English: Depiction of the matrix of the DFT for N=8. Each element is represented by a picture of its location in the complex plane in relation to the unit circle. The values are also coded in the shading of the unit disk. Generated by the following Python code.

from numpy import linspace,ones_like,real,imag,cos,sin,pi,exp,arange,mod
import matplotlib.pyplot as plt  
from matplotlib import cm

def fft_matrix_viz(n):
    L = 1.
    vsep = -1.
    lwfac = 4
    markersize = 75./float(n)/2 *lwfac
    i = complex(0.,1.)
    t = linspace(0,L,501,endpoint=True)
    kvals = range(0,n)#-n//2,n)
    colormap = cm.get_cmap('hsv') # a cyclic colormap
    nc = len(kvals)
    cmrotation = 0.95 #0.8  #0.05
    colors = colormap( mod( linspace(0,1,nc,endpoint=False) + cmrotation, 1 ) )  # get nc colors from the colormap
    # pastelize colors
    p = 0.55
    colors = (1-p)*colors + p*ones_like(colors)
    # darken colors
    d = 1 #0.9
    colors *= [d,d,d,1]
    clockcolor = '#888888'
    fontsize = 20*4/n*1.5*2
    figsize = 20
    plt.figure(figsize=(figsize,figsize))

    theta = linspace(0,2*pi,100)
    c = cos(theta)
    s = sin(theta)
    def plotZ(x,y,Z):
        radius = 0.35 
        tickfraci = 0.15
        tickfraco = 0.05
        color = colors[(j*k)%len(kvals)]
        plt.fill(x+radius*c, y+radius*s,color=color,alpha=.75)  # fill unit circle
        plt.plot(x+radius*c, y+radius*s,color=clockcolor,lw=lwfac,alpha=1)#0.35)  # draw unit circle
        for ticktheta in linspace(0,2*pi,n,endpoint=False):  # draw ticks
            tc,ts = cos(ticktheta),sin(ticktheta)
            plt.plot([x+(1-tickfraci)*radius*tc,x+(1-tickfraco)*radius*tc],[y+(1-tickfraci)*radius*ts,y+(1-tickfraco)*radius*ts],color=clockcolor,lw=lwfac,alpha=1)#0.35) 
        plt.plot([x,x+radius*real(Z)],[y,y+radius*imag(Z)],color=clockcolor,lw=lwfac,alpha=1)#0.35) # draw radius
        plt.plot(x+radius*real(Z),y+radius*imag(Z),'o',markersize=markersize,color='#505050',alpha=1) # mark Z

    plt.subplot(1,1,1,aspect=1,frameon=False)
    for k in kvals:
        zd = exp(-2*pi*i*k*arange(n)/float(n))
        for j,Z in enumerate(zd): plotZ(j,vsep*k,Z)
        plt.text(-0.5,vsep*k,str(k),va='center',ha='right',fontsize=fontsize)

    for j in range(n):
        plt.text(j,vsep*(min(kvals)-0.50),str(j),ha='center',fontsize=fontsize)
    plt.xlim(-.5,n+.5)
    plt.ylim(vsep*(max(kvals)+1),vsep*(min(kvals)-1))
    plt.xticks([])
    plt.yticks([])

    plt.savefig(f'dft_visualization_rev2_n{str(n).zfill(4)}.svg',bbox_inches='tight')

fft_matrix_viz(8)