The instability of spherical flames in quiescent premixtures is considered using the linear theory developed by Bechtold and Matalon, which includes hydrodynamic and thermodiffusive instabilities. Ranges of unstable wavenumbers are presented for different Markstein numbers as the flame propagates, characterized by the flame Peclet number. Hydrodynamic instabilities arise at the larger wavelengths and cascade down through a range of ever–decreasing wavelengths to be stabilized at the smallest wavelength by thermodiffusive effects. It would appear that, in practice, the associated cell formation lags somewhat behind what is predicted by the theory and an attempt is made to allow for this. With it, a fractal analysis is employed with the two limiting unstable wavelengths as inner and outer cut–offs. The ever–increasing surface wrinkling as the flame propagates creates a larger surface area and consequent flame acceleration. The analysis yields an expression for the flame speed after a critical Peclet number has been attained. The flame speed increases as the square root of elapsed time and the analytical expression is in agreement with the results of measurements of large explosions. The fractal analysis is probably valid because of the large length–scales and small flame stretch rates, unlike those in many turbulent flames in engineering applications, where the flame stretch rate usually reduces the burning rate. The mechanisms for the creation of turbulence are discussed, including a brief speculation of the repercussions for deflagration to detonation transitions in large–scale explosions, including supernovae.