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THThis is hyperbolic space — a space with continual and constant negative curvature. When viewed from this perspective, the entire infinite expanse of the upper sheet is seen within this finite disk. Each of the lines within this surface is entirely straight — a geodesic — straight from the perspective within, and yet appearing curved without. From this vantage point, every line is a part of a circular arc that meets the boundary of the disk at a right angle. This is the Poincaré disk. As we can see, it is simply a matter of perspective. Due to the continual expansion of the space, we can produce exotic tessellations — here we see five squares meeting at each vertex, a Euclidean impossibility. Hyperbolic space doesn’t just exist within the third dimension — we can also have hyperdimensional hyperboloids, giving rise to the possibility of voluminously expanding manifolds.
Appendix: Geometric Notes
In the video, two perspectives are shown. The one with the circular arcs is the Poincaré disk, while the other is the Klein disk. The Klein disk produces straight lines, or chords, and is a result of viewing from the vantage point of the origin — right between the two sheets of the hyperboloid. The Poincaré disk, on the other hand, produces circular arcs and is the result of viewing from the vantage point of the vertex of the lower sheet.
Also, there is a point in the animation where a cone is drawn around the hyperboloids to demonstrate that hyperboloids are asymptotic to cones.
The hyperboloid in this animation is called a 2-hyperboloid, embedded in three-dimensional space. We can also have a 3-hyperboloid embedded in four-dimensional space, or higher versions. These are manifolds — spaces of curvature existing within higher-dimensional spaces.
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