The physical phenomenon behind the 3D-SIM technique can be compared to what happens when you scan a printed picture.
During the process of scanning, the original picture is subdivided into a pattern of points (the pixels). If you scan a picture that is printed, it already consists of pixels. Now, if the points that are scanned are not exactly “in line” with the points that are printed, if they are shifted against each other, then you get a new pattern, that looks different. This is the so-called Moiré effect. The Moiré pattern is usually more coarse, bigger than the original pattern. (In everyday life, this effect can be seen when someone is wearing clothes with a fine pattern on TV. Instead of the original pattern you see “dancing” lines.)
In the new microscpy technique of 3D-SIM, the laser beam that is used for imaging is divided into several components, which are focused on a certain spot in the sample. There, the waves interfere, which leads to the formation of a pattern, which is different from the pattern of the sample, i.e. the substructures of the sample. Both patterns are so small that they cannot be imaged, but the combination of these two patterns leads to the formation of a coarser Moiré pattern, similarly to the example of scanning a printed picture. This Moiré pattern can be registered through the microscope. Since the inteference pattern is known, in theory, you can calculate the substructure from the Moiré pattern. However, in order to gain enough information to be able to calculate and visualize the substructures, it is necessary to image the sample from different angles. (Like in mathematics, where an equation system that contains more variables than equations cannot be solved.)