We show that identifying the intersection of a turbulent flame and a tomographic plane to a random self-avoiding planar walk yields d(F) = d-1 + 1/3 for the fractal dimension (dF) Of the front surface. This suggests that d(F) = 7/3 in 3-dimensional ambient space (d = 3) could have a geometrical origin, not directly related to the k(-5/3) Kolmogorov scaling law.