Final answer:
The Euler characteristic χ(T) of a torus T can be calculated using a simple triangulation. For example, using a 3x3 grid converted into a triangulation on the torus, the Euler characteristic is determined to be 0. This is consistent with the known properties of a torus in topology.
Step-by-step explanation:
The question requests a calculation of the Euler characteristic χ(T) for a torus T using a given triangulation. The Euler Characteristic is a topological invariant that for a triangulation of a surface is defined as χ = V - E + F, where V is the number of vertices, E is the number of edges, and F is the number of faces.
To triangulate the torus, we can imagine a square grid on the surface of the torus where each square is divided into two triangles. Wrapping this grid around the torus so that the opposite edges of the grid are identified gives us the desired triangulation. As a simple example, consider a grid of 9 squares (3x3 grid), which results in 18 triangles, 9 vertices (after identification), and 27 edges (after identification).
The Euler characteristic χ(T) for the torus using the simple 3x3 grid triangulation would then be:
χ(T) = V - E + F = 9 - 27 + 18 = 0
This calculation shows that the Euler characteristic for a torus is always 0, reflecting its topological nature.