A photo of the MO-Labs “Clebsch diagonal surface with colored lines”. To purchase the model used in the photo and others, follow the links to our producer “shapeways” provided below.

Clebsch’s diagonal surface contains 27 real straight lines. The fact that any smooth cubic contains 27 straight lines (possible imaginary) was known since 1849. The British lawer/mathematician Arthur Cayley first found that there may only be finitely many lines. When he wrote a letter about this to his friend George Salmon, he almost immediately received the answer saying that their number is exactly 27.

These 27 lines have an interesting intersection structure. Cayley noticed this at a very early stage already, but he realized that it would be much easier to deeply understand this structure with a reasonable notation at hand. It was the Swiss Ludwig Schläfli who published such a notation for the 27 lines in 1858, based on his discovery of the so-called “double six”. These are two sets of six lines a_i and b_j, plus 15 lines c_{ij} with i<j. The a_i do not meet any other a_k, the b_j no other b_l, but a_i meets b_j if and only if i \neq j. Two such intersecting a_i and b_j with i \neq j span a plane. This plane intersects the cubic surface in a cubic curve which splits into a_i, b_j, and a third line which is the line c_{ij}.

MO-Labs sculptures used: Clebsch’s diagonal surface with two planes

To purchase this exhibit size version, just use the “buy now”-link provided below. You will be sent to our MO-Labs shop on the “shapeways” website, so that you can let “shapeways” produce the object for you and send it to you in really short time, where ever you are located in the world.

We have designed a large variety of other Clebsch diagonal surface sculptures, in one color, or with differently colored lines, or even together with two related planes. We provide versions in different sizes and at very different prices, from tiny and cheap to large and more valuable.

Here are some posts presenting models of the Clebsch diagonal surface:

  • Clebsch and Klein in the family room - MO-Labs photo series: Math sculptures in context

Clebsch and Klein in the family room

It was back in the 1872 Göttingen, Germany, at a meeting of the scientific society. Alfred Clebsch and Felix Klein each presented a model of a cubic surface. Our modern versions of these historical - nowadays quite famous - sculptures are the main figures in our photo.

  • Clebsch diagonal surface in the family room

The Clebsch diagonal surface in the family room

Our modern version of Clebsch's historical - nowadays quite famous - diagonal surface model is the main figure in our photo from the series "math sculptures in context". It is the pure white version with the 27 straight lines.


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