Recent Highlights

3d printing at school

3d printers provide an easy to use infrastructure to create objects for class room demonstration. High-school students delve into geometry when given the chance to constructing their own items.

Introduction to 3d printing »

Puzzles

Basic geometric concepts are communicated in the form of mathematical puzzles. The mathematical foundations are embraced upon constructing new puzzles based on the same principles.

Segmentation of volumes »

Models and Animations

3d visualization by computer graphics has revolutionized teaching the subject of geometry. Printed models provide haptic feedback, and animations help to translate temporal evolution in solid models.

Vibrating strings »

School Mathematics

Polytopes and 3d printing. Students see Maths materialize.

Codes used by 3d printers are based on triangulations or other dissections of the surface area. They are solely based on the positions of the body's edges. Thus, the printer code for polytopes offers unique opportunities to connect stereoscopic conception and coordinates in R³. In workshops at high schools the students see their ideas take form via 3d printing.

Symmetries. Pupils and parents discover mathematics.

Feedback at home is an important determinant in mathematics education. In an outreach project parents and pupils are guided towards an appreciative interaction about a mathematical topic.

Segmentation of Volumes. Turning Laws into Puzzles.

Mathematics sets rules on regular and quasi-periodic tilings of the plane, and the segmentation of higher-dimensional volumes and spaces. It is enlightening to phrase these laws in form of mathematical puzzles. This approach of teaching is practices with teachers in the seminar Mathematics Education on Fundamental Ideas.

University Mathematics

Model Sponsors. Students care for a historical model.

Students explore the historical context of a model, and report their results in the seminar Mathematics Education on Fundamental Ideas.

Scanning and fitting. Students explore cubic surfaces.

The Göttingen collections hosts a wealth of plaster models of singular cubic surfaces. The singular features can only approximately be represented in a plaster model, they cause problems in scanning the models, and special care is needed at these points to generating mechanically stable 3d prints file from the scans. An approach to achieve viable models from our 3d printer was established in the seminar Singularities of Clebsch Diagonal Surfaces.

Cubic surface with 4 real double points, model 150

Extending the Collection. Students design new models.

Many models in the Göttingen Collection of Mathematical Models and Instruments have been build as part of diploma theses in the late 19^th and early 20^th century. In the past winter didactics students have designed two new models: A model showing the poles of the Riemann Zeta function, and an illustration of the emergence of fractal structures by multiple reflections in four spheres.

$fractals in four reflecting spheres$

Applied Mathematics

Phase-space visualization. The gist of scientific models.

Mathematics lectures in differential equations often focus on models that can be solved analytically. Physics lectures in theoretical mechanics augment this approach by perturbation theory around these solutions. The seminar Singularities of Geometrical Models in Physics and Mathematics rather emphizises topological approaches. Hence, one can classify the space of solutions of differential equations, and pinpoints the role of bifurcations in changing the qualitative structure of solutions. The theory was mostly developed based on historical models. Where appropriate however, students engineered new models.

Singularities and Physics. Intrinsic Limits of Mathematical Models.

Singularities persist for example in the solution of the wave equation. For this and many other physical models they provide a scaffold of the quantitative modeling of observations. On the other hand, singularities are unphysical mathematical idealizations. How are these contradicting observations reconsiled in real-world models? The seminar Singularities of Geometrical Models in Physics and Mathematics brought together Masters students in the programs of Didactics of Mathematics and the Physics of Complex Systems to explore this question from different perspectives.