The author presents most known numeral systems (ways of representing natural numbers) in lambda calculus, classified by whether the term use their bound variables exactly one time (linear), at most one time (affine), or multiple times (non-linear). He illustrates some numerals in each system with a graphical notation that strongly reminds me of interaction nets [1], a computational model closely related to lambda calculus. The notation they use for lambda terms is rather non-standard and somewhat confusing.

[1] https://en.wikipedia.org/wiki/Interaction_nets

This is beautiful art

I think I lack context to see what this is about. The line graphs are pretty though, and I'd like to understand more.

I didn’t understand that notation. Can someone please explain?

This should be "numerals"