# Cubic surfaces

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An algebraic surface is one of the form *f*(*x*,*y*,*z*) = 0 where *f*(*x*,*y*,*z*) is a polynomial in *x*, *y* and *z*. The order of the surface is the degree of the polynomial. A surface of order one is a plane. A surface of order two is called a quadric surface and consists of surfaces such as ellipsiods and hyperboloids. These include cones, cylinders and paraboloids. The surface whose history we are interested in for this short article is a surface of order three which is called a cubic surface.

In 1849 Salmon and Cayley published the results of their correspondence on the number of straight lines on a cubic surface. It was Cayley who, in a letter to Salmon, first showed that there could be only a finite number of straight lines on a cubic surface while it was Salmon who then proved that there were exactly 27 such straight lines in general. At the end of his 1865 treatise *The Geometry of Three Dimensions* Salmon described how the two had collaborated over finding the Cayley-Salmon theorem.

Steiner already knew of Cayley-Salmon theorem about 27 straight lines when he started his own work on cubic surfaces. He wrote an important article which gave results that allowed a purely geometrical treatment of cubic surfaces. He proved in 1856 that:-

The nine straight lines in which the surfaces of two arbitrarily given trihedra intersect each other determine, together with one given point, a cubic surface.

He introduced the notion of a "nuclear surface" and investigated its properties. Many results on cubic surfaces were stated by Steiner without proof and we shall comment later how Cremona and Rudolf Sturm proved many of these ten years after Steiner's paper.

Clebsch described the plane representations of various rational surfaces, he was especially interested in that of the general cubic surface. Using the Hessian surface, he gave the first proof that any given cubic surface could be written in the pentahedral form which had been proposed by Sylvester. Other results on cubic surfaces were proved by Clebsch which included: there exists a covariant of order nine which intersects the cubic surface in exactly 27 lines; and every smooth cubic surface can be represented in the plane using four plane cubic surfaces through six points and vice-versa.

It was Steiner who communicated to Schläfli the Cayley-Salmon theorem on 27 lines on a cubic surface. In 1858 Schläfli became the first to classify the cubic surfaces with respect to the number of real straight lines and tritangent planes on them, finding that there were exactly five types in his classification. Schläfli then found 36 "double sixes" on this surface. He divided cubic surfaces into 23 species according to the nature of their singularities in 1863 and he published the classification in his paper *On the distribution of surfaces of the third order into species, in reference to the presence or absence of singular points and the reality of their lines*. In his lengthy *Memoir on Cubic Surfaces* Cayley presented Schläfli's complete classification of cubic surfaces into 23 distinct species and he also added further investigations of his own.

In March 1866 Cremona published *Memoire de géometrie pure sur les surfaces du troisième ordre*. In this memoir he established many of the properties that had only been stated by Steiner. He also established connections between the Cayley-Salmon theorem on 27 lines on a cubic surface and Pascal's Mystic Hexagram:-

If a hexagon is inscribed in any conic section, then the points where opposite sides meet are collinear.

For his memoir Cremona was awarded a share of the Steiner Prize. He shared the Prize with Rudolf Sturm who studied third degree surfaces in their projective representations and also proved theorems stated, but not proved, by Steiner.

In 1869, at Clebsch's suggestion, Christian Wiener constructed plaster of Paris models of cubic surfaces which, together with other models of surfaces he had constructed, were exhibited in London in 1876, Munich in 1893, and Chicago also in 1893. Klein investigated cubic surfaces in 1870 and his work shows a special concern for geometric intuition regarding spatial constructions.

Karl Geiser's great uncle was Steiner so he set out on his mathematical career already having links to one of the important figures in the development of the theory of cubic surfaces. Perhaps, therefore, it is not surprising that he should make his most important research contribution on cubic surfaces. One of his results explains how the 28 double tangents of the plane quadric are related to the 27 straight lines of the cubic surface.

Le Paige spent his whole career at the University of Liège where he worked on the theory of algebraic forms, a topic whose study had been initiated by Boole in 1841 and then developed by Cayley, Sylvester, Hermite, Clebsch and Aronhold. In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work. He is best known for his construction of a cubic surface given by nineteen points. Starting from the construction of a cubic surface given by a straight line, three groups of three points on a line, and six other points, Le Paige was led to the construction of a cubic surface given by a line, three points on a line and twelve other points. By means of this construction he then constructed a cubic surface given by three points on a line and sixteen other points, finally arriving at a cubic surface given by nineteen points.

The last person we will mention in this short history of the study of cubic surfaces is Gino Fano. Fano studied with Klein in 1893 and did an Italian translation of Klein's *Erlanger Program* (1872), which gave his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations. It was later in his career that Fano became interested in cubic surfaces and algebraic surfaces in general. Already by this time, however, interest in topics of this type had somewhat declined.

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**Article by:** *J J O'Connor* and *E F Robertson*