Visualizing the sphere and the hyperbolic plane: five projections of each

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Recommends user76284 · 2015-08 · Mathematics Stack Exchange · on “What hyperbolic space *really* looks like”

There are several models of hyperbolic space [LINK] that are embedded in Euclidean space [LINK] . For example, the following image depicts the Beltrami-Klein model [LINK] of a hyperbolic plane: [LINK] where geodesics [LINK] are represented by straight lines. The following image, on the other hand, depicts the Poincare model [LINK] of the same hyperbolic plane: [LINK] where geodesics are represented by segments of circles intersecting the boundary of the disk orthogonally. Both of these models capture the entire $n$-dimensional hyperbolic space in a disk (or more generally, a Euclidean $n$-ball). I was thinking about what hyperbolic space would "really look like" from the perspective of an ob

Recommends Magma · 2019-08 · Mathematics Stack Exchange · on “Does there exist an area-preserving map from the hyperbolic plane to the Euclidean plane?”

There does in fact exist an area-preserving map, as demonstrated in this video at 11:20 [this video] : the Lambert azimuthal equal-area projection. The idea is that you take polar coordinates of the hyperbolic plane and map them to polar coordinates of the euclidean plane via a map $(r, \theta) \mapsto (f(r), \theta)$ where $f$ is chosen such that area is preserved. Let's derive $f$ : Let $h(r)$ be the area of a hyperbolic ball of radius $r$ , and $e(r)$ be the area of a euclidean ball of radius $r$ . Assuming that the hyperbolic plane has curvature $-1$ , it is true that $h(r) = 2\pi(\cosh(r)-1)$ and $e(r) = \pi r^2$ . We want to ensure $h(r) = e(f(r))$ . Substitution and rearrangement yields $f

Recommends Timothy · 2019-11 · Mathematics Stack Exchange · on “How to teach a kid geometry”

Sometimes kids learn something better when you get them interested in it. I'm guessing they tend to get excited by rare and unusual things. Maybe the teacher could play only the part of the YouTube video Visualizing the sphere and the hyperbolic plane: five projections of each [this video] with the orthographic projection of the sphere, muted. After that, they could then bring to the attention of the students that that is a tiling that is not possible on a flat plane. Then they might get all fascinated about the fact that a sphere has different geometry than a flat plane. However, it's probably just spherical geometry and not Euclidean geometry that that would get the students interested in at fi