*The photo above shows our “45 cubics, series I (ball cut)” in the exposition “Forms and Formulas” at Lisbon, Portugal, 2012-2016.*

Ccubic surfaces have been classified several times in the past. The first reasonable classifications were developed by Ludwig Schläfli in 1858 and 1863.

There is a modern version by Knörrer and Miller from 1987. Their method of classification yields almost the same result. The most interesting part consists of the classification of all irreducible cubic surfaces with only finitely many singularities and lines. They found 45 different types of these. Oliver Labs carefully created equations for each of those types such that all main features of each of the surface can be visualized in a single image or model, called “KM 01”, …, “KM 45”. Here is the article in PDF.

Our series “45 cubics, series I (ball cut)” presents exactly this classification. As the name of the series suggests, each of the cubic surfaces has been cut by a ball in order to get a finite object. This gives the whole series a quite coherent design.

This complete series of all 45 cubics with only finitely many singularities and lines was first exhibited during the exposition “Forms and Formulas” at Lisbon, Portugal, from 2012-2016.

- See the category “exhibit Forms and Formulas” for more details on this exposition and other MO-Labs models presented there.

However, if you prefer a more “elegant” look, you may be interested in our series II (cylinder cut) which we will present in another post.

### Links to purchase parts or all models from this series

Below, you may choose your preferred subset of the series of all 45 types of cubic surface models (series I, ball cut). For your convenience, we provide links via which you may put a complete set of several cubic surface models into your shapeways shopping cart:

#### Buy all 45 cubics

Put all 45 models from “45 cubics, series I (ball cut)” into your cart (these are 45 different cubic surface models of a size of approximately 7.7cm):

**PUT ALL 45 CUBICS SERIES I (BALL CUT) INTO YOUR SHAPEWAYS CART.**

#### Buy 6 carefully selected cubics

If you are just looking for a few of these, here are 6 cubic surface sculptures, carefully selected by Oliver Labs. To put these 6 objects from “45 cubics, series I (ball cut)” into your shapeways cart, just click the following link:

**PUT 6 SELECTED CUBICS SERIES I (BALL CUT)
INTO YOUR SHAPEWAYS CART: 01, 6, 11, 27, 43, 44.**

These are the 6 models included in this set:

- KM 01: a smooth cubic with 27 straight lines. This is NOT the Clebsch diagonal cubic, but a slightly more general cubic, so that that 27 lines have a more general intersection structure.
- KM 06: a cubic with 1 conical singularity (A1) and 21 straight lines. Comparing this model with number 01, one can see how the singularity forces 12 lines to coincide in pairs.
- KM 11: a cubic with 2 conical singularities (A1).
- KM 27: a cubic with 4 singularirities and 9 straight lines. We call this cubic “Cayley/Klein cubic” because it was Arthur Cayley who first studied this cubic in a publication, and it was Felix Klein who presented the first model of it and who based his deformation theory of cubic surfaces on this surface.
- KM 43: a cubic with 3 cuspidal singularities and 3 straight lines. We call this surface “Salmon’s cubic” because George Salmon was the first to realize that 3 is the maximum possible number of cusps on a cubic.
- KM 44: a cubic with 1 conical singularity (A1) and one higher biplanar singularity (A5) 2 straight lines. We call this surface “Schläfli’s cubic” because Schläfli was the first to realize that a cubic may contain an A5 and an A1 singularity, but no higher A-singularity, in particular no A6.

#### Buy each cubic individually

To put each of the 45 models from “45 cubics, series I (ball cut)” invididually into your shapeways cart, you may use the following links:

- KM01
- KM02
- KM03
- KM04
- KM05
- KM06
- KM07
- KM08
- KM09
- KM10
- KM11
- KM12
- KM13
- KM14
- KM15
- KM16
- KM17
- KM18
- KM19
- KM20
- KM21
- KM22
- KM23
- KM24
- KM25
- KM26
- KM27
- KM28
- KM29
- KM30
- KM31
- KM32
- KM33
- KM34
- KM35
- KM36
- KM37
- KM38
- KM39
- KM40
- KM41
- KM42
- KM43
- KM44
- KM45

Below you will find other posts related to cubic surfaces:

#### 45 cubics, series II (cylinder cut), 15cm

The complete series of 45 types of cubic surfaces with only finitely many singularities and lines. We provide links to put a complete set of several cubic surface models into your shopping cart.

#### A math vase of degree 3 without bottom

A math vase of degree 3 without bottom. It has been created by rotating a graph of a polynomial of degree 3 about an axis.

#### A math vase of degree 3

A math vase of degree 3. It has been creating by rotating a graph of a polynomial of degree 3 about an axis.

#### Cubic surface KM 42 pendant

Cubic surfaces are a math model classic from the 19th century. We provide one of our favourite examples (cubic surface KM 42) in the form of a pendant.

#### The Clebsch diagonal surface: 27 lines only

The sculpture we present here is a 3D-printed modern object consisting of the 27 lines only, and a thin part of the surface as a border.

#### A 3d graph of a cubic function

This visualizes a 1-parameter family of cubic functions or a 3d graph of a function in one variable in a 3d-coordinate system.

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