A sextic surface is one defined by a polynomial equation of degree 6. The Barth sextic, drawn above by Craig Kaplan, is the sextic surface with the maximum possible number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\).
The Zamolodchikov tetrahedron equation, illustrated above by J. Scott Carter and Masahico Saito, is a fundamental law governing surfaces embedded in 4-dimensional space. It also arises purely algebraically in the theory of braided monoidal 2-categories.
This is an image of the Clebsch surface, created by Greg Egan. The Clebsch surface owes its fame to the fact that while all smooth cubic surfaces defined over the complex numbers contain 27 lines, for this particular example all the lines are real, and thus visible to the eye. However, it has other nice properties as well.
This animation by Greg Egan shows 27 lines on a surface defined by cubic equations: the Clebsch surface. It illustrates a remarkable fact: any smooth cubic surface contains 27 lines.
This is the Enneper surface, as drawn by Greg Egan using Mathematica. It’s a minimal surface, meaning one that necessarily gets more area if you warp any small patch of it. A soap film will make a minimal surface if it doesn’t enclose any air. But the Enneper surface intersects itself: it’s immersed in 3d space, but not embedded. So, you can’t make it with soapy water!
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