robinhoustonI don't remember coming across the Heptagon Numbers before. Very nice!
I should also link Scott Vorthmann's talk, from which the original screenshot was taken: https://youtu.be/Zz4jR0HGIc0
Scott Vorthmann - Algebra and Serendipity: The Beautiful Mathematics of vZome - CoM February 2025
And Mathologer has a video on the topic. I haven't watched it, but I trust him. https://youtu.be/cCXRUHUgvLI
Way beyond the golden ratio: The power of AB=A+B (Mathologer masterclass)
Michael Kinyon@robinhouston Steinbach wrote a follow up a few years later. Toward the end, he mentions a problem with rational approximants he wasn't able to resolve.
https://archive.bridgesmathart.org/2000/bridges2000-35.html#gsc.tab=0
@ProfKinyon IIUC he says he does not know whether this pattern continues. But that's not exactly about rational approximants: are you alluding to something else?
@robinhouston I was referring to the (somewhat vague) discussion starting on p. 42 and ending at the top of the next page. I actually missed the sentences on p. 38 where he talks about the uniqueness of optimally proportional sections. I admit I haven't read the paper carefully.
Return of Aadmaa@robinhouston neat! I'm going to challenge one of my kids to try proving these, if she dare, at least the first one. assuming I can prove them to myself this evening.
Øystein@aadmaa2 @robinhouston I was able to prove all of them right now, but some with more difficulty than others. The first and last equation on the top row fell out immediately from the figure I drew, while the remaining three equations required a bit of algebraic manipulation (and a bit of trial and error to find those algebraic manipulations).