Why does this Moiré pattern look like this?
You'll need to understand the sampling theorem. In short, each signal has what we call a spectrum¹, which is the Fourier transform of the signal as it comes in time domain (if it is a time signal), or spatial domain (if it is a picture. Since the Fourier transform is bijective, a signal and its transform are equivalent; in fact, one can often interpret the Fourier Transform as change of basis. We call that "conversion to frequency domain", since the Fourier transform's values for low ordinates describe the things that change slowly in the original (time or spatial) domain signal, whereas high-frequency content is represented by Fourier transform values with high position.

Generally, such spectra can have a certain support; the support is the minimal interval outside of which the spectrum is 0.

If you now use an observing system whose ability to reproduce frequencies is limited to an interval that is smaller than said support (which often is infinite, by the way, and always is infinite for signals that have finite extension in time or space), you can not represent the original signal with that system.

In this case, your picture has a certain resolution – which is, in the end, the fact that you evaluate the value of your function at discrete points in a fixed, non-infinitesimal spacing. The inverse of that spacing is the (spatial) sampling rate.

Thus, your picture cannot represent the original signal – it's simply mathematically impossible that the mapping of underlying function to pixels is truly equivalent to the original function, since we know that in this case, the total range of frequencies representable by your evaluation at discrete points ("sampling") is half the sampling rate, and thus, something must go wrong with the part of your signal's spectrum that is above half the sampling rate.

What happens is, in fact, that the spectrum gets aliases – every spectral component at a frequency $f_o \ge \frac{f_\text{sample}}{2}$ gets "shifted" down by $n\cdot f_\text{sample},\, n\in \mathbb Z$, so that $|f_o-nf_\text{sample}| &lt;  \frac{f_\text{sample}}{2}$. In effect, that leads to "structure" where there (feels like) shouldn't be some.

Take the "large" structures from your picture that I've painted green:



It certainly looks like there is low-frequency content here - but in reality, it's just the high-frequecy content at frequencies $&gt;\frac{f_\text{sample}}2$ that got aliased to low frequencies, since it was close to an integer multiple of the sampling rate.

So, yes, you can predict the artifacts that happen to a 2D signal when being sampled by comparing its Fourier transform to the bandwidth offered by the sampling rate. 


¹ this might be different from the spectrum as used in linear algebra to describe the Eigen-properties of operators.