Mathematical interpretations of geometric principles

From: mk_thisisit

Mathematics is widely considered to be full of beauty and elegance, a concept comparable to the concise and profound nature of Japanese Haiku poetry [00:00:21]. Just as Haiku is characterized by its concise form, deep content, and aesthetic beauty [00:01:13], similar qualities are found in mathematical expressions [00:02:14].

Elegance in Scientific Formulas

Examples of such elegant “haikus” in science include:

• Einstein’s E=mc², known for its concise form, deep content, and undeniable elegance [00:02:47].
• The Schrödinger equation, which describes the evolution of a particle in quantum physics [00:02:55].
• Boltzmann’s entropy formula, representing the second law of thermodynamics [00:03:05].
• Einstein’s field equations, which describe the geometry of spacetime [00:03:15].

These formulas, much like Haiku, embody a profound meaning within a simple structure, highlighting the elegance and beauty in mathematics and its role in understanding the universe.

Euler’s Formula: The Quintessence of Mathematical Elegance

Euler’s formula ($e^{i\pi }+1=0$) is often regarded as the most elegant and beautiful formula in mathematics [00:03:42]. Its significance stems from its ability to combine five fundamental mathematical constants: 0, 1, Pi ($\pi$), the imaginary unit ($i$), and Euler’s number ($e$) [00:05:19]. These constants originate from diverse areas of mathematics, including trigonometry ($\pi$), complex numbers ($i$), and mathematical analysis/differential calculus ($e$), yet they converge harmoniously within this single equation [00:05:32].

Interpreting Geometric Principles through Coordinates

The Pythagorean Identity and the Unit Circle

The trigonometric identity ${\sin }^2(x)+{\cos }^2(x)=1$ is a foundational example of a mathematical “haiku” that can be understood geometrically [00:06:01]. Its essence is derived directly from the Pythagorean theorem [00:06:25].

Consider a right-angled triangle where the hypotenuse has a length of 1. If the two legs have lengths of $\cos (x)$ and $\sin (x)$, then by the Pythagorean theorem, $(\cos (x){)}^2+(\sin (x){)}^2=1^2$, simplifying to ${\cos }^2(x)+{\sin }^2(x)=1$ [00:06:46]. This means the distance from one end of the hypotenuse to the other is exactly 1 [00:07:05].

The introduction of a coordinate system, a fundamental discovery by Descartes, allows for a deeper geometric interpretation [00:07:15]. If one end of the hypotenuse is placed at the origin (0,0) of the coordinate system, the other end must lie on a circle with a radius of 1 [00:07:51]. The coordinates of any point on this unit circle are precisely $(\cos (x),\sin (x))$ [00:08:05]. Thus, the identity ${\cos }^2(x)+{\sin }^2(x)=1$ directly describes a circle with a radius of 1 [00:08:23].

In this context, ‘x’ represents the length of the arc along the unit circle, starting from the positive x-axis and extending to the point $(\cos (x),\sin (x))$ [00:08:43]. For example, a quarter of the circle has an arc length of $\pi /2$, leading to coordinates $(0,1)$ which are $(\cos (\pi /2),\sin (\pi /2))$ [00:09:12]. Half the circle has an arc length of $\pi$, corresponding to coordinates $(-1,0)$ or $(\cos (\pi ),\sin (\pi ))$ [00:09:49]. This property is used to define the number $\pi$ formally in mathematics [00:09:37].

Operations on Points in a Coordinate System

The coordinate system enables operations on points in a plane that were not obvious to ancient Greek mathematicians [00:10:37]:

• Addition of Points: Points (a,b) and (c,d) can be added by summing their corresponding coordinates: $(a+c,b+d)$ [00:10:50]. Geometrically, this corresponds to vector addition, where the sum of two vectors is the diagonal of the parallelogram formed by them [00:11:03].

• Multiplication of Points (Complex Numbers): To multiply points, a new notation is introduced where a point $(a,b)$ is written as $a+ib$, replacing the comma with a plus sign and ‘b’ with $ib$ (where ‘i’ is a new symbol) [00:11:52]. When multiplying $(a+ib)(c+id)$, the product involves an $i^2$ term [00:13:14]. To define multiplication meaningfully, the value $i^2=-1$ is assigned [00:13:22]. This is the definition of the imaginary unit ‘i’, which allows for the multiplication of complex numbers, effectively enabling the multiplication of points on a plane [00:14:03].

Exponential Function and its Relation to Trigonometry

The exponential function, $e^x$, is fundamentally defined by the property that its derivative is equal to itself ($d/dx(e^x)=e^x$) [00:15:56]. This function can be expressed as an infinite series: $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$ [00:16:22]. Taking the derivative of this series term by term results in the original series, confirming its defining property [00:17:42].

Similarly, sine and cosine functions also have derivative properties: $d/dx(\sin (x))=\cos (x)$ and $d/dx(\cos (x))=-\sin (x)$ [00:18:43]. These functions can also be expressed as infinite series:

• $\sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$ (contains only odd terms) [00:18:59]
• $\cos (x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$ (contains only even terms, with alternating signs) [00:19:01]

Euler’s profound discovery was the connection between these seemingly disparate series: $e^{ix}=\cos (x)+i\sin (x)$ [00:19:54]. By substituting $ix$ into the series for $e^x$ and using the property $i^2=-1$, $i^3=-i$, $i^4=1$, etc., the terms naturally separate into the series for cosine (even powers of $ix$) and sine (odd powers of $ix$), with the correct alternating signs provided by powers of $i$.

When $x=\pi$, Euler’s formula becomes: $e^{i\pi }=\cos (\pi )+i\sin (\pi )$ [00:20:05] Since $\cos (\pi )=-1$ and $\sin (\pi )=0$, this simplifies to: $e^{i\pi }=-1+i(0)$ $e^{i\pi }=-1$ Which can be rearranged to $e^{i\pi }+1=0$ [00:20:11].

This derivation demonstrates the deep interplay between mathematics and physics, showcasing how abstract mathematical concepts like complex numbers and infinite series can elegantly describe geometric principles (like the unit circle) and unify fundamental constants.