Knowledge of the fractal properties of premixed flame surfaces can potentially be used to help develop turbulent combustion models. Here, direct numerical simulations of low Damkohler number flames are used to analyse the fractal nature of the flames. Two sets of data are considered: (i) thermochemical hydrogen-air turbulent premixed plane-jet flames with detailed chemistry and (ii) thermonuclear flames in type la supernovae. A three-dimensional box counting method is used to investigate fractal dimension of the flame surface, characterising the self similarity of flame fronts. In the premixed flames, the fractal dimension is found to vary in time between 2.1 and 2.7. The supernovae flames in distributed combustion regimes yield fractal dimension about 2.7. The results for the maximum fractal dimensions are higher than previously reported. They are explained theoretically by a Reynolds number similarity argument which posits that the high Reynolds number, low Damkohler number limiting value of the fractal dimension is 8/3. Also tested is Mandelbrot's fractal additive law which relates the fractal dimension determined in two dimensions, which is typical of experimental measurements, to that in three dimensions. The comparison of the fractal dimension in both two-dimensional and three-dimensional spaces supports the additive law, even though the flames considered do not formally satisfy isotropy. Finally, the inner-cut off is extracted from the hydrogen flames and found to be consistent in order of magnitude with Kolmogorov scaling. (C) 2013 The Combustion Institute. All rights reserved.