How Mathematical Curves Enable Advanced Communication

A simple geometric idea has been used to power advances in information theory, cryptography and even blockchain technology.

Given a collection of points in space, can you find a certain type of curve that passes through all of them? This question — a version of what’s called the interpolation problem — has interested mathematicians since antiquity. 

But while the work has generated a lot of excitement among pure mathematicians, interpolation has practical consequences that extend far beyond the realm of geometry. Interpolation is central to storing and communicating electronic data, constructing cryptographic schemes, and more. It’s why you can scratch a CD and still hear music, or get a QR code dirty and still scan it. It’s why space missions like the Voyager program could send clear digital images back to Earth. It’s why a cluster of computers can perform a complex computation even if one of those computers malfunctions.

These applications all rely on a strikingly beautiful and conceptually straightforward use of interpolation: so-called Reed-Solomon codes, and the codes that build on them.

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