Following a suggestion of , the antennae of the Submillimeter Array, a radio-wave astronomical observatory on Mauna Kea in Hawaii, are arranged on four nested Reuleaux triangles. Placing the antennae on a curve of constant width causes the observatory to have the same spatial resolution in all directions, and provides a circular observation beam. As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the Fourier transform of the signal from the array. The antennae may be moved from one Reuleaux triangle to another for different observations, according to the desired angular resolution of each observation. The precise placement of the antennae on these Reuleaux triangles was optimized using a neural network. In some places the constructed observatory departs from the preferred Reuleaux triangle shape because that shape was not possible within the given site.