Circles inscribed in regular polygons in hyperbolic geometry

Question

Does the radius of a circle matter when determining the number sides of a regular polygon in hyperbolic geometry? The sides must be tangent to the circle. Can't I just use an equilateral triangle every time?

Answer 1

In Euclidean geometry, the radius does not matter — one can superscribe a regular $n$-gon around a circle of radius $r$, for every $n\ge 3$ and every $r>0$.

But in hyperbolic geometry, $n$ must grow with $r$. The reason is the divergence of geodesics from one another, a hallmark of hyperbolic geometry. When trying to circumscribe an $n$-gon, we pick $n$ equally spaced points on the circle and draw geodesics tangent to the circle at those points. The issue is that when $r$ is large enough, those geodesics never meet each other and no polygon is formed.

This is apparent in the Beltrami–Klein model of the hyperbolic plane, where geodesics are straight line segments which appear to end at the ideal boundary (in red below). The picture shows a failed attempt to circumscribe a square around the blue circle.

This circle is too bid for $n=4$ to work; one needs more sides.

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By the way, the special case $n=3$ is an illustration of the definition of a $\delta$-hyperbolic space (which is a concept generalizing the hyperbolic plane): a geodesic triangle cannot be "fat", in particular it cannot contain a large disk inside of it.