To see a wave you must sample it at least twice per cycle. Sample any slower and a fast wave masquerades as a slow one — the wagon wheel that turns backward on film. Below the wall, the machine reconstructs a world that was never there. Slide the sample rate (or tap) through the limit.
The Nyquist–Shannon sampling limit. A pure tone of frequency f₀ can be reconstructed only if the sample rate fs exceeds 2·f₀ (the Nyquist rate). Below it the samples are indistinguishable from a lower-frequency alias at |f₀ − fs·round(f₀/fs)| — a real fast signal reconstructed as a slow lie, the same effect that spins wagon wheels backward. The true sine, the sample points, and the reconstructed (possibly aliased) sine are drawn live. A fail-loud self-check throws unless a rate above 2·f₀ recovers f₀ exactly and a rate below it yields a different apparent frequency.
One tone and ideal point-sampling (real sensors integrate over a pixel and pre-filter); f₀ is fixed. The Nyquist rate and the alias frequency are computed exactly.