Times Tables, Mandelbrot and the Heart of Mathematics
Mathologer4 HN recommenders · recommended in 4 months, 2017–2023
explainer · general · evergreen · 2017–2023 on HN
Mathologer shows how times table diagrams on a circle produce cardioids and other curves connected to the Mandelbrot set. He explains the patterns and links to a tool for making your own diagrams.
HN recommends it as the clearest explanation of the patterns behind modular multiplication diagrams and times table art.
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4 comments from Hacker News
Grasping the concept of number regardless of base system is an interesting exercise (linked to modular arithmetic), see Mathologer on times tables in any number base: ▶ this video
Check out some of the png files to view the art! This YouTube video explains the algo behind what you're seeing. ▶ this video
How does that work? ... Explanations invited. Hold the 1 fixed. Notice in both sequences the remaining numbers are 3 interleaved ordered sets, in multiples of 2. Then watch this (just happened to post it two hours ago too)... "Times Tables, Mandelbrot and the Heart of Mathematics" [video] ▶ this video link NB: And once again we find cycloids hiding underneath it all.
There’s not really a reasonable continuous analog, IMO (unless you just want to see a single solid blob, like in your picture, but that hides all of the interesting part of the structure). But there are moiré-like patterns. Check out ▶ this video for some insight based on a related type of diagram (related to a single row at a time from the OP’s diagrams).