The day's running long, so I'm logging a few thoughts and notes on Gudermannian relativity here. When I started trying to derive formulas in hyperbolic relativity, as I mentioned, I was inspired by Brian Greene's explanation that all objects move at the constant speed of light and can transfer their light speed from the time dimension, (where it points when an object is at rest), to the space dimension if they'd like, (leading to time dilation, slower time), in the rest frame. I built diagrams like the following, (picture 1), which showed what I imagined the relationship to be in terms of circular trigonometry.
However, this never got me to the proper relationships which I knew involved the hyperbolic tangent. What I was missing was the inverse Gudermannian which gives a relationship between circular and hyperbolic angles, (picture2). The super-cool bit is that sigma shown in picture 2 corresponds to the actual speed that the travelling object would measure for itself, (referred to by Brehme as speedometer velocity and by Walter as proper speed ).
This brings us to a really nice mathematical time sink posted by +John Baez yesterday. Check out the post that discusses the inverse Gudermannian and zigzag numbers, and don't forget the comments wherein you'll be treated to references to the Catalan numbers and the Borel transform!
1. Previous post on Brian Greene's constant speed interpretation of special relativity
2. John Baez's super-interesting post!
3. Walter on proper speed[not open access]