“Hyperbolic” Polysphericons
“Hyperbolic” polysphericons differ from the common polysphericons with respect to the curvature of their surfaces. The concept of curvature is therefore necessary to make the distinction.
As seen in the article “Polysphericons” on this website, polysphericons are constructed from bodies of revolution. They are therefore locally flat (sectional curvature is zero). The following polysphericons however are locally hyperbolic. This means that the sectional curvature is negative. Figuratively, the surface of hyperbolic polysphericons locally looks like the famous potato chip depicted below.
Famous potato chip with negative Gaussian curvature. 
Normal to the surface indicated by red arrows. 
Circles which approximate the curvater. They are located both above and below the surface.
Figurative Explanation of negative curvature:
To explain negative curvature, one can make use of the concept of Gaussian curvature which is well-suited to the potato chip situation:
The grid shows the two principal curvatures. These are the lines which are curved most strongly with respect to an axis which stands vertically (like a small tower in the valley) on the surface. The value of this curvature is determined by the radius of the circle which approximates this line. The smaller the circle, the stronger the bending of the line and hence the value of the curvature is further away from zero (when trying to fit a circle to a flat surface, one needs an infinitely big circle, the curvature is hence zero).
Looking at the two perpendicular lines of the grid, one realizes that the centers of the approximating circles lie above and below the surface, respectively. This shows that the bending of the grid lines corresponds to negative curvature. By definition the Gaussian curvature is the product of the two principal curvatures. This product is negative for the chip, since the two bendings are in different directions and hence positive and negative.
“Hyperbolic” Polysphericons
On the left: locally hyperbolic sphericon. On the right: “normal”, locally flat sphericon 
On the left: locally hyperbolic dual (6,1)-polysphericon. On the right: dual (6,0)-polysphericon (=body of revolution from a hexagon with edges bent inwards). 
On the left: locally hyperbolic even (6,1)-polysphericon. On the right: body of revolution from a inwards bent hexagon. 
On the left: uneven (5,1)-polysphericon. On the right: body of revolution from a pentagon with inwards bent edges.
The edges of the polysphericons are obviously problematic. Since there is no normal axis on these edges, one cannot properly define curvature at these singularity points. In fact, there is no solid which is globally hyperbolic without these problems. When looking at the chip, one can imagine that a solid like this without edges would have to be infinitely big.This stands in contrast to a sphere which can be finitely big without boundary. The sphere has positive Gaussian curvature after all.
More about hyperbolic geometry
Hyperbolic geometry is very interesting and area of active research. Lots of concepts of Mathematics can be applied to hyperbolic geometry and hence justify the necessary work to overcome the unfamiliarity of human brains with that type of geometry.
If you want to know more about hyperbolic geometry, you can find introductory information (outside of HITS website) for further reading below :
Mit dem Laden des Videos akzeptieren Sie die Datenschutzerklärung von YouTube.
Mehr erfahren
- J.Milnor – Hyperbolic Geometry: The first 150 years : https://projecteuclid.org/download/pdf_1/euclid.bams/1183548588
- F. Klein – Über die sogenannte nicht-Euklidische Geometrie https://gdz.sub.uni-goettingen.de/id/PPN235181684_0004?tify={%22pages%22:[586],%22panX%22:0.76,%22panY%22:0.618,%22view%22:%22info%22,%22zoom%22:0.396}
- Introduction from a machine learning perspective: http://hyperbolicdeeplearning.com/simple-geometry-initiation/
- Introduction to hyperbolic geometry with focus on the artist M.C. Escher https://pointatinfinityblog.wordpress.com/2018/02/19/life-on-the-poincare-disk/
powered by Borlabs Cookie