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27 lines on cubic surface
27 lines on cubic surface
This surface is called the Clebsch diagonal surface and the defining equation is given by homogeneous coordinates as
X03 +X13 +X23 +X33 +X43 = 0
X0 +X1 +X2 +X3 +X4 = 0
There are 27 lines on the surface. Among them there is pair of 6 lines which do not intersect each other. They are indicated as
1, 2, 3, 4, 5, 6 ; 1', 2', 3', 4', 5', 6'
on the model. Moreover, there are 15 lines indicated as
ij, i<j, i,j=1, ..., 6.
They intersect with the lines i, j, i', j'.
The configuration of these 27 lines was studied by Schläfli.
High resolution picture of the Clebsch diagonal surface made in alminium (by Yamada Seiki Co. Ltd.)
