Geodesic
On the Continued Fractions of Conjugate Quadratic Irrationalities
Canadian mathematical bulletin, Tome 23 (1980) no. 2, pp. 199-206

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Let 1 be the simple continued fraction (SCF) of an irrational number x. The partial quotients ai which we shall sometimes refer to as the "terms" of the SCF are integers and, for i ≥ 2, they are positive. If x is a quadratic irrationality then, by Lagrange's Theorem, the right side of (1) becomes periodic from some point on. 
Herzog, Fritz. On the Continued Fractions of Conjugate Quadratic Irrationalities. Canadian mathematical bulletin, Tome 23 (1980) no. 2, pp. 199-206. doi: 10.4153/CMB-1980-027-x
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     title = {On the {Continued} {Fractions} of {Conjugate} {Quadratic} {Irrationalities}},
     journal = {Canadian mathematical bulletin},
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     year = {1980},
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     doi = {10.4153/CMB-1980-027-x},
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