Basically, Snells law tells us how a ray of light will behave as it moves from one optical medium to another as it crosses the interface of the two media at a specific angle.
Consider the diagram below. Here we have two media of refractive index n(1) and n(2) in intimate contact separated by an interface of infinitesimal extent (thick black horizontal line). We have a ray of light moving from one medium to the other (from n(1) to n(2)), such that the ray incident upon the interface makes an angle Q1 with respect to the normal (i.e. the line perpendicular to the interface; the dashed vertical line). Note i use Q here in substitution for the Greek symbol theta. If the refractive indices of the two media are different, then the incident ray will be refracted upon exiting the interface, with the angle of incidence dependent upon the ratio of the two refractive indices as well as the angle of incidence. This is described mathematically as n(1)sin(Q1) = n(2)sin(Q2).
So, if we take your original diagram, and assume that the interface is a horizontal line bisecting G2 through its central point, then a ray of light projected from G1 towards this centre point will make an angle of (Q1) = 43.33 degrees (as per your figures) with the normal to the interface. If we assume that this is a model for diffraction of light between two media, and that the diffracted ray makes an angle of (Q2) = 31.95 degrees (as per your figures), and if we know the refractive index of n(1), we can calculate the refractive index n(2) using Snells law.
For the figures given, i calculated n(2) at 0 C (since this was the value for air I had at the time), I got n(2) = 1.297.
For "moist air" at -40C, at 1 atm and humidity of 50%, then the range of values that the refractive index of air can take for visible light (700 nm to 350 nm) is from 1.000336677 to 1.000349269. (Note that humidity at this temperature has little effect on the refractive index). You can play around with the figures here [www.wolframalpha.com]
If we use these ranges of refractive index for air at -40, as well as your angular figures, then we can calculate n(2) to have a range from 1.297161 to 1.297177. So regardless, the value doesn't change to 4 decimal places, and is 1.2972.
The refractive index of ice at 1 atm, and at -40, in the same visible range (700 nm to 350 nm) is between 1.3069 and 1.3249 (again calculated from here [www.wolframalpha.com]). This is why I said it is close and not exact (its about 1% out).
I we work out the deviation of the refracted angle assuming that in your diagram the upper medium is air, and the lower medium is ice at -40C, and assuming the incident angle is correct, then we would expect the refracted angle Q2 to have the range of 31.2053 (for 350 nm) to 31.6841 (for 700 nm). So your refractive angle is two big for an exact match. Of course there could be an error in the incident angle. And one could back calculate what the range of incident angles would be assuming that the refracted angle is correct (44.496 for 350 nm and 43.737 for 700 nm).
However, if there are errors then they will likely be in both incident and refracted measurements, but because they are coupled, then all we can say is that the range of the ratio of the two sines i.e. n(1)/n(2) = sin(Q2)/sin(Q1) will be between n(1)/n(2) = 0.75504 and 0.7654271 for 350 nm to 700 nm range respectively (or n(2)/n(1) = 1.3244 to 1.30646). Finding the minimum error in both angles Q1 and Q2 is then tricky involving what amounts to systematic trial and error.
The logical conclusion from all this is as follows. Using your values for the angles, and your hypothesis that the Giza represents forms of water and air at a specific temperature (-40 C), then it COULD encode refractive properties of light of these optical media. I say could because the values are not exact, but they are are close. However, to accept such a hypothesis as correct, other evidence from the archaeological records would be required to support such a conclusion, as well as accepting that the alignment is out considering other alleged precisions that these structures have. I say this because you are basing one hypothesis (that the layout represents encodes refraction of specific media) upon an initial unconfirmed hypothesis that the specific structures encode elemental information.
I still remain to be convinced, but I do think that it is interesting.
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