A nested series of 72o-18o-90o
“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 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 72o,
18o and 90o :
Now sin(18o) = (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 18o, will equal 1 (one).
The crop artists next drew an imaginary, long thin line (marked in
red), so as to bisect both that 18o
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 9o and 9o 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 45.885o
N, longitude 40.140o 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 90o angle (labelled in white) with our
hypotenuse of length 2Φ on the left. It then proceeds horizontally, all
the way to the original 90o angle from our landscape
“triangle” (shown in the first slide above). There it divides that 90o
right-angle into two smaller angles of 18o and 72o
(labelled in white).
Now we have created a second, somewhat-smaller right-angled triangle
with three internal angles of 72o, 18o and 90o.
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 72o, 18o and 90o.
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
by a similar
geometrical process, which involves the repeated subdivision of a large
square into many smaller rectangles or squares
wikipedia). If we divide each term of that infinite series by
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
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
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
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
Please study this crop picture as well, and try to learn something new
(Dr. Horace R. Drew)
P.S. A geometrical derivation for sin (18o) = (sqrt (5) -1) / 4 = 1 / 2Φ is given here (see mathforum).