This blog thread lists lots of strange number properties... https://www.quora.com/What-is-the-most-beautiful-number-and-why
Is there a field of mathematics that deals with these strange number properties? I thought maybe this was "number theory" but a quick look at the definition of that subject makes me think it is not the right name.
A fair amount of mysticism seems mixed into the blog comments, but despite that there are lots of weird ratios and patterns described.