From: mk_thisisit
Professor Dawid Kielak from Oxford University presented a lecture on “Haiku in mathematics,” exploring the elegance of mathematics [00:00:04]. He aims to explain what mathematical elegance means to mathematicians by using an analogy with Japanese poetry, specifically Haiku [00:00:26].
Haiku in Mathematics
Haiku, a form of Japanese poetry, is defined by three properties [00:01:08]:
- Conciseness: It is very concise, always consisting of only three lines [00:01:13].
- Deep Content: It is meant to inspire a deeper understanding of reality, often related to Japanese culture and Zen Buddhism [00:01:30].
- Beauty/Aesthetics: The connection between the concise form and the deep content must be beautiful and provide aesthetic pleasure [00:02:00]. This exact kind of elegance is also found in mathematics [00:02:11].
Examples of “Haiku in science” include [00:02:42]:
- Einstein’s E=mc² [00:02:47], which is concise, has deep content, and possesses a certain elegance [00:02:50].
- Schrödinger’s equation, describing particle evolution in quantum physics [00:02:55].
- The second law of thermodynamics (Boltzmann entropy does not decrease) [00:03:05].
- Einstein’s field equations, which describe the geometry of space-time [00:03:15].
Euler’s Formula: The Most Elegant Formula
Euler’s formula, e^(iπ) + 1 = 0, is considered perhaps the most elegant and beautiful formula in mathematics [00:03:42]. Leonhard Euler was an outstanding Swiss mathematician, recognized as the most productive mathematician in history by the sheer volume of pages written [00:03:46].
The formula e^(iπ) + 1 = 0 is believed to meet the three criteria of a “Haiku” [00:04:16]:
- Concise Form: The form is undoubtedly concise [00:04:22].
- Deep Content: It contains profound meaning [00:04:25].
- Elegance: The relationship between its concise form and deep content is highly elegant [00:04:33].
Core Components
Euler’s formula is considered extremely beautiful because it combines five basic constants of mathematics [00:05:15]:
- 0 and 1 [00:05:21].
- π (Pi): A constant from trigonometry [00:05:24].
- i (Imaginary Unit): A constant from complex numbers [00:05:24].
- e (Euler’s number): A constant from mathematical analysis and differential calculus [00:05:28].
These three numbers (π, i, e) come from different areas of mathematics but meet in this single formula [00:05:32].
Understanding the Elements
1. The Unit Circle and Trigonometry (sin²(x) + cos²(x) = 1)
The identity sin²(x) + cos²(x) = 1, known from school mathematics [00:06:01], can be understood through the Pythagorean theorem [00:06:33]. In a right-angled triangle, if the legs have lengths cos(x) and sin(x), the hypotenuse squared is cos²(x) + sin²(x) [00:06:57]. If this sum equals 1, the hypotenuse has a length of 1 [00:07:03].
A fundamental discovery in science, introduced by Descartes, was the introduction of the coordinate system [00:07:13]. When a coordinate system is placed at one end of the hypotenuse, the other end of a hypotenuse of length 1 lies on a circle with a radius of 1 [00:07:51]. The coordinates of any point on this unit circle are precisely (cos(x), sin(x)) [00:08:02]. Thus, sin²(x) + cos²(x) = 1 means that a circle with a radius of 1 is described in the coordinate system by points (cos(x), sin(x)) [00:08:23].
The value ‘x’ in cos(x) and sin(x) represents the length of the arc along the circle, starting from the horizontal axis, reaching the end of the hypotenuse [00:08:46]. For a circle with radius 1, its circumference is 2π [00:09:03].
- Moving 1/4 of the circumference (arc length π/2) leads to the point (0,1) [00:09:12], so cos(π/2) = 0 and sin(π/2) = 1 [00:09:27].
- Moving 1/2 of the circumference (arc length π) leads to the point (-1,0) [00:09:49], so cos(π) = -1 and sin(π) = 0 [00:10:08].
2. Addition and Multiplication of Points in the Plane
While Greeks would say adding points on a plane doesn’t make sense [00:10:37], the coordinate system allows for it [00:10:46]. Two points (a,b) and (c,d) can be added by summing their respective coordinates: (a+c, b+d) [00:10:50]. Geometrically, this corresponds to vector addition [00:11:05].
To multiply points, a new notation is introduced where (a,b) becomes a + ib [00:11:52]. For this multiplication to consistently result in another point on the plane, the imaginary unit ‘i’ is introduced, defined by the property that i² = -1 [00:13:30]. This defines how to multiply points on the plane, which is the foundation of complex numbers [00:13:43].
Using this new notation, the point at (-1,0) can be written as cos(π) + i sin(π) = -1 + i*0 = -1 [00:14:39].
3. The Exponential Function (eˣ)
The number ‘e’ is defined by a specific infinite sum (series): e^x = 1 + x + x²/2! + x³/3! + … [00:15:02]. The key property of the exponential function e^x is that its derivative is equal to itself (d/dx (e^x) = e^x) [00:15:52]. This property uniquely determines the function [00:15:52].
This property can be observed by taking the derivative of its infinite sum [00:16:15]: d/dx (1 + x + x²/2! + x³/3! + …) = 0 + 1 + 2x/2! + 3x²/3! + … = 1 + x + x²/2! + … [00:16:40]. Thus, the derivative of the sum is the sum itself, confirming it as e^x [00:17:57].
Similarly, sine and cosine functions also have specific infinite sum representations (Taylor series):
- d/dx (sin(x)) = cos(x) [00:18:43]
- d/dx (cos(x)) = -sin(x) [00:18:47]
By examining their series expansions, it is observed that sine contains only odd terms (x - x³/3! + x⁵/5! - …) and cosine contains only even terms (1 - x²/2! + x⁴/4! - …) [00:19:01]. The alternating minus signs in cosine and sine series are crucial [00:19:07].
4. The Derivation of Euler’s Formula
Euler discovered that by substituting ‘ix’ for ‘x’ in the series definition of e^x, the alternating signs of cosine and sine naturally emerge due to the property i² = -1 [00:19:41].
This leads to Euler’s Identity: e^(ix) = cos(x) + i sin(x) [00:19:54].
Now, substitute x = π into Euler’s Identity: e^(iπ) = cos(π) + i sin(π) [00:20:05]. We know that cos(π) = -1 and sin(π) = 0 [00:10:08]. So, e^(iπ) = -1 + i(0) [00:20:11]. e^(iπ) = -1 [00:20:11].
Rearranging the equation yields the elegant Euler’s Formula: e^(iπ) + 1 = 0 [00:20:14].
This demonstrates the deep connection between fundamental mathematical constants and different branches of mathematics within a single, concise expression.