duality of the Fourier transform
The forward and inverse Fourier integrals look almost the same — they differ only by the sign of the exponent and a constant. This near-symmetry has a powerful payoff: every transform pair you know can be read backwards to give you a second pair for free. That mirror trick is duality, and it roughly doubles the size of any transform table you have memorized.
Concretely, suppose f(x) has Fourier transform F(k). Duality says that if you now feed the function F (treated as a function of the variable x) into the transform, you get back the original f, reflected and scaled: the transform of F(x) is 2 pi f(minus k) (the exact constant and reflection depend on the convention). The reason is plain from the formulas: swapping the roles of x and k turns the forward integral into the inverse one. So the rectangular-pulse-to-sinc pair immediately hands you the sinc-to-rectangle pair; the Gaussian, being its own transform, is the fixed point where duality acts trivially.
Duality is more than a labor-saving device; it is the structural reason the time and frequency domains stand on equal footing, with no preferred direction. Every theorem comes in a dual pair: a shift in time mirrors a modulation in frequency, differentiation in one domain mirrors multiplication by the variable in the other, and convolution in one domain mirrors ordinary multiplication in the other. Learning the transform well means learning to flip any statement through this symmetry and recognize its twin.
Since the rectangle transforms to a sinc, duality says a sinc transforms (up to a constant and a reflection) back to a rectangle. So a brick-wall ideal low-pass filter, flat in frequency, has the unrealizable sinc as its impulse response in time.
Read one pair backwards and you get its dual: rectangle-and-sinc serves both the windowing and the ideal-filter stories at once.
Duality only swaps the variables cleanly when the constants and exponent signs are tracked carefully — the reflection f(minus k) and the factor of 2 pi are easy to drop, and the symmetric 1/sqrt(2 pi) convention is the one where duality looks tidiest.