# Understanding symmetry in DFT magnitude plot

I am trying to intuitively understand the mystery of why the DFT of a complex vector with real values produces apparent symmetry in the magnitude plot (see the second plot here for example). In DFT $$X[k]$$ of a $$N$$-dimensional complex vector with real entries $$x[t]$$, the relation $$|X[k]|=|X[N-k]|$$ holds for each frequency $$k=0,1,2,\dots N-1$$. It should be the case that the frequencies $$k$$ and $$N-k$$ should trace the same sinusoid, right?

My reasoning is that $$e^{i\frac{2\pi kt}{N}}$$ ($$t=0,1,2,3,\dots$$) traces a unit circle in counterclockwise direction on the complex plane while $$e^{i\frac{2\pi (N-k)t}{N}}$$ ($$t=0,1,2,3,\dots$$) traces a unit circle in clockwise direction on the complex plane. It is evident if you look at real parts of $$e^{i\frac{2\pi kt}{N}}$$ and $$e^{i\frac{2\pi (N-k)t}{N}}$$ for each $$t$$:

(Assume $$N=6$$ and $$k=1$$)

$$\cos\left(\frac{2\pi k}{N}\cdot0\right)=\cos\left(\frac{2\pi (N-k)}{N}\cdot0\right)=1$$

$$\cos\left(\frac{2\pi k}{N}\cdot1\right)=\cos\left(\frac{2\pi (N-k)}{N}\cdot1\right)=\frac12$$

$$\cos\left(\frac{2\pi k}{N}\cdot2\right)=\cos\left(\frac{2\pi (N-k)}{N}\cdot2\right)=-\frac12$$

etc...

How do I call this weird phenomenon? Aliasing?

Also, I am wondering how do I graph this result? This is my attempt: https://www.desmos.com/calculator/w6gl0cqwwa and this is not what I had in mind. Frequencies $$k$$ and $$N-k$$ should trace the same sinusoid

Intuition is in the eyes of the beholder but this property pops out directly from the definition of the DFT

$$X[k] = \sum_{n=0}^{N-1}x[n] \cdot e^{-j2\pi\frac{kn}{N}}$$

First of all, it follows directly from the definitions that both $$x[n]$$ and $$X[k]$$ are periodic with $$N$$, i.e. $$X[k] = X[k + mN], m \in \mathbb{Z}$$

Hence we have

$$X[N-k] = X[-k] = \sum_{n=0}^{N-1}x[n] \cdot e^{j2\pi\frac{kn}{N}}$$

If the input is real, i.e. $$x[n] \in \mathbb{R}$$ then we see directly that

$$X[-k] = X^{*}[k]$$

The DFT at a negative frequency is the complex conjugate of the value at the positive frequency (and vice ver