# Oliver Labs

mathematician/artist
MO-Labs
Ingelheim am Rhein, GERMANY

Each piece of mathematical art I create is based on a carefully chosen math concept or a concrete formula, often related to math history.

The fascinating aesthetics of an object is usually inherently related to the mathematical content behind the piece, in combination with the modern production methods I use.

For each individual piece, I choose the method of production allowing for the best possible representation of the mathematical shape or concept I wish to visualize. E.g., for surface sculptures, I use a 3d printing process allowing for wall thickness close to zero; for extremely fragile objects, I use the laser-in-glass method without any stability restrictions. Always on the edge of what is currently possible technically.

Rational points on Clebsch's Cubic
12 x 8 x 8 cm
glass (laser-in-glass)
2019

Alfred Clebsch's model of his cubic has been well-known since its presentation in 1872. All over the world, there are lots of university math departments that own a copy of it.

The laser-in-glass model shows a new aspect of this classic: We do not visualize the whole surface, but only the finitely many points not on the 27 lines and with rational coordinates with numerator and denominator bounded in modulus by 100.

It is not so clear one can find many such points on it; indeed, a random point will typically not work, and a version of Fermat's famous last theorem states that there are no non-trivial rational points on some other cubic!

This object is the result of a collaboration with Tim Browning and Ulrich Derenthal.

A cubic with 27 straight lines - cylinder cut
15 x 9 x 9 cm
white plastic (3d print)
2016

The 150-year-old classical model of Alfred Clebsch's diagonal surface is famous in large part because this beautifully curved surface contains exactly 27 straight lines. This is a well-known fact. But many non-specialists believe it is the only such example, although Arthur Cayley and George Salmon showed in 1849 that ANY smooth cubic contains exactly 27 straight lines, at least in complex projective space.

Here, I show a different cubic surface containing 27 straight lines all of which are real. The aim is to put a focus on the astonishing classical result by Cayley and Salmon which also works for many other cubic surfaces besides the best-known one.

This math sculpture is number one from my series of 45 cubic surfaces.