pi-th root of unity and Tan 1

While thinking about the "Pith root of unity", I found (cos 1)+(sin 1)i which is actually the (2*pi)-th root of unity. The one-radian angle marks out an equilateral wedge: both straight segments and the curved part are all the same length. Whereas sin(1) and and cos(1) are not especially interesting continued fractions, their ratio, the tangent of 1 radian, the slope of the ray to the 2*pi-th root of unity, does have a regular continued fraction:

In[3]:= ContinuedFraction[Tan[1],100]

Out[3]= {1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 1, 27, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43, 1, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 63, 1, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 75, 1, 77, 1, 79, 1, 81, 1, 83, 1, 85, 1, 87, 1, 89, 1, 91, 1, 93, 1, 95, 1, 97, 1, 99}