Elegance and beauty in mathematics

From: mk_thisisit

Mathematics is often described as being full of beauty and elegance, a concept that can be difficult for non-mathematicians to grasp [00:00:21]. Professor Dawid Kielak from Oxford University uses an analogy with Japanese Haiku poetry to explain this mathematical elegance [00:00:04] [00:00:26] [00:00:31].

Haiku as an Analogy for Elegance

Haiku is a form of Japanese poetry distinguished by three key properties [00:01:08] [00:01:11]:

1. Conciseness: Haiku is very concise, typically consisting of only three lines [00:01:13] [00:01:17].
2. Deep Content: While brief, Haiku aims to inspire a deeper understanding of reality, often related to Japanese culture and Zen Buddhism, prompting reflection and meditation [00:01:30] [00:01:37] [00:01:43] [00:01:49].
3. Aesthetic Beauty: The connection between the concise form and the deep content must be beautiful and provide aesthetic pleasure [00:02:00] [00:02:05] [00:02:07]. This unique blend of brevity, profundity, and beauty is precisely the “elegance” found in mathematics [00:02:14] [00:02:15].

Haiku in Science

Many fundamental scientific formulas can be considered “Haiku” due to their concise form, deep content, and inherent elegance [00:02:42]. Examples include:

• Einstein’s equation E=mc² [00:02:47].
• Schrödinger’s equation, which describes particle evolution in quantum physics [00:03:00].
• Boltzmann’s entropy principle, stating that entropy does not decrease (the second law of thermodynamics) [00:03:05].
• Another of Einstein’s equations, which describes the geometry of spacetime [00:03:15].

Euler’s Formula: A Mathematical Haiku

Leonhard Euler, an outstanding and highly prolific Swiss mathematician from the 18th century, formulated what is perhaps the most elegant and beautiful formula in mathematics [00:03:42] [00:03:46] [00:03:48]:

$e^{i\pi }+1=0$ [00:04:04]

This formula exemplifies the three features of a Haiku:

• Concise Form: It is undoubtedly concise [00:04:19].
• Deep Content: It possesses profound content [00:04:25].
• Elegance: The relationship between its concise form and deep content is highly elegant [00:04:33].

Euler’s formula is considered profoundly beautiful because it combines five fundamental constants of mathematics: 0, 1, $\pi$ (Pi), $i$ (the imaginary unit), and $e$ (Euler’s number) [00:05:15] [00:05:19]. These three numbers ($e$, $\pi$, and $i$) originate from distinct areas of mathematics—trigonometry, complex numbers, and mathematical analysis/differential calculus—yet they converge beautifully in this single formula [00:05:32].

Deconstructing Euler’s Formula

Understanding the elegance of Euler’s formula requires exploring its constituent parts, which can be thought of as “translating” from the “language of science” [00:04:55].

1. The Unit Circle and Trigonometry

A simpler “Haiku” in trigonometry is the identity: ${\sin }^{2}x+{\cos }^{2}x=1$ [00:06:01]. This formula, derived from the Pythagorean theorem, describes the relationship between the sides of a right-angled triangle [00:06:25] [00:06:33]. If the legs of a right triangle are $\cos x$ and $\sin x$, then the hypotenuse squared is ${\cos }^{2}x+{\sin }^{2}x$, which equals 1, meaning the hypotenuse has a length of 1 [00:06:46] [00:06:53].

The profound implication of this formula becomes clear with the introduction of the coordinate system by Descartes [00:07:10] [00:07:15]. By placing one end of the hypotenuse at the origin (0,0), the other end must lie on a circle with a radius of 1 [00:07:43] [00:07:56].The coordinates of any point on this unit circle are precisely $(\cos x,\sin x)$ [00:08:05] [00:08:09]. This gives a full description, or parameterization, of the unit circle [00:08:40]. The variable $x$ in $\cos x$ and $\sin x$ represents the length of the arc along the unit circle from the positive x-axis to the point $(\cos x,\sin x)$ [00:08:46] [00:08:51].

The number $\pi$ is defined through this property: A quarter of the unit circle’s circumference has a length of $\pi /2$ [00:09:12] [00:09:16]. Thus, the point at the top of the unit circle, with coordinates $(0,1)$, also has coordinates $(\cos (\pi /2),\sin (\pi /2))$ [00:09:21] [00:09:29]. Extending this, a half-circle traversal leads to the point $(-1,0)$, corresponding to $(\cos \pi ,\sin \pi )$ [00:09:49] [00:10:08].

2. Complex Numbers

The coordinate system opened up new possibilities, such as adding points on a plane [00:10:20] [00:10:37]. While the Greeks believed this was impossible, Descartes’ system allows for it by simply adding corresponding coordinates: $(a,b)+(c,d)=(a+c,b+d)$ [00:10:46] [00:10:50].

To facilitate multiplication of points, a new notation is introduced where $(a,b)$ is written as $a+ib$ [00:11:52]. When multiplying $(a+ib)(c+id)$, a term $i^2$ appears [00:13:14]. To make multiplication meaningful and ensure the result is still a point on the plane (i.e., in the form of $X+iY$), mathematicians defined $i^2=-1$ [00:13:18] [00:13:22] [00:13:30]. This introduced the “imaginary unit” $i$, leading to the concept of complex numbers [00:13:43] [00:13:53].

With this new notation, the point on the unit circle corresponding to $\pi$ can be written as $\cos \pi +i\sin \pi =-1+i\cdot 0=-1$ [00:14:31] [00:14:51].

3. The Exponential Function ($e^x$)

Understanding the meaning of raising a number to an arbitrary power (like $\pi$ or a square root) requires defining the exponential function, $e^x$ [00:15:21] [00:15:39]. A key property of $e^x$ is that its derivative is equal to itself ($d/dx(e^x)=e^x$) [00:15:52] [00:16:02].

This function can be represented as an infinite sum (a Taylor series) [00:16:15]: $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots ={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$ [00:16:22] [00:17:46]. Taking the derivative of this series indeed reproduces the original series, confirming its unique property [00:17:42].

Similarly, sine and cosine functions also have their own infinite series representations, which can be derived from their derivative properties ($d/dx(\sin x)=\cos x$ and $d/dx(\cos x)=-\sin x$) [00:18:43] [00:18:52]: $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$ [00:19:01] $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$ [00:19:01]

4. Connecting the Pieces: Euler’s Identity

Euler’s genius was in recognizing the similarity between the series for $e^x$ and those for $\sin x$ and $\cos x$ [00:19:21]. The alternating signs in the sine and cosine series, and the presence of only odd or even terms, are crucial [00:19:27]. By substituting $ix$ into the series for $e^x$, and leveraging the property $i^2=-1$: $e^{ix}=1+ix+\frac{(ix{)}^2}{2!}+\frac{(ix{)}^3}{3!}+\frac{(ix{)}^4}{4!}+\dots$ $e^{ix}=1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+\dots$ [00:19:41] Separating the real and imaginary terms: $e^{ix}=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots \right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots \right)$ [00:19:54]

This reveals Euler’s formula: $e^{ix}=\cos x+i\sin x$ [00:19:54].

To arrive at the final elegant identity, substitute $x=\pi$: $e^{i\pi }=\cos \pi +i\sin \pi$ [00:20:05] Since $\cos \pi =-1$ and $\sin \pi =0$: $e^{i\pi }=-1+i\cdot 0$ $e^{i\pi }=-1$ [00:20:08]

Rearranging gives the celebrated Euler’s Identity: $e^{i\pi }+1=0$ [00:04:04] [00:20:14]. This formula beautifully unifies five fundamental mathematical constants from different domains, demonstrating the profound interconnectedness and inherent elegance within mathematics.