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