Göttingen Collection of Mathematical Models and Instruments

Regular basic division according to Schoenflies

Model 331

B III 1-11


Regular basic division according to Schoenflies.
331: 1 block, 3 pieces
331a: 1 block, 0 pieces.


Schoenflies' series in Göttingen's collection consists of eleven models illustrating different regular partitions of the three-dimensional Euclidean space. Each of them consists of several single pieces which can be composed into a larger block without gaps as well as a precomposed block made of about 12 single pieces.

There are two different types of models. The first one involves single pieces which are all congruent to one another, the second one consist of such congruent pieces as well as their mirror images. The single pieces of the models are marked in a way that facilitates the composition into larger blocks. A face marked with a C will require a congruent piece next to it, an S indicates the need of a mirrored piece at this place.

Although the number of configurations in space is infinite, they can all be characterized by finitely many space groups, in particular 230, that are generated by different symmetry operations (translation, rotation around an axis, twisting, reflection through a point, reflection across a plane, glide reflection).

The deduction of all 230 crystallographic space groups was done by Arthur Moritz Schoenflies and Jewgraf Stepanowitsch Fjodorow in the year 1891, after some ground work had been done by Leonhard Sohncke in the year 1876.

Manufactured by Schoenflies in the year 1891 in Göttingen. The price at that time for the whole series added up to 160 Goldmark, each single piece cost 50 Pfennig extra.

Text written by: Franziska Gelin, Lea Renner and Ulla Wege

Translated by: Lea Renner

Showcase of this model is Case number 24


Separataband M1 im Mathematischen Institut p. 311.

Schoenflies, Arthur(1877). Krystallsysteme und Krystallstructur, Berlin, (Habilitationsschrift, Universität Göttingen).

Schoenflies, Arthur. Zehn Modelle zur Darstellung von regulären Gebietseinteilungen des Raumes.