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.
This object looks like a piece of art. But in fact, it is "just" a mathematical curve, called 7/3 torus knot. Imagine a donut, and then tie a cord around it in some interesting way.
Cubic surfaces are classics from the 19th century. This particular cubic is smooth and has tetrahedral symmetry. All 27 lines are real, but only 24 are visible in the model because 3 are infinitly far away.
This so-called Boy surface represents a fascinating example of a non-orientable surface. The first such surface was constructed by Werner Boy in his dissertation in 1902.
In the 19th century, Olaus Henrici constructed a model of a quite symmetric cubic surface. This is a modern variant of it, allowing also to look "inside" of it from the bottom.
This version of Clebsch's famous diagonal surface model features colored lines and two additional planes. One intersects the surfaces in a line and a hyperbola, the another one in three lines.
Our modern version of Klein's historical cubic surface model with four singularities is the main figure in our photo from the series "math sculptures in context". It is the pure white version with its 9 straight lines.