Complex Numbers, $\small \pi$, Fourier Transform, Eigenvectors
This video,
[Source: YouTube; discussion]
describes the mathematics behind one of Richard Feynman’s lectures on the elliptical orbits of planets.
 Feynman discusses the algebraic completeness of complex numbers here.
 See also Reflecting on complex numbers (again)
Interestingly, I later encountered the same (conceptual) description at about the 3minute mark in this lecture, on the Fourier transform (“An animated introduction to the Fourier Transform, winding graphs around circles.”),
[Source: YouTube]
This page, Understanding the Fourier Transform [mirror] describes the Fourier transform very succinctly.
[Image source. Click image to open in new window.]That figure is explained in the accompanying blog posts.
Basically, whenever you see $\small i$ and $\small \pi$, think “circles and rotation.”
Euler’s formula ties in with $\small i$ and rotation and eigenvectors, as discussed in the really excellent blog post Intuitive Understanding of Euler's Formula:

Euler's identity $\small e^{i \pi} = 1$ emerges from a more general formula, $\small e^{ix} = \cos(x) + i \sin(x)$ relating an imaginary exponent to sine and cosine (and, where plugging in $\small \pi$ gives 1).
[Aside: that formula formed the basis of the embeddings in RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space (Feb 2019).]
Euler’s formula also comes up in my background discussion on graph signal processing. These complex numbers manifest later in that discussion, when I use GNU Octave to determine some eigenvalues in my various examples: I get “complex” eigenvectors of the form $\small 0.14037 + 0.55581i$ (which is reminiscent of Euler’s formula, $e^{i\phi} = cos\phi + (i)sin \phi$):
octave:>> c = [5,8,16; 4,1,8; 4,4,11] c = 5 8 16 4 1 8 4 4 11 octave:>> [eigenvectors, eigenvalues] = eig(c) eigenvectors = 0.81650 + 0.00000i 0.14037  0.55581i 0.14037 + 0.55581i 0.40825 + 0.00000i 0.71521 + 0.00000i 0.71521  0.00000i 0.40825 + 0.00000i 0.28742 + 0.27790i 0.28742  0.27790i eigenvalues = Diagonal Matrix 1.00000 + 0.00000i 0 0 0 3.00000 + 0.00000i 0 0 0 3.00000  0.00000i octave:>> eig(c) ans = 1.00000 + 0.00000i 3.00000 + 0.00000i 3.00000  0.00000i octave:>>
Stated in [Imaginary Numbers] Complex Eigenvalues,

"Complex numbers are added like vectors:

$\small x + iy + u + iv = (x + u) + i(y + v)$

$\small z * w = (x + iy)(u + iv) = xu  yv + i(yu  xv)$.

$\small 1/z = 1/(x + iy) = (x  iy)/(x^2 + y^2)$."
… hinting (as we already know) at the relationship between vectors (e.g. vector space, models) and imaginary numbers.
That same source includes these statements,

"Geometric interpretation. If $\small z = x + iy$ is written as a vector $\small \begin{align} \begin{bmatrix}x \\ y\end{bmatrix} \end{align}$, then multiplication with an $\small y$ other complex number $\small w$ is a dilationrotation: a scaling by $\small w$ and a rotation by $\small arg(w)$."
"Unit circle. Complex numbers of length $\small 1$ have the form $\small z = e^{i\phi}$ and are located on the unit circle. The characteristic polynomial $\small f_A(\lambda) = \lambda^5  1$ of the matrix

$\small \begin{align} \begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\\
0 & 0 & 1 & 0 & 0 \\\
0 & 0 & 0 & 1 & 0 \\\
0 & 0 & 0 & 0 & 1 \\\
1 & 0 & 0 & 0 & 0
\end{bmatrix} \end{align}$
These concepts are elaborated in Geometry of the Eigenvectors of Plane Rotation, which states the following. [A good background before reading this material is Intuitive Understanding of Euler's Formula,]

"Rotation of a twodimensional vector $\small (x,y)$ in the Cartesian plane can be expressed as a matrix multiplication, where the rotated vector $\small (x',y')$ is equal to the product of the rotation matrix $\small M$ and the original (column) vector $\small (x,y)$. For rotation by a counterclockwise angle $\small A$, the antisymmetric $\small 2x2$ matrix $\small M$ has the cosine of $\small A$ in the diagonal elements, the sine of $\small A$ at lower left, and the negative of the sine of $\small A$ at upper right. ...
"For an arbitrary angle $\small A$, however, $\small M$ has the two complexconjugate eigenvalues $\small c = e^{iA}$ and $\small c^* = e^{iA}$. These complex eigenvalues have meaning only if the setting is generalized to allow $\small x$ and $\small y$ to be complex, so that the products of $\small c$ and $\small x$ and of $\small c$ and $\small y$ are again elements of the vector space. The product of $\small c$ and a complex number $\small z$ is itself equivalent to a rotation of $\small z$ in the complex plane by the angle $\small A$. To see this, represent $\small z$ in polar coordinates as magnitude times exponential of the product of $\small i$ and the angle; then multiplying by $\small c$ just adds $\small A$ to the angle. Thus, multiplication of the complex eigenvector $\small (x,y)$ by the eigenvalue $\small c$ or $\small c^*$ is equivalent to rotation of the complex $\small x$ and $\small y$ by the angle $\small A$ (for $\small c$) or the angle $\small A$ (for $\small c^*$) in the respective complex $\small x$ and $\small y$ planes.
"The matrix $\small M$, however, is real. Thus, for complex $\small x$ and $\small y$, the product of $\small M$ and $\small x$ or $\small y$ can be viewed as the product of $\small M$ and the real vector $\small Re(x,y)$ plus $\small i$ times the product of $\small M$ and the real vector $\small Im(x,y)$, where $\small Re$ and $\small Im$ denote the real and imaginary parts, respectively. Each of these real products is a rotation by $\small A$ in a real twodimensional plane: the first is the rotation by $\small A$ in the plane of the real parts of $\small x$ and $\small y$, and the second is the rotation by $\small A$ in the plane of the imaginary parts of $\small x$ and $\small y$. Each of these is equivalent in form to the original rotation of the original real vector $\small (x,y)$.
"The eigenvalue condition in the complex setting can therefore be viewed as a commutation relation: rotation by $\small A$ of the vector of real parts of $\small x$ and $\small y$ in the plane $\small Im(x,y)=(0,0)$, followed by rotation by $\small A$ of the vector of imaginary parts of $\small x$ and $\small y$ in the plane $\small Re(x,y)=0$, gives the same result as rotation of the complex $\small x$ by $\small A$ (or $\small A$, for $\small c^*$) in the plane $\small y=0$, followed by rotation of the complex $\small y$ by $\small A$ (or $\small A$, for $\small c^*$) in the plane $\small x=0$. ...
"The vector space in the complex setting is effectively four dimensional, with independent real and imaginary parts for each of $\small x$ and $\small y$. This makes the geometry of the rotations by $\small M$ and $\small c$ or $\small c^*$ difficult to depict or visualize. ..."
[Image source. Click image to open in new window.]
Sources and Additional Reading
 Rotation matrix
 Euler’s rotation theorem
 How to get the eigenvectors from eigenvalues in a rotation matrix?
 Vectors in Three Dimensions and Transformations
 [local copy] Eigendecomposition [Ch. 2 (Linear Algebra) excerpt, from Goodfellow, Bengio & Courville, Deep Learning (2016]
Imaginary Numbers
 Complex Eigenvalues
 Geometry of the Eigenvectors of Plane Rotation
 Intuitive Understanding of Euler's Formula