = -1 [00:14:32].

3. Mathematical Analysis and Euler’s Identity

The number e (Euler’s number) is introduced through the exponential function e^x [00:15:40].

• A key property of e^x is that its derivative is equal to itself [00:16:02].
• This property can be demonstrated through its infinite series expansion: e^x = 1 + x + x²/2! + x³/3! + ... [00:16:22]. Taking the derivative of this series reproduces the original series [00:17:31].
• Similarly, sine and cosine functions also have infinite series expansions [00:18:43].
  • sin(x) involves only odd terms with alternating signs.
  • cos(x) involves only even terms with alternating signs.
• Euler discovered a profound connection between e^x, cos(x), and sin(x) by substituting ix into the series for e^x. Because i² = -1, i³ = -i, i⁴ = 1, etc., the powers of i naturally introduce the alternating signs needed for the sine and cosine series [00:19:39].
• This leads to Euler’s Identity: e^(ix) = cos(x) + i sin(x) [00:19:54].
• When x = π, the identity becomes e^(iπ) = cos(π) + i sin(π) [00:20:07]. Since cos(π) = -1 and sin(π) = 0, this simplifies to e^(iπ) = -1 [00:20:11].
• Rearranging, we get Euler’s Formula: e^(iπ) + 1 = 0 [00:20:14].

This demonstration concludes by showing how diverse mathematical concepts converge into a single, elegant formula, much like a concise Haiku opens up a vast depth of understanding [00:20:19].