Clebsch Diagonal Surface

The Clebsch Diagonal Surface is certainly one of the most famous surfaces in mathematics. It was described by Alfred Clebsch in 1871. It is a very special example of the so-called cubic surfaces which is highly symmetric and on which there are 27 lines in a very special position.

Clebsch Diagonal Surface, height: 299mm
Clebsch Diagonal Surface, height: 299mm

 

Every smooth cubic surface contains exactly 27 lines as was shown already in 1849 in a letter-exchange by Arthur Cayley and George Salmon, but here these lines can actually all be seen and have a high symmetry and interesting intersection properties, e.g. on the Clebsch Diagonal Cubic it happens 10 times that three of the 27 lines meet in a point (called Eckhardt Point, see the lower central part of the picture).

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