This paper proposes an interpolatory ternary subdivision for triangular meshes that produces C^2 continuous limit surfaces for regular meshes while achieve C^1 continuity with bounded curvature at extraordinary vertices. Prior to this paper, there is no practical interpolatory subdivision scheme which exhibits C^2 continuous for triangular meshes. This subdivision scheme splits each triangle into nine by inserting two E-vertices onto each edge and a F-vertex onto each face. The regular mask is derived from interpolatory ternary subdivision curves and the irregular masks are established based on Fourier transformation. Finally, we demonstrate the visual quality of surfaces refined by our new ternary subdivision scheme with several examples.