To perceive two dots as such, the observer must be able to recognize a space between them. The two dots in an umlaut in small print are, of course, closer together than two stars in the night sky. The perceived distance between the two dots also plays a key role. This depends on the viewing angle at which the observer sees the two dots. A human eye with normal vision can resolve a viewing angle of 1/60 of a degree.
In other words, if you want to see two dots on a sheet of paper, you can simply move closer and thereby increase the viewing angle, or you can bring two stars closer to you with a pair of binoculars to also increase the viewing angle. If this is no longer enough, you have to get a magnifier or even a microscope. However, "zooming in" is not necessarily beneficial with the latter as another resolution limit applies — which was discovered by Ernst Abbe 138 years ago. It states that the resolution also depends on the wavelength of the light.
Abbe's resolution limit of the optical microscope of around 200 nanometers was long considered insurmountable. That is until optical tricks were used to bypass it. Through the skilled use of fluorescence, it is now possible to resolve much smaller structures — down to 20 nanometers. However, it is usually enough if you move a bit closer to read the fine print.
February 23, 2011