A photo of the MO-Labs “Clebsch diagonal surface with two planes” model shot in the MO-Labs office, on a table made from a beautiful 100 year-old piece of wood found in a barn. 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.
One of the simplest ways to count this number starts with one line, and considers all planes through this line. It turns out that in five special positions the plane cuts the surface in two more lines besides the fixed one. This is the main ingredient of the argument.
Our model shows two such planes: One passing through a line and intersecting the surface in a smooth conic section (a hyperbola, in fact), and another one passing through a line and intersecting the surface in two more lines (thus in three lines in total). Such planes are called triply tangent planes, or tritangent planes for short, because they are indeed tangent to the surface in the three intersection points of the three lines.
This sculpture also appears in our article “Straight lines on models of curved surfaces” (to appear in 2017 in Mathematical Intelligencer). For all our math sculptures mentioned in this article see the corresponding category article “Straight lines on models of curved surfaces”.
To purchase this exhibit size (20.5 cm / 8.1 in) version or a slightly smaller version (17.5 cm / 6.9 in) of Clebsch’s diagonal surface with two planes shown in the photo above, 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 of our posts related to the Clebsch diagonal surface: