We study the nucleation of carbon nanotubes based on the hypothesis that a cylindrical seed can be formed by the spontaneous rolling of a small graphite fragment under thermal fluctuations. We calculate the vibrational spectra of fragments of different sizes by means of a tight-binding model, and compare the spectra to that of an infinite graphite sheet, as a function of temperature. The spectra of finite-size fragments contain an increasing density of low-frequency, large amplitude modes with increasing temperature; it is shown that such modes tend to fold over the fragment into a cylindrical shape. Finally, direct tight-binding molecular-dynamics simulations of the high-temperature fluctuation of parallel graphite fragments demonstrate that nanotube seeds closed at one end can spontaneously form.