# 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?

-
You want to know if research mathematicians ever use complex numbers to study geometry and vice versa? –  Jack M Feb 24 at 22:06
I suppose, yes. Also if there's some underlying connection between the two past the idea that complex numbers are points on the complex plane. –  Michael T Feb 24 at 22:08
Perhaps what you really wanted to ask was why complex numbers have connections to geometry? –  Jack M Feb 24 at 22:30
I was thinking that maybe there might be more connection between the two than just the complex plane / complex number relationship, but I suppose not. –  Michael T Feb 24 at 22:35

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.

-
Are there any "natural" analogues to complex multiplication in $\Bbb{R}^3$ or higher dimensions? –  supercat Feb 24 at 22:51
The quaternions fill a similar role for 3D geometry. In higher dimensions there are Clifford algebras, however, they have dimension $2^n$ for $\mathbb R^n$. –  LutzL Feb 25 at 0:49

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 understand this much, but in the question I was asking for "real math" applications –  Michael T Feb 24 at 21:44
What do you mean by "real math"? –  Emin Feb 24 at 21:46
By that I mean things that a graduate student or research professor would interest themselves in. I understand the connection between complex numbers and the complex plane, but I am interested to see if there are more applications beyond that. –  Michael T Feb 24 at 22:01