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I’d put a big magnet on a two-axis carriage built on extruded aluminum bars. Simple! There might be a better solution, but I’m the wrong kind of engineer to work it out.

One axis radial, the other linear?

Like an indexable stepper-motor, with another stepper attached that uses a belt-drive, like an inkjet printer?

Looks like it does use one radial axis and one linear [0]

[0] https://sisyphus-industries.com/about/

Link: https://sisyphus-industries.com/customize/

Beyond how the machine works, I'm wondering how he designed the patterns.

I bet a healthy dose of trial and error, plus software to draw it out without dedicating hours.