The Dodecacup

Indeed, the shape of the cup used in a liquid cymatics experiment determines the resulting standing wave. If we use a triangular cup the standing wave possibilities are limited by the specific triangular boundary; thus, various geometries are more likely to appear, and other geometries are filtered due to the lack of resonance. For example, it is very unlikely to achieve a square-like standing wave in a triangular cup, since the boundary conditions of the fluid's elastic membrane at the surface limit the way the mechanical oscillations interact with the system.

This is why we mostly use circular cups in our experiments. The circle is the ultimate polygon, offering all possible geometries of standing waves for the specific fluid/cup combo.

Here is an example of a dodecahedral (12-sided) cup engineered by our own Erik Larson! Notice how the overal geometry of the standing wave is three-fold ,but because the cup has a defined 12-sided geometry, the edges of the triangular geometry are "shaved", resulting in this unusual 12-sided standing wave with a predominantly three-fold geometry.
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The cup's geometry matters!