Googling your ratio 5.828427.. finds a seemingly related discussion[1] which identifies it as the square of the "silver ratio" i.e (1 + sqrt(2))^2[2,3]
The explanation at [1] looks ok to this non-mathematician, but best view it there I think!
1: https://math.stackexchange.com/questions/1651227/pythagorean...
2: (see calculator section at) https://www.google.com/search?q=%281+%2B+sqrt%282%29%29%5E+2