This constant, denoted with the Greek letter phi (φ), is called the golden ratio and it's one of the most well-known constants in mathematics. It's the average of 1 and the square root of 5.
The ratio between two consecutive terms of the Fibonacci sequence converges to φ. In fact, even if we started with any positive real number a_0 and a_1, and make a Fibonacci sequence with it (a_n = a_(n-1) + a_(n-2)), the ratio a_n/a_(n-1) converges to φ.
Rectangles whose ratio of the long and short sides equal to the golden ratio are called golden rectangles. They are known to be aesthetically pleasing, and are used in the design of many pieces of architecture.
The distance between this number and its square is 1. The distance between this number and its reciprocal is also 1. Both of these can be proven using the fact that the golden ratio (φ) satisfies the equation φ2-φ-1=0.