• Size-XS: about 3-5cm (1-2in) and usually not more than 8 € (incl. VAT) for the white strong and flexible pastics material. Your unique opportunity to hold a piece of math in your hands, for a very low price.
• Size-S: about 7.5cm (3in) and usually between 12 € and 20 € (incl. VAT) for the white strong and flexible pastics material. These objects are still affordable, but already large enough to be really used. We present 45 cubic surfaces of that size in an exhibition in Lisboa, Portugal.
• Size-M: about 13cm (5in) and usually between 25 € and 45 € (incl. VAT) for the white strong and flexible pastics material. This size is perfect for passing around in seminars or lectures.
• Size-L: about 20cm (8in) and usually between 70 € and 190 € (incl. VAT) for the white strong and flexible pastics material. These Math Objects fit perfectly in your book shelf. The size is also large enough to be shown in lectures or seminars with up to 30 participants.
• Size-XL: even larger and more expensive for the white strong and flexible pastics material. This size looks simply impressive. We present six Math Objects of height 30cm (12in) in an exhibition in Lisboa, Portugal, and many visitors were just flashed by these shapes.

Our series of 45 types of cubic surfaces, however, is based on a more modern topological classification of cubic surfaces by Knörrer and Miller from the 1980s. For each their 45 types, we give one representative on which all lines and all singularities of the surface can be seen.

See the cubic surfaces section in our Sculpture Shop.

Singularities of surfaces are very special points. In most parts of a surface, the shape looks quite smooth, but in some it might not; such points are likely to be singularities.

The particular shape of the surface depicted in our logo is determined by the equation \(x^2-y^2=z^5\). Actually, the logo only shows its silhouette which is essentially the intersection of the surface with the plane \(y=0\), namely the plane curve \(x^2=z^5\):

The MO-Labs-Logo-M-Curve.

The related Math Objects in our sculpture shop are:

• 27.54 €: An A4-Singularity, height: 81mm (3.2in).
• 30.24 €: Deformation of an A4-Singularity, height: 81mm (3.2in).

Attention: These prices include German VAT (19%). Depending on your country, there might be lower or higher costs. Just click on the links above to see the exact prices in your case on our Sculpture-Shop-Site.

Some related links to external websites:

• A Deformation of an A4-Singularity, Day 8 in the Advent Calendar of Geometrical Animations 2006. By Duco an Straten and Oliver Labs.

A Smoothed Kummer Surface, 30cm (11.8in), 2012 by MO-Labs

The symmetry is one of the striking geometric features of our Math Object. Kummer surfaces are not necessarily so symmetric, but the symmetric version is certainly one of the most classical ones. The surface actually has the symmetry of the tetrahedron, i.e. those symmetry planes which reflect a regular tetrahedron on itself also reflect the Kummer Quartic on itself.

Related Math Objects in our sculpture shop:

• 18.24 €: A Smoothed Kummer Surface, height: 6.9cm (2.7in).
• 162.07 €: A Smoothed Kummer Surface, height: 19.2cm (7.6in).

Attention: These prices include German VAT (19%). Depending on your country, there might be lower or higher costs. Just click on the links above to see the exact prices in your case on our Sculpture-Shop-Site.

Some related links to external sites:

• The Wikipedia Site on K3-Surfaces.
• The Wikipedia Site on Kummer Surfaces.
• Stephan Edrass’ site on Kummer Surfaces.
• E.E. Kummer on the MacTutor History of Mathematicians Archive.

Clebsch Diagonal Surface, height: 299cm

Every smooth cubic surface contains exactly 27 lines as was shown already in 1849 in a letter-exchange by Arthur Cayley and George Salmon, but here these lines can actually all be seen and have a high symmetry and interesting intersection properties, e.g. on the Clebsch Diagonal Cubic it happens 10 times that three of the 27 lines meet in a point (called Eckhardt Point, see the lower central part of the picture).

The related Math Objects in our sculpture shop:

• 28.04 € : The Clebsch Diagonal Surface, no lines shown, height: 119mm (4.7in).
• 29.07 € : The Clebsch Diagonal Surface, all 27 lines shown, height: 119mm (4.7in).
• 111.60 € : The Clebsch Diagonal Surface, no lines shown, height: 199mm (7.8in).
• 114.82 € : The Clebsch Diagonal Surface, all 27 lines shown, height: 199mm (7.8in).

Attention: These prices include German VAT (19%). Depending on your country, there might be lower or higher costs. Just click on the links above to see the exact prices in your case on our Sculpture-Shop-Site.

Some related links to external sites:

• Illustrating the Classification of Real Cubic Surfaces (PDF), by Stephan Holzer and Oliver Labs.
• Kubische Flächen und die Coblesche Hexaederform (in German, PDF), Diploma Thesis of Oliver Labs.
• The Wikipedia Site on the Clebsch Diagonal Surface.
• Alfred Clebsch on the MacTutor History of Mathematics Archive

In this blog, we will present background information on our Math Objects and news from the world of mathematical models.

One of these is the Deutsches Museum (German Museum of Science and Technology at Munich, Germany). Two of our computer generated math images are part of the permanent collection and are displayed in the so-called Mathematical Cabinet of the Museum:

Our images are part of the touring exhibition project called Imaginary; see the Imaginary-website for impressions from the events.

In one of the images, you see in the back some smaller models. These are 45 so-called cubic surfaces; for every type in a classification by Knörrer/Miller we show one example. These Math Objects have a height of approximately 7cm (2.8in).

For highly fragile geometries, we prefer, however, the laser-in-glass method: we produced four so-called world-record-surfaces using this technique for the exhibition.

An interesting aspect of this exhibition is the fact that our new models are displayed in connection with historical ones and also with interactive screens.

The team of curators and organizers of the exhibition consists of four locals from Lisbon and one external member: Ana Eiro, Cristina Luis, Susanna Napoles, Jose Francisco Rodrigues, Oliver Labs.

The exhibition “Formas e Formulas” is taking place at the “Museu Nacional de História Natural e da Ciência”, Rua da Escola Politécnica, 56-58, 1250-102 Lisboa (Portugal). The website of the exhibition also presents a large set of photos.