The successive actions of the generators of the modular group SL(2, Z) on its fundamental domain produce a tessellation of the upper half-plane H. Each tile is a curved triangle bounded by circular arcs. In this paper, we study the Euclidean curvatures of these arcs and uncover a remarkable arithmetic structure: every curvature is an integer, and each such integer is either odd or divisible by 8. Moreover, we prove the converse—every odd integer and every multiple of 8 occurs as a curvature in the tessellation. This paper is also available as viXra preprint 2604.0023.

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