Sampling — from smooth to points
To store or draw a continuous signal, a computer can only keep a finite number of samples — its value read off at evenly spaced spots. Join those samples back up and you get a stand-in for the original. Too few and it's a crude sketch; add more and the reconstruction closes in on the smooth truth.
This is every pixel grid, every mesh, every data series. Slide the sample count.
Aliasing — when too few samples lie
Sampling has a sharp limit. Push the signal's frequency past half the sample rate — the Nyquist limit — and the samples no longer describe it. They quietly trace a different, slower wave that was never there. That phantom is aliasing: the moiré in a photo of a screen, the wheels that spin backwards on film, the jaggies on a thin diagonal.
The sample rate is fixed. Raise the frequency and watch the lie appear.
Quantization — discrete values
Sampling chops up position. Quantization chops up value. Allow only a fixed set of levels and every reading snaps to the nearest one, so a smooth ramp becomes a staircase. Few levels give visible banding; many levels and the steps vanish into apparent smoothness.
This is colour depth, bit rate, contour intervals, 3D-print layer heights. Change how many levels are allowed.
Time steps — discretizing motion
Continuous motion has to be marched forward in finite time steps. The simplest rule — Euler's — just follows the current direction for one step and repeats. On a circular orbit that direction always points slightly outward, so big steps spiral away from the truth. Shrink the step, or use a smarter rule, and the path closes back onto the circle.
This is the heart of every physics simulation. Set the step size and the method.
The discretization studio
Here's the whole trade-off in one place. A smooth, continuous field sits underneath; you decide how finely to break it apart. Resolution sets how many cells carve up space; levels set how many values each cell may take; reconstruction decides whether the cells stay blocky or are blended back toward smooth. Coarsen it and you get a buildable, storable, low-cost approximation; refine it and it converges on the continuous original. Find the setting that's faithful enough — that judgement is computational design in miniature.