Really wonderful work, and wonderful article. As someone interested in more intuitive digital tools for exploring, understanding and articulating pure mathematics, this methodical breakdown of representations of algebraic numbers is an absolute delight.

Folks interested in another angle on the field of algebraic numbers, which the author begins to enumerate in this article, might enjoy these two visualizations of the field, along with the C that was written to generate them.

https://commons.m.wikimedia.org/wiki/File:Algebraicszoom.png

https://en.m.wikipedia.org/wiki/File:Leadingcoeff.png

It would be quite a fun project to make either of these visualizations “explorable” a la Bret Victor as an application of Frederik’s work. Since the set of algebraic numbers is countable, it seems to me to a scrubbing UI across the points in the color visualizations, joined to the enumeration that Calcium seems to be able to output, and rendered dynamically as the cursor moved around the image, would be a neat demo.

Also Conway and Guy’s “The Book of Numbers” has an extraordinarily playful, intuitive and focused introduction to algebraic numbers that doesn’t require even an undergraduate math background to grok, for folks coming to this post wondering what algebraic numbers are.

https://www.amazon.com/Book-Numbers-John-H-Conway/dp/0387979...

> the example of the quartic number 3.1416...

The second 1 in there is a typo. A Freudian slip for π perhaps?

Is someone procrastinating on writing their thesis?