Steiner's Surface

Recall the discussion of double points on quartics from Quartics with Tetrahedral Symmetry. Kummer showed [1] that such surfaces paramaterised by $\lambda, \mu$ have 16 double points (counting those in complex projective space) when $\lambda = (3\mu - 1)/(3 - \mu)$. Note that $\lambda = \mu = 1$ satisfies this equation; this particular case is known as the "Steiner surface", or "Roman surface", given this name because Steiner discovered it while he was visiting Rome in 1844. This model is of particular interest as it provides a concrete depiction of projective space. Click the image here for an animation of Steiner's surface.

Steiner's Surface

(Copyright © 2007 "Keyi")

The surface again has tetrahedral symmetry, which is unusually symmetric for such a manifestation of the real projective plane. The equation defining Steiner's surface simplifies to $$x^2 y^2 + y^2 z^2 + z^2 x^2 - xyz = 0$$Note that swapping the variables around (e.g. changing the places of $x$ and $y$) still yields the same surface. This gives some insight into why the surface exhibits so much symmetry.

Steiner's surface represents an "immersion" of the real projective plane into familiar 3-dimensional space ($\mathbb{P}^2$ into $\mathbb{R}^3$). Note that model IX 7 also represents $\mathbb{P}^2$ (in a different way). An immersion is a mostly well-behaved map, but self-intersects at some points. A familiar example of an immersion is the Klein bottle in $\mathbb{R}^3$, which must cut itself in order to be displayed in 3-dimensional Euclidean space. The surface self-intersects in three lines consisting of singular points (the three lines with "sharp" edges on the model). At these lines, different "sheets" of the surface intersect. In the centre, there is a "triple point", and at each of the six endpoints of the singular lines we have a "pinch point". A pinch point, also known as a "Whitney singularity" or "branch point", is a singularity around which any neighbourhood must be self-intersecting.

At each point on Steiner's surface, there are infinitely many conics lying on the surface which pass through that point [1]. Steiner's surface can in fact be realised by "splicing together" three hyperbolic paraboloids, then "smoothing edges out". This feature is described in good detail in this interesting Wikipedia article.

The surface is also "non-orientable", in the sense that it only has one side (like a Möbius band or Klein bottle). Orientation is an important concept in differential geometry, crucial for studying integration on manifolds. This means that if you are travelling around on the surface, it is possible (by passing through the singular lines) to travel around such that you reappear at your starting point, but on the opposite side of the surface. The non-orientability of Steiner's surface is not so easily seen from the model. 

Sadly, our model is badly damaged, but see here for an intact model in Göttingen