What are the “real math” connections between Euclidean Geometry and Complex Numbers?

Some background: I am a high school student and I am very interested in math. I have done a lot of the extracurricular learning I have done is through doing math problems from various competitions, whether that be the Crux Mathematicorum or the AIME. In doing these problems, I have found various ways that complex numbers can be used to help solve a geometry problem, and vice versa. For example, expressing coordinates as complex numbers and rotating them by multiplying by $e^{k \pi}$ for some rational number $k$ is something I have seen commonly, and I have seen a derivation of Heron’s formula using complex numbers.

However, something I would like to know about is the connection between the two in the “real” modern math world. Are the two fields deeply and fundamentally intertwined, or do they just provide each other with tools that can be used to aid each other?

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From the modern perspective, Euclidean geometry is the study of the topological space

$$\Bbb{R}^n = \{ x_1, \ldots, x_n \; \mid \; x_i \in \Bbb{R} \}$$

together with the bilinear form

$$\langle \cdot, \cdot \rangle: \Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}_{\ge 0}$$

defined by

$$\langle \vec{x}, \vec{y} \rangle = \big\langle (x_1, \ldots, x_n), (y_1, \ldots, y_n) \big\rangle = x_1y_1 + \cdots + x_ny_n$$

This “dot” product defines distances and angles via

$$\| \vec{x} \| = \sqrt{\langle \vec{x}, \vec{x} \rangle} = \sqrt{x_1^2 + \cdots + x_n^2}$$

and

$$\cos \theta = \frac{\langle \vec{x}, \vec{y} \rangle}{\sqrt{\langle \vec{x}, \vec{x} \rangle \langle \vec{y}, \vec{y} \rangle}} = \frac{\langle \vec{x}, \vec{y} \rangle}{\| \vec{x} \| \| \vec{y} \|}$$

From the perspective of transformations (a decidedly modern take), certain functions

$$\phi: \Bbb{R}^n \to \Bbb{R}^n$$

are more interesting: conformal maps preserve angles (these are also called similarity transformations):

$$\langle \phi(\vec{x}), \phi(\vec{y}) \rangle = \langle \vec{x}, \vec{y} \rangle \qquad \text{for all } \vec{x}, \vec{y} \in \Bbb{R}^n$$

and certain conformal maps, called isometries, preserve distances as well (these are also called congruence transformations):

$$\| \phi(\vec{x}) \| = \| \vec{x} \| \qquad \text{for all } \vec{x} \in \Bbb{R}^n$$


In the plane $(n = 2)$ there is a nice characterization of all isometries as translations, rotations, reflections, and glide reflections. (Throw in dilations to get conformal maps.)

By equipping $\Bbb{R}^2$ with the imaginary unit $i$ $(i^2 = -1)$, it becomes the complex plane, and all of these isometries can be written as rational functions $\phi: \Bbb{C} \to \Bbb{C}$. (Note that the complex structure is really essential to define rotations.) EDIT: You need complex conjugation, too, in order to write down any orientation-reversing map, such as a reflection.

In that sense, the complex numbers are the perfect algebraic object to capture the Euclidean structure of $\Bbb{R}^2$ and express these maps in elegant formulas.

Complex numbers can be written in the plane just as $R^2$ where x axis represents the real part and the y axis represents the imaginary part of a complex number. Than by knowing some rules in complex numbers, it is easy to solve geometric problems, since they are represented in $R^2$ plane.

I recommend looking at the following three relatively cheap books, which I’ve listed in order from the most elementary to the least elementary:

Complex Numbers and Geometry by Liang-shin Hahn

Introduction to the Geometry of Complex Numbers by Roland Deaux

Geometry of Complex Numbers by Hans Schwerdtfeger