  A method is demonstrated to rapidly calculate the shapes and properties of
quasi-axisymmetric and quasi-helically symmetric stellarators. In this
approach, optimization is applied to the equations of magnetohydrodynamic
equilibrium and quasisymmetry, expanded in the small distance from the magnetic
axis, as formulated by Garren and Boozer [Phys. Fluids B, 3, 2805 (1991)]. Due
to the reduction of the equations by the expansion, the computational cost is
significantly reduced, to times on the order of 1 cpu-second, enabling wide and
high-resolution scans over parameter space. In contrast to traditional
stellarator optimization, here the cost function serves to maximize the volume
in which the expansion is accurate. A key term in the cost function is $||
\nabla\mathbf B ||$, to maximize scale lengths in the field. Using this method,
a database of $5\times 10^5$ optimized configurations is calculated and
presented. Quasisymmetric configurations are observed to exist in continuous
bands, varying in the ratio of the magnetic axis length to average major
radius. Several qualitatively new types of configuration are found, including
quasi-helically symmetric fields in which the number of field periods is two or
more than six.
