The funny patterns we see when regular patterns overlap at an angle or at different frequencies are called moiré. Here you can generate all sorts of moirés to see what's going on. The word "see" is important. Basically everything gives a moiré but the human eye has a threshold below which it's not visible. By using a trick of Fourier Transforms, we can calculate how strong (or weak) the moiré can be.
So, enjoy playing with grids, lines and dots of various frequencies and angles. But on a slow device don't get too ambitious too quickly!
If you've started this app for the first time, you will see a classic moiré effect. The pattern of verticle lines in the top-left graphic is overlayed with the same lines at an angle of 15°. The eye sees this as producing a set of black and white lines at an angle of 7.5°. The calculated moiré is shown in the bottom-right graphic. The aim of this app is to get you used to simple moiré effects then build up to those typically found in printing.
The first thing to try is moving the Angle 2 slider. As the angle gets higher the moiré gets weaker - a key feature of moiré. Also try changing %, the thickness of the line
Set Angle 2 to 0; there is no moiré. Now change Frequency 1. You get a very strong moiré. Frequencies are relative (the absolute frequency is chosen to look OK within the limits of a simple app), so setting Frequency 1 to 0.5 means that the frequency of the reference has been halved, with the others staying constant.
Returning to the original settings, see what happens with a Grid and then with Dots - where we are starting to look more like printing. Although a 15° moiré is strong with the Grating, with Grid and Dots it is visually weak. The frequency of the moiré is too high to bother us. Which is a good thing because this is one of the angles we are forced to choose between some of our CMYK colours (taking into account that 0-->15 is the same as 0-->75 because 0 is equivalent to 90.
The startup default is that the 4 colours are KKKK. You can select any of the variants of CMYK at any time.
Before adding more colours/angles (by clicking the relevant Active box), a warning about speed. The analysis involves a couple of FFTs (using the wonderful TurboMaze routines from Anthony Liu) which slow things a bit, but the biggest drag is creating lots of dots. Even on a fast laptop, do all the setting up of angles etc. using the Grating setting, then swap to Dots to see the effect.
You can set up a 4-colour print and see the nice rosette when you have the classic 0, 15, 45, 75 angles. A rosette is clearly a moiré but its visual intensity is low, which is why we are at ease with it. Imperfections in the rosette show up rather easily. It is arguable whether the unsatisfactory look is moiré or not. Because the 15° moiré is somewhat more visible than others, choosing a weak pair of colours (e.g. ones that don't feature too much in the print) can help further reduce the visibility.
The X effect
Finally, what about the X options? In screen printing, the mesh imposes its own square influence on the print and at some combinations (with a 7.5° angle being a common offset), a moiré can be highly visible. Similarly, in flexo, the hexagonal pattern on the anilox roller can give a worrying moiré. The X option allows you to explore these possibilities. The Frequency X setting is relative to the main frequency and has a larger range to cover a wide set of possibilities, though with obvious numerical limitations (plots become meaningless if the value is greater than 2).
The Rule of 4
One of the fundamental rules of moiré is that it is visible if the frequency of the original is more than 4 times the frequency of the moiré. If you return to the original Grating setting with just the second angle active and set the angle to 10°, you can count ~5 lines in the moiré image, and given that the original has 32 lines that is 32/5=6.4, with a strong moiré. If you go to 16° you find ~9 lines, and 32/9 = 3.6, which is below 4; and the moiré is very light. At 15° there are 8 lines (32/8 = 4) and you are right on the border. Although this rule is not infallible, it is a remarkably good rule of thumb.
It is an obscure route from the Arabic word for cloth from an Angora goat, mukhayyar, through the patterns one sees in fine fabrics to our word moiré, but that is where the word comes from.