A nested series of 72^o-18^o-90^o “half-golden triangles” were drawn within a distinctive landscape feature of similar shape near Fastovetskaya, Russia on June 19, 2019. Here is a truly good example of how to learn “interesting aspects of geometry” by studying crop circles!  

One day after a much-publicized but poorly-made “lesson in geometry” was planked in crops by two busloads of students at Blaesheim near Strasbourg in France (see Alsace comments   or Alsace articles  ), a real and much-more clever “lesson in geometry” was kindly drawn for our worldwide education, by the E.T. crop artists near Fastovetskaya in Russia. Would you like to understand how it works? 

First, by careful inspection of Earth’s surface from the air (or perhaps from space), they found a suitable landscape feature for their new field drawing just outside of Fastovetskaya, Russia. There we can see (using Google Earth) a right-angled triangle with three internal angles of 72^o, 18^o and 90^o : 

Now sin(18^o) = (opposite / hypotenuse) = 0.309016994 = 1/ 2Φ exactly, where the golden ratio phi Φ = 1.61803399. Thus if the length of our hypotenuse equals 2Φ, then the length of our shortest side, opposite to the smallest angle of 18^o, will equal 1 (one).  

The crop artists next drew an imaginary, long thin line (marked in red), so as to bisect both that 18^o angle (on the right), and also the short opposite side (on the left). This thin, dashed red line creates two new, half-width angles of 9^o and 9^o on the right, and splits the short opposite side into two equal lengths of 1/2 and 1/2 (labelled in yellow).  

Finally they drew a new crop picture exactly where indicated by a small yellow circle in the slide above: most of the way along that imaginary red line to the left. Let us try to understand now why they drew it there, and what they were trying to teach us!  

In the next slide, we can see a real aerial photograph of that large triangular field, just south of Fastovetskaya at latitude 45.885^o N, longitude 40.140^o E. Now our hypotenuse of length 2Φ appears at upper left, while our short opposite side, which has been bisected into two equal lengths of 1/2 and 1/2, appears at lower centre. The long, thin red line which they used to bisect our short opposite side (in the slide above) is vertical.  

Finally, they drew a new crop picture near the lower-central part of that large, slightly-red, landscape “triangle”. It includes four shiny, flattened circles of varying size, along with several thin lines as indicators or markers:  

If we wish to understand what this new crop picture is trying to tell us, then we will need to draw two more long, thin, red dashed lines! The larger of those two lines creates a 90^o angle (labelled in white) with our hypotenuse of length 2Φ on the left. It then proceeds horizontally, all the way to the original 90^o angle from our landscape “triangle” (shown in the first slide above). There it divides that 90^o right-angle into two smaller angles of 18^o and 72^o (labelled in white).  

Now we have created a second, somewhat-smaller right-angled triangle with three internal angles of 72^o, 18^o and 90^o. Since the length of our new “hypotenuse” (labelled in yellow below) is just (1/2 + 1/2) = 1, then the length of our new “short opposite side” (on the far left) becomes 1/2Φ (labelled in yellow), by a rule of “similar triangles”.  

Very interesting, is it not? Next let us create a third, thin red line, which lies perpendicular to our second, thin red line, just discussed above. It connects down to where our first, long, “bisecting” red line connects to a short opposite side, which has two lengths of 1/ 2 and 1/ 2. Once again, we have created another new, even-smaller right-angled triangle with three internal angles of 72^o, 18^o and 90^o. 

Since the length of our third new “hypotenuse” (labelled in yellow) is only 1/2, then the length of our third new “short opposite side” (on the left near the centre) becomes just 1/4Φ (labelled in yellow).  

This process of creating new “half-golden triangles” could continue indefinitely, by drawing even more thin red lines, each shorter than the previous one. The lengths of our new “short opposite sides” would then decrease as an infinite series equal to 1 + 1/2Φ + 1/4Φ + 1/8Φ + 1/16Φ, etc. = 1.618, so as to regenerate by convergence the golden-ratio phi Φ.  

Any infinite series 1/2 + 1/4 + 18 + 1/16, etc. converges naturally to 1.000 by a similar geometrical process, which involves the repeated subdivision of a large square into many smaller rectangles or squares (see wikipedia). If we divide each term of that infinite series by Φ, then we get a total of 1/Φ. Next if we add 1 to give (1 + 1/Φ) = 1.61803399, then we regenerate the golden-ratio phi Φ exactly. Here we have subdivided a large triangle into many small triangles, rather than squares or rectangles, to get that interesting result.  

Let me ask you please, dear students of geometry in France, or other interested followers of crop-circles worldwide:  

Did you learn anything new from this kindly-offered “lesson in geometry”?  

I know that I did, and I have a Caltech Ph.D. in physical science!  

Who do you think made it: “two men with rope and boards” or someone far more intelligent?  

Do you wish to remain ignorant, like black slaves in the Old South (because it was “against the law” to teach them), or would you like to learn more? Why was the Christian Bible only written in Latin or Greek for thousands of years, so that regular people would not be able to read it? 

Another beautiful crop picture, which coded for the golden ratio Φ to ten digits 1.61803399, appeared near Oare in Wiltshire, England on June 21, 2010 (see cropcircles.lucypringle.co.uk). Please study this crop picture as well, and try to learn something new and worthwhile!   

Red Collie (Dr. Horace R. Drew) 

P.S. A geometrical derivation for sin (18o) = (sqrt (5) -1) / 4 = 1 / 2Φ is given here (see mathforum).