Complex numbers · Fourier analysis · Euler's formula
Roots of unity
Every DFT template value is one of the N points Wᴺʲ on the unit circle; frequency k steps through them k at a time, points half a turn apart are negatives, and squaring maps the 8-point lattice two-to-one onto the 4-point lattice.
Slide the frequency k to trace the orbit the template at k takes through the 8th roots of unity, and toggle squaring to watch each point map to its square, landing the 8 points on the 4 ringed points of the half-size lattice.