The Geometric Viewpoint

geometric and topological excursions by and for undergraduates
 

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Curvy Television

Television screen size became a status symbol in the American household soon after TV’s adoption
Televisionin the 1940’s, and remains so today. There are many equivalent ways to characterize screen size; today’s standard is by the diagonal length of the screen. Everyone understands that this length is related in a familiar way to a screen’s area. With today common 16:9 aspect ratio (horizontal length:vertical length), the screen area {A(l)=\alpha l^2} where {l} is the diagonal length and {\alpha=337/144}. In other words, the area varies with the square of the diagonal length.

This polynomial relationship, however, is a feature of euclidean space. TV screens placed on hyperbolic and spherical surfaces lead to surprising differences in area relative to diagonal length. Curvature will play an important role in illustrating the discrepancy between euclidean, spherical, and hyperbolic geometries. To see this, let us imagine covering our television screen with a piece of paper. Using Ilmari Karonen’s imagery from Stack Exchange we shall lay a large sheet of paper over our euclidean television, and cut it down to size.

Now imagine that the surface of our TV is lying on a sphere. We will try to lay our cut-to-size sheet over the screen, a curved surface. To anyone who has used paper-mache, it’s immediately apparent that the paper will need to wrinkle and fold up around itself to fit to the surface. There is simply too much paper and too little sphere to go around. What is happening? The intuitive answer is that there is simply not enough space on our spherical surface. This simple fact hints at a sobering idea for cartographers; there exists no flat map that is locally isometric to the Earth at any point. No matter how carefully you restrict your vision, a flat map will be always be wrong. Gauss formally proves this in his Theorema Egregium. It turns out to be a consequence of the constant positive curvature of spherical space, as opposed to zero curvature in euclidean space.

Tangent plane intersecting a surface to create a curve.

Tangent plane intersecting a surface to create a curve.

Curvature, also know as Gaussian curvature {K} at a point {\alpha} on a surface, is defined as the product {\kappa_1 \cdot \kappa_2 }, where {\kappa_1} and {\kappa_2 } are the principal curvatures of {\alpha}. Finding the principal curvatures at {\alpha} can be visualized by the following method: First, find the normal vector to the surface at {\alpha}. Next, look at the intersection of a plane containing the normal vector with the surface. This intersection is a curve on the surface. As you rotate the plane around the normal vector, the intersection curve will change. The principal curvatures are the maximum and minimum curvature of the intersection curve at {\alpha} as you rotate the plane.

On a plane, the intersection of a normal plane to a point on the surface will be a line. Since the curvature of any line is zero, {k_1 \cdot k_2 = 0} at all points, resulting in constant zero curvature as we have already mentioned. Similarly, on a sphere, any plane containing a normal vector will intersect the sphere along a great circle (geodesic) of the sphere. Since any two great circles have the same (non-zero) curvature, we see that {\kappa_1 \cdot \kappa_2 = \beta > 0} at all points, i.e. a sphere has constant positive Gaussian curvature.

Hyperbolic Spaces locally look like a saddle point.

Hyperbolic Spaces locally look like a saddle point.

Hyperbolic space challenges this definition of curvature. How can we use our method on a surface which cannot be embedded in euclidean 3-space? One way to visualize hyperbolic space is as a collection of points each of which locally looks like a saddle point. If we take a plane containing a normal vector to a saddle point, and intersect it with the saddle, we see that the curve on the saddle radically changes as we rotate the plane. The maximal and minimal curvatures of the intersection curves will be positive and negative respectively. Therefore, {K= \kappa_1 \cdot \kappa_2 < 0}. Since each point of hyperbolic space locally looks like an identical saddle, we see that hyperbolic space has constant negative curvature.

Largest Rectangle possible in Hyperbolic space

Largest rectangle possible in hyperbolic space

What happens when we embed our television in a hyperbolic surface? The first thing to realize is that {2\pi} is the maximum area of a rectangular screen. This is similar to how a spherical rectangle is bounded by the size of the sphere. However, we do not want to restrict the size of our television. Bigger TV’s are always better! Instead, we will investigate the area of circular televisions, or hyperbolic disks, which have no upper bound. If we calculate the circumference of a hyperbolic circle in terms of its radius {r}, we find that {Circ(r) = 2\pi\sinh(r)}. Now we can simply integrate this value from {0} to {R} to find the area {A(R)} of a hyperbolic disk, {\int_0^R 2\pi\sinh(r) dr = 4\pi\sinh^2(\frac{1}{2}R)}. This can be rewritten as {A(R) = \pi(e^{R}+e^{-R})-2\pi}, where {R} is the radius of the disk.

Notice that in hyperbolic geometry, circular area grows exponentially with the radius of a disk. Our TV’s area, therefore, will grow much faster as we increase its radius than it would in euclidian space. In other words, there is more space in a hyperbolic disk than in a euclidean disk of the same radius. This is the opposite of our result from spherical geometry, and is a good example of the connection between Gaussian Curvature and the “amount of space” in a given surface. The book Crocheting Adventures with Hyperbolic Planes is a good reference for more information about curvature and the hyperbolic plane.

The connection between curvature and space has been used by physicists to investigate our universe. A major question in the field of astronomy is whether or not our universe, on a large scale, has positive, zero, or negative curvature. One method to test this theory is to calculate the density of galaxies at different distances from our own galaxy. A constant density would be reminiscent of a flat (zero-curvature) universe. However, if we observed a growing density of galaxies as we looked further away, this would hint at negatively curved space, because we are observing “more space” with our instruments, and hence we would see more galaxies. Similarly, a shrinking density of galaxies would suggest a positively curved universe. Using this method, physicists have been able to rule out the possibility of a large curvature in our universe, however the difficulty of accurately measuring densities has restricted the ability to make a stronger claim.

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