10/2/2023 0 Comments Hyperbolic planeSo what happens when you have a hyperbolic surface sitting inside three-dimensional space? Well, all that extra surface area has to go somewhere, and things naturally "crinkle up". But the upper hemisphere has a surface area of $2 \pi \approx 6.28$.īy contrast, in hyperbolic space, a circle of a fixed radius packs in more surface area than its flat or positively-curved counterpart you can see this explicitly, for example, by putting a hyperbolic metric on the unit disk or the upper half-plane, where you will compute that a hyperbolic circle has area that grows exponentially with the radius. In flat space, this circle would enclose an area of $\pi^3/4 \approx 7.75$. Off the top of my head, I don't know what the formula is for a circle inscribed on the sphere (a positively-curved surface) is, but we can get an indication that circles in positive curvature enclose less area than in flat space as follows: the upper hemisphere on a sphere of radius 1 is a spherical circle of radius $\pi/2$, since the distance from the north pole to the equator, walking along the surface of the sphere, is $\pi/2$. In flat (Euclidean) space, we all know that the formula is given by $A(r) = \pi r^2$, so that there is a quadratic relationship between the radius of your circle and the area enclosed. One way to detect the curvature of your surface is to look at what the surface area of a circle of a given radius is. My guess as to why this shows up again and again (and I am certainly not a biologist here, so this is only speculation) is that hyperbolic space manages to pack in more surface area within a given radius than flat or positively curved geometries perhaps this allows lettuce leaves or jellyfish tentacles to absorb nutrients more effectively or something.ĮDIT: In response to the OP's comment, I'll say a little bit more about how these relate to hyperbolic geometry. For instance, you can see some characteristically hyperbolic "crinkling" on lettuce leaves and jellyfish tentacles:![ Algebraically, it is isomorphic to the free product of three order-two groups (Schwartz 2001).Maybe this isn't the sort of answer you were looking for, but I find it striking how often hyperbolic geometry shows up in nature. The real ideal triangle group is the reflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Real ideal triangle group The Poincaré disk model tiled with ideal triangles Note that in the Beltrami-Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models, is not conformal i.e. In the Beltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is circumscribed by the boundary circle. In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles. In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. Thin triangle condition The δ-thin triangle condition used in δ-hyperbolic spaceīecause the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any hyperbolic triangle, this fact is important in the study of δ-hyperbolic space. If the curvature is − K everywhere rather than −1, the areas above should be multiplied by 1/ K and the lengths and distances should be multiplied by 1/ √ K. R = ln 3 = 1 2 ln 3 = artanh 1 2 = 2 artanh ( 2 − 3 ) =, with equality only for the points of tangency described above.Ī is also the altitude of the Schweikart triangle. The inscribed circle to an ideal triangle has radius.In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties:ĭistances in an ideal triangle Dimensions related to an ideal triangle and its incircle, depicted in the Beltrami–Klein model (left) and the Poincaré disk model (right) An ideal triangle is the largest possible triangle in hyperbolic geometry.An ideal triangle has infinite perimeter.The interior angles of an ideal triangle are all zero.All ideal triangles are congruent to each other.Ideal triangles have the following properties: The vertices are sometimes called ideal vertices. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Type of hyperbolic triangle Three ideal triangles in the Poincaré disk model creating an ideal pentagon Two ideal triangles in the Poincaré half-plane model
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