TITLE: Stirring a square cup results in a strange pattern. What's the math behind it?
QUESTION [10 upvotes]: I came across this picture earlier today: https://imgur.com/gallery/qo8F4e3 and it left me kind of baffled.  Can anyone explain the mathematics behind the reason why this is happening?

REPLY [3 votes]: Had the cup been circular we would see vortex shaped upper surface deeper at the center than at edges with polar circular symmetry as there is no radial component. When boundary is a regular polygon of $n$ sides we can see standing wave interference patterns caused by vibration frequency due to  radial and circumferential component streamline flow interfering to form $n$ spiral arms. 
An attempt at modeling should include the two interactions.
EDIT1:
If such a cup is placed and rotated sufficiently fast on a turntable I suspect it may give rise to some similar pattern, that is if viewing at all is possible.