The quintic with 15 cusps is a so-called world record surface: To our knowledge, it is not known if there may be a quintic with more than 15 cusps although it is known that a quintic cannot have more than 20 cusps.
The Barth Sextic is probably the most famous example of the sometimes so-called world record surfaces. This post shows a smoothed variant of it.
A Togliatti quintic surface is a so-called world record surface. Among all quintic surfaces, it has the maximum possible number of singularities, namely 31. Our model is a smoothed version of such a surface.
The shape of our "four pillows meet pendant" is given by a single mathematical equation. Have you ever seen a pendant like this before?
Our "six pillows' secret pendant" is a very special piece of math jewelry. Its shape is given by a single equation. Have you ever seen a pendant like this before?
The photo shows a smoothed Kummer surface in steel (inflated with bronze). This post also features links to plastic versions of this shape. The Kummer surface is a classic from the 19th century; our model is a smoothed version of it.