Mathematics at the Deutsches Museum – Heidelberg Laureate Forum – SciLogs

“In this house, anyone can do what they want.” With this classic saying, Oskar von Miller, founder and influential designer of the “German Museum for Masterpieces of Science and Technology” in Munich, introduced “education-oriented hands”. as a rule ”almost 100 years ago. to the point. Aside from the internal contradictions: it is extremely helpful when visitors can play with the exhibits – but not with the real ones, of course! After all, the museum also has the classic task of preserving historical artifacts and exhibiting them for posterity.

After seven years of construction, the traditional walls have been partially restored from July 8, 2022. The renovation was divided into two stages, ensuring that at least half of the museum is accessible at all times; The “future initiative” (“zukini” in inner jargon) now continues with the other half. The newly opened faculties include “Image Writing Codes” (including print and cryptology), “Photo and Film”, “Electronics” and “Mathematics” that share one wing of the house.

The latter in particular occupies an extreme position in the field of tensions between participation and the preservation of history. Photographers and filmmakers have an impressive collection of old cameras and imaging accessories to offer, in the printing department there are vintage printing machines, in the electronics department you can still see lamps, transistors, capacitors and resistors in person, their modern counterparts long ago immersed in the anonymity of a boring microchip . (Yes, it makes the old man nostalgic for the electronics pack of his youth.) And math?

The team from Munich didn’t have much for that. No wonder: this science is actually about abstract objects. Sometimes this can be realized in models which also helps a lot for understanding; but traditionally professionals do not see this as their main job. There was even a current associated with the name of Bourbaki that wanted to throw all views out of mathematics for the sake of purity – well, it was going on to some extent. However, at the end of the 19th century, models made of plaster appeared, made of incised and interlocking paper discs or stretched threads, which impressively depict special surfaces in three-dimensional space. And curator Katja Rasch is clearly enthusiastic about these historic sites.

Special screens in front of the projection lamp (for example) allow only a fan of light to emerge, which lies entirely in one plane. Targeting a cone modeled from wires, it highlights special curves and explains why they are called “conical sections”. The individual parts of this exhibit ended up in the warehouses of various departments, because no one knew what to do with them, and they were put back together during the current exhibition.

Photo: German Museum

Surfaces that have a minimum surface area among their peers, provided that some curve in space (“boundary curve”) belongs to the surface. For visualization purposes, it is usual to create a wire edge curve and dip it into a soapy solution. When you pull it out carefully, a soapy peel hangs on the wire frame, which is shaped like a minimal surface, since the minimal surface area minimizes the energy generated by surface tension.

Photo: German Museum

Crystallographer from Marburg Elke Koch and her colleague Werner Fischer used an innovative material instead of soap: women’s stockings. Yes, their flexibility also causes them to shrink into a form that uses as little energy as possible. But whether and under what circumstances it is actually a form of a minimum surface (it looks at first glance) has not been quickly clarified by the current experts.

What about the current facilities? Well, if they are to be exposed to the public, they have to be indestructible, which turns out to be an extremely restrictive requirement. This is evidenced, among others, by wooden blocks with which you can arrange various symmetrical patterns.

The lines on the blocks are arranged in such a way that they always connect to those on an adjacent block, no matter how twisted the blocks are. A symmetrical pattern remains symmetrical, albeit with a different symmetry, by performing a symmetry operation on the blocks. Example: in every second column, turn all the blocks 90 degrees clockwise.

A more recent discovery is Gömböc (Hungarian for “fat”). It is a massive, convex body which, like a cylinder, has only one stable equilibrium position, but not because there is a hidden weight inside that forces it into that position. Rather, the mass distribution is completely homogeneous. Until a few years ago, it was debated whether such a body could even exist. Now he can be seen in person at the exhibition – but behind the glass, he cannot be moved to observe his behavior in a standing position. The risk of losing a good piece would be too great.

Photo: German Museum

Could there be a body that casts a shadow like a triangle in one direction, a square in the other, and a circle in the third? Actually a standard task in descriptive geometry; this solution is known and implemented at the exhibition by a solid wooden block which almost fits into three appropriately shaped openings. It turns out that there are even two fundamentally different solutions to this problem! And professionals are wondering if there are more, maybe even a whole continuum of shapes that interpolate between the two solutions.

A block of wood (right) can be imagined how it is formed in this way: On the circular top surface of the cylinder, which is as wide as it is tall, a diameter is drawn and a segment is made along the plane that passes through it through the diameter and the point on the lower surface of the cylinder, as well as a similar cross-section through the diameter and the opposite point on the lower surface. On the other hand, the 3D printed black block is a somewhat elongated tetrahedron with two semi-cones attached at the front and back – not round cones, but shaped in such a way that the bottom surface that is cut obliquely to the cone axis has a round edge.

Due to the modest availability of the material (and limited space), a comprehensive math review was out of the question. Instead, the exhibition organizers switched to “interesting tidbits with an emphasis on geometry.” And this is where the marvelous weapon of this exhibition comes in: a touchscreen with an appropriate program.

After all, abstract things – by definition – cannot be touched; but what you can do with your own fingers with pictures of these things is very close. A regular geometric solid appears on the screen; I can not only rotate it in all directions, but also more or less radically cut corners with a virtual knife. In the extreme case, only one point remains from each edge and e.g. the cube turns into an octahedron or vice versa. I can not only put a knife on the corners, but also on the edges, and thus gradually transform a platonic solid into a semi-regular (Archimedean) solid and much more.

© Jürgen Richter-Gebert

My finger turns into a virtual brush (color and stroke width adjustable) and makes a modest stroke on the touchscreen. Not only the line appears in place, but also shifted, rotated and mirrored reflections of the same, making a regular ornament, according to the symmetry group that I can also choose. This works not only in the plane, but also on the ball – yes, you can cover the surface of the ball with lots of congruent, slightly curved tiles without gaps. The iOrnament app on which current development is based also offers decorations in the hyperbolic plane. It is a structure in which there is not only one parallel to a straight line passing through a given point, as in an ordinary plane (this is the famous Euclidean axiom of parallels), but infinitely many. You can’t see it in its true beauty, but there is a distortion that maps the entire hyperbolic plane to the disk (Poincaré mapping). And in this form, the patterns look extremely decorative.

© Jürgen Richter-Gebert

An abstract object allows a direct comparison of a physical representation with a virtual one: Sierpinski’s tetrahedron. Replace a regular (regular) tetrahedron with a quadrilateral halfway along the edge of the tetrahedron, or which means the same, cut a regular octahedron from the center of the tetrahedron, leaving four small tetrahedrons. Repeat this procedure with small tetrahedra, then with the resulting 16 small tetrahedra and so on ad infinitum. The result of the boundary process is a fractal with the strangest properties (zero volume, finite area, infinite sum of all edge lengths); and of course no representation can grasp it. It quickly reaches its limits due to the stability of the material or the size of the pixels on the screen. But still: A close-up of Sierpinski’s tetrahedron hangs on the ceiling, in a glass case so that pieces of a sensitive object do not fall on the visitors’ heads, and it can also be admired on the screen.

(Top left photo: Deutsches Museum) Sierpinski’s tetrahedron in person (left) and virtually. If the gaze is directed from the center of one edge to the center of the opposite edge of the original tetrahedron, then Sierpinski’s tetrahedron, like any of its approximations, is opaque – somewhat surprising for an object that is actually just a point cloud.

I have to admit, the virtual image looks better. And so says someone who is concerned with the creation of physical geometric bodies with great intensity.

All these programs are the work of one person: Jürgen Richter-Gebert, professor of mathematics at the Technical University of Munich and – not only with tactile programs – significantly involved in the concept of the exhibition. When it comes to ornaments, his tireless activity is reflected in the iOrnament app. More projects can be found on his science-to-touch website.

After all, the touchscreen exhibits and hands-on stands served as a key argument in pulling the rope by square meters for mathematics. “People have a lot to play, and the museum needs to have enough kindergartens to be attractive.” Anyone who sees math as a serious matter and does not tolerate trickery may wrinkle their nose; but if it serves to find the truth …

Leave a Comment