To capture a wave with dots, you must sample at least TWICE its highest frequency. Sample slower and something eerie happens: a fast wave masquerades as a slow one — ALIASING, the same trick that makes wagon wheels spin backwards in movies. It’s why CDs sample at 44.1 kHz (just over twice the 20 kHz of hearing). Slide the sample rate down and watch the true wave collapse into a fake one.
The Nyquist–Shannon sampling theorem: a signal band-limited to a maximum frequency fₛ is fully determined by samples taken at any rate fₛ > 2fₛ₃ₓ; that threshold 2fₛ₃ₓ is the Nyquist rate. Sample too slowly and frequencies above fₛ/2 fold back (ALIAS) to a lower apparent frequency |f − round(f/fₛ)·fₛ|, indistinguishable from a genuine low tone — irreversibly corrupting the signal (the wagon-wheel effect). Real systems place an anti-alias low-pass filter before the sampler. It sets every digital audio/video rate and the resolution of any measurement. A fail-loud self-check throws unless the alias formula matches for an over-Nyquist tone. ◆ real signal math, node-verified.
The theorem assumes a strictly band-limited signal and ideal reconstruction; real anti-alias filters aren’t perfect, so a small guard band is used. The 2× bound and the alias-fold formula are exact.