Clebsch Diagonal Surface with Configuration: 31 Spheres and 1 Sylvester Pentahedron .
The Clebsch Diagonal Surface is a non-singular (i.e. smooth) Cubic Surface originally discovered and documented by Alfred Clebsch (1833 - 1872). The Clebsch Diagonal is a historically important model and was of great interest to early machine designers and mathematicians.
This model is a recreation of the original plaster model made by hand in the 1800's under the direction of Felix Klein. This is a FULL SIZE 1:1 version.
It shows the 27 possible lines on a smooth cubic surface and is the "god-head" of the series of 23 types of singularities possible on a Cubic Surface (that are also shown on Shapeways).
All the lines are numbered with imprinted numbering of the complex configuration. In total there are 36 possible ways to number the lines.
This series of Clebsch models show various aspects of the complex configuration of geometric objects on this famous surface. Please see the model description to make sure which model contains which objects in the configuration. We'd also be happy to customize a model for you.
27 total lines: 15 Diagonal Lines and 12 Horizontal Lines (Shafli's "Double Six")
30 points where 2 lines of ruling intersect
24 within the model and 6 at infinity
10 points where 3 lines of ruling intersect (Eckardt Points)
7 within the model and 3 at infinity
The Sylvester Pentahedron
Additionally there are three "passages" which can be seen in the animation below. Three are "holes" that can be easily seen. Three are within the model at the "ears" and the seventh is somewhat vertical at the "waist".
For additional documentation or other models or sizes please contact the author. Note that this models is the same as the Clebsch Diagonal Surface with Imprinted Numbers - with the addition of some of the geometric objects in the Configuration.