A pedagogical project, called the ARPAM project, has been founded in an attempt to foster a rapprochement between the general public and the mathematical sciences. It is based on the presentation of mathematical concepts and facts through some buildings called follies. This paper describes a folly devoted to the illustration in the physical world of two important concepts, those of singularity and bifurcation, linked through the fundamental concept of stability. The use of properties of some fluid flows on certain surfaces is the main tool which is used to exhibit these concepts and mathematical facts. The artistic attraction of this folly comes from the multiple effects of light through the various uses of the fluid elements.