Screenshots from the program jReality Exhibit.
The sculpture of the Matheon Bear stands in front of the Maths building of the TU Berlin. The bear is interesting from a mathematical point of view, because of the pattern painted on it. The periodic pattern of circles and Matheon logos filling the plane forms the basis. The mathematical challenge was to apply this pattern such that the forms distorted as little as possible. Actually, you can see that all of the angles appearing in the pattern also appear on the bear. We can say that the image of the pattern on the bear is “conformal”.
Schwarz Minimal Surface
Minimal surfaces are surfaces that have the curvature features of physical soap films. The copy illustrated here was found by Karl Hermann Amandus Schwarz in the 19th century and periodically fills the entire space in the manner of a crystal grid. What you see here is, strictly speaking, not at all a smooth surface, but consists of many circular disks contacting each other in a certain way. Such “discretizations” of smooth surfaces have recently played an important part in architecture, if the question is to create curved surfaces from flat elements.
Minimal Surface Site
Here, you see a so-called Enneper surface onto which two (downwards turning) duct-shaped outgrowths were grafted. The fine art of creating minimal surfaces is to construct minimal surfaces which, without boundaries, extend towards infinity without self-intersection. The first requirement is met, here. The piece shown of the whole surface can be continued indefinitely. But very soon self-intersections occur. In this respect, this surface may well be taken from the apprenticeship workshop of a minimal surfaces constructor, but it is beautiful all the same.
Soap films consist of a material which resists being stretched. However, you can bend it in any direction with no effort. Willmore surfaces, in contrast, consist of a material which does not resist stretches, but develops elastic counter force against bending. What you see here is a piece of a closed torus with the Willmore feature, which was found by Matthias Heil on the basis of a theory of Babich and Bobenko.
The Tetranoid is part of a class of surfaces that have the curvature features of soap bubbles. In mathematical terms this reads: The Tetranoid has a constant mean curvature. The four “legs” of the Tetranoid in reality keep on going to infinity. The existence of the Tetranoid (just like the existence of similar surfaces with arbitrary symmetry based on platonic bodies) was demonstrated by Nicholas Schmitt; he also calculated the surface.
Helicoid with Handles
One of the best known minimal surfaces is the Helicoid which you will recognise from spiral staircases and car park ramps. It is, indeed, possible to connect different levels of the Helikoid with each other without destroying the minimal surface feature or making the surface intersect itself. This connecting piece is called a “handle“ in mathematical terminology. Depending on where you are such a handle looks like a hole in the floor or ceiling, or like a column connecting the levels. This surface with two handles was found and calculated by Markus Schmies.
보이(Boy)의 곡면은 뫼비우스 띠의 경계에 원판의 경계를 붙여서 만들어집니다. 이 조작이 부드럽게 가능하다는 것은 수학자 Werner Boy가 1903년에 증명하였습니다. 보이의 곡면은 자기교차하는 것을 제외하고는 각 점에서 부드러운(smooth) 형태를 갖습니다. 여기 보이는 형태는 이 중 평균곡률(mean curvature)이 최소화되는, 즉 “불필요한 돌출이 없는” 특성을 갖고 있습니다. 따라서 이는 보이 곡면의 가장 “아름다운” 표현이라는 것을 수학적으로 ‘정확하게’ 이야기할 수 있는 것입니다. 이 매개화는 Robert Bryant와 Robert Kusner가 찾아냈습니다.