It looks like math-fun mangled the formula. The index of refraction is sqrt(2-(r/R)^2). -- Gene On Wednesday, March 20, 2019, 9:19:25 AM PDT, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote: The Luneburg lens is a ball of radius R whose index of refraction varies as √2-(r/R)2. A parallel beam of light is exactly focused, in the approximation of geometric optics, to a point on the surface opposite the direction of entry. -- Gene On Tuesday, March 19, 2019, 8:28:44 PM PDT, Keith F. Lynch <kfl@KeithLynch.net> wrote: Decades ago, it was generally believed that no solid could be transparent to light for more than a few meters. Today it's known that non-transparency in transparent is due to a small handful of causes, including crystal boundaries and microscopic inclusions of opaque material. Today we've learned that extremely pure amorphous glass fibers can be transparent for tens or even hundreds of kilometers. That raises the question of what the moon would look like if it were perfectly transparent. Assume it's the same size it is now, in the same orbit, and is perfectly spherical. Assume it's immune to micrometeoroids, tidal stress, and radiation damage. Spheres don't make very good lenses, but that's only because they typically have a constant refractive index. If the refractive index varies smoothly with the distance from the center, it could have a perfectly sharp focus. How would the refractive index have to vary with depth for the moon to have a focal length equal to the distance to Earth's surface? (Is it practical to make (small) lenses on this principle today? If not, why not?) If the problem is underspecified, I'll further specify that the refractive index at the moon's surface should be 1, the same as the surrounding vacuum. That will prevent the surface from being reflective, since it's sudden changes in the refractive index that cause reflections. This would revolutionize astronomy, since it would turn the Earth- moon system into a giant telescope. You'd need to take very brief photos due to the speed of Earth's rotation. Either than or take photos (or naked eye observations) from a very fast jet that compensates for both Earth's and the moon's motions. If your first thought is that is that the moon would be a burning glass, leaving a trail of destruction during every solar eclipse, think again. In a week I will explain why that wouldn't happen unless someone beats me to it, which I hope they will.