What are Golomb Rulers?

Imagine a normal ruler, and make a series of notches on it at a few of the numbered points. Now, calculate the pairwise distance (meaning, if you have 0 1 4 9 11, you need the difference between 0 and 1, 0 and 4, 0 and 9, 0 and 11; the difference between 1 and 4, 1 and 9...) for all points.

If every difference you got is unique- that's a Golomb Ruler!

A normal ruler next to a Golomb ruler. Each row of the table compares its number against the other numbers. Notice how there's no duplicates among the numbers in red

Variants

These Golomb Patterns (or, non-patterns) don't just exist as flat lines. They exist in rectangular grids, honeycomb grids, circular coordinates... There are even specific forms, such as the Costas array (proposed by mathematician John Costas in the 1960s), a rectangular grid with exactly one mark in each row and column.

Applications

For such a specific mathematical idea, these patterns have applications in a wide range of fields. Such as:

That last item is one we want to highlight. Getting better pictures of the cosmos is a noble, but difficult, endeavor. We can build bigger telescopes- but just to get a resolution similar to the Hubble from the ground, you'd need a kilometer wide receiver. Or, we can establish multiple small receivers, and create a distributed telescope. In 2019, a team of scientists used this idea at a whole new scale to capture the first ever image of a black hole.

For a distributed system to work, each aperture would have to be positioned to capture a non-redundant portion of all the data.

These non-redundant configurations map near-perfectly to the mathematical concept of a Golomb Pattern. And that is where you- our citizen scientists- come in. Computing these patterns by brute force is exponentially difficult, but people have a lot of success finding them by trial-and-error. You can play the game and contribute to the known repository of Golomb Patterns. Your contribution could help scientists understand our universe in a new way.

But why Citizen Science?

Why couldn't we just pay to have a super-computer crunch the numbers on the best Golomb patterns? Because the problem of constructing Golomb-related patterns is speculated to be NP-hard (see some of the papers below for further reading on that), meaning it is difficult and expensive for a computer to generate new patterns. But, it's easy to check if a pattern is a valid Golomb patterns, and people (citizen scientists like you) are great at coming up with them!

Further reading