My dog Phi Phi

I wanted a nice region bounded by two curves (who wouldn't?), so I graphed a parabola and a reciprocal function passing through (0,1):

y = -x² + 1
y = 1/(x + 1)

The second intersection looked vaguely familiar... 0.61803398875... where had I seen that before?

Oh yeah, the golden ratio!  Phi = 1.61803398875...

An interesting fact is that Phi - 1 = 1/Phi ... are there any other numbers that have this quality?