The dimension (D) of aircraft trajectories is fundamental in interpreting airborne data. To estimate D, we studied data from 18 trajectories of stratospheric aircraft flights 1600 km long taken during a "Mach cruise" (near constant Mach number) autopilot flight mode of the ER-2 research aircraft. Mach cruise implies correlated temperature and wind fluctuations so that approximate toDeltax(z)(H) where Z is the (fluctuating) vertical and x the horizontal coordinate of the aircraft. Over the range approximate to3-300 km, we found H(z)approximate to0.58+/-0.02 close to the theoretical 5/9=0.56 and implying D=1+H(z)=14/9, i.e., the trajectories are fractal. For distances <3 km aircraft inertia smooths the trajectories, for distances >300 km, D=1 again because of a rise of 1 m/km due to fuel consumption. In the fractal regime, the horizontal velocity and temperature exponents are close to the nonclassical value 1/2 (rather than 1/3). We discuss implications for aircraft measurements as well as for the structure of the atmosphere.