A signal has two faces of one object: its shape in time, and its content in frequency. Neither is more real; each is the other, fully — and you can pass between them and back with nothing lost. Draw in the time panel; the frequency panel is its exact dual, recomputed live. This is duality you can measure.
◧ time — x[n] (draw here)
frequency — |X[k]| ◨
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round-trip error idft(dft(x)) vs x
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Parseval energy conserved
N=64
samples (both faces)
drag in the time panel to reshape x[n] · the spectrum and both invariants update live
two faces, nothing lost
The discrete Fourier transform is an exact, invertible map: idft(dft(x)) = x to machine precision, so no information is created or destroyed crossing the seam — time and frequency are one object seen two ways. And Parseval: the energy Σ|x|² equals (1/N)Σ|X|², conserved exactly. That is what makes this a true DYAD and not a mere comparison — the two sides are provably the same thing.
green · measured live The DFT/IDFT run in your browser on the signal you draw. Round-trip error ~1e−15 (machine epsilon) and Parseval conserved to ~1e−9 — displayed live, node-verified before shipping. The invertibility is not asserted; it is computed in front of you.
amber · the framing Calling time and frequency "the two faces of one object" is the honest reading of an exact theorem, but "faces" is still a chosen word. The magnitude spectrum shown discards phase (kept internally for the round-trip) — the picture is |X[k]|, not all of X[k].
red · scope This is a finite DFT (N=64), not the continuous Fourier transform; edge effects and sampling are real. It is a clean instance of duality, not a claim that every duality is invertible — most are not. Here, this one provably is.
DYAS · the Fourier Dyad · idft(dft(x)) = x, exact · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0