This polygonal pattern is known as the rose-window instability. It’s formed between two electrodes – one a needle-like point, the other flat – separated by a layer of oil. The pointed electrode’s voltage ionizes the air nearby, creating a stream of ions that travel toward the flat electrode below. Oil is a poor conductor, however, so the ions build up on its surface until they’re concentrated enough to form a dimple that lets them reach the lower electrode. At higher voltages, the electrical forces driving the ions and the gravitational force trying to flatten the oil reach a balance in the form of the polygonal cell pattern seen above. Smaller cells form near the needle electrode, where the electrical field is strongest and the temperature is highest, as revealed in thermal and schlieren imaging (lower images) that shows a warm stream of gas impacting there.
As a final note, I’ll add that the latest in this research comes from a paper by a Pakastani teenager. It’s never too early to start contributing to research! (Image and research credit: M. Niazi; via NYTimes; submitted by Kam-Yung Soh)