![[Clebsch Cubic surface]](../images/cubic3.jpg "A Euclidean form of the Clebsch cubic surface")
I spent a good part of the
afternoon creating this image (click to enlarge) of one of
the remarkable inhabitants of the platonic heaven: the Clebsch cubic
surface; specifically, this is the Euclidean form of the latter which
has the greatest possible group of symmetries (24 of the 120
symmetries of the Clebsch cubic are realized as Euclidean
isometries).
Every smooth cubic surface has twenty-seven lines on it (sometimes
poetically known as Solomon's seal
; I do not know who coined
the term or whether it is related to the plant Polygonatum
biflorum, which also goes by that name). But in general they
exist only as complex lines and might not all be realized: the number
of real lines can be three, seven, fifteen or twenty-seven, and
on the Clebsch cubic all twenty-seven lines exist in a real sense.
You can only see twenty-four lines in the picture (can you?),
however, because the last three lines are away at infinity.
Furthermore, it is possible for three lines on a cubic surface to all meet in a single point, in which case the point in question is known as an Eckardt point (this is a remarkable feature, and while all cubic surfaces have lines on them, not all have Eckardt points, even in the complex sens): the Clebsch cubic surface is unique in that it has ten Eckardt points, and all are real (on my particular Euclidean realization, four are the vertices of a regular tetrahedron, two of which can be clearly seen, and six are at infinity).
Another way to represent a smooth cubic surface (at least one which
has all twenty-seven lines real) is as a set of six points in the
(projective) plane (in general position, that is, such that no three
are aligned and all six do not line on a common conic). It is not
easy to describe precisely the relation between the six marked (or
blown up
) points and the cubic surface[#], but it is quite easy to explain
how the twenty-seven exceptional lines are seen: consider the six
marked points, the fifteen lines connecting any two of them, and the
six conics going through five of the marked points — now
6+15+6=27, and they correspond exactly to the exceptional lines on the
cubic surface; and even intersection is preserved if we agree that
intersection at the marked points in the plane is only taken into
account when it is tangential[#2]. Eckardt points are also easily
seen that way: when three lines defined by three pairs of marked
points meet in a common point, that point is an Eckardt point; also
when the conic through five of the six marked points has a tangent at
one of said marked points which goes through the sixth, then that
tangent direction is an Eckardt “point”. Under this
correspondance, the Clebsch cubic is the most remarkable configuration
of six points in the plane, namely, a regular pentagon and its center.
The ten Eckardt points are then obvious.
So one of the answers I might give when asked what my thesis is
about is: six points in the plane
.
[#] Each point on the
cubic surface corresponds to a point on the plane and, if it is one of
the six marked points, a line direction throught that point. (This is
what is meant by blowing up
: replacing a marked point by the
set of all directions through that point.)
[#2] For example, given three of the six marked points, the three lines connecting them are thought not to intersect; however, each of them intersects the two marked points which it joins. This is in accordance with the idea that the marked points have been replaced by the set of directions through them.