On the Catalan number wikipedia page, scroll down to "A convex polygon with n + 2 sides..." to see the polygon dissection: https://en.wikipedia.org/wiki/Catalan_number

From my brief reading of it, it seems like the interesting bit here is the development of the Hypercatalan numbers as the coefficients of the infinite sums of roots of polynomials. Some partial results for special cases of the Catalan numbers and roots had been found in the past, but the full understanding of the structure they call the Geode enabled generalization of the previous findings.

> A UNSW mathematician has discovered a new method to tackle algebra’s oldest challenge – solving higher polynomial equations.

See also author's blog https://njwildberger.com/ and youtube channel Insights into Mathematics https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ

Also the same author has a blog, with a post about this paper: https://njwildberger.com/ - blog also notes he's about to retire, too! Wow

I had trouble finding a pdf link to the paper to download a regular pdf instead of using their in-browser viewer. It is here:

https://www.tandfonline.com/doi/pdf/10.1080/00029890.2025.24...

additionally he has another channel "Wild Egg Maths" where he delves into more advanced topics.