study is a continuation of the analysis of the Manton Drove Crop Circle.
While this study does not offer new insight for the polar clock's
date/time, the design maintains focus on the role of Pi in this crop
This geometry promotes the theory that a squared circle can be proven if
these dimensions are reflected in two adjoining sides of a geometric
object: 1/2 the square root of Pi and 1/2 the square root of 2. The
inscribed tan quadrilateral, named "Karotumus" (for its tasty discovery;
accent on the second syllable), appears to include these dimensions.
Intuitively, the 1000-units-diameter blue corral (the circle) should
limit the infinity of Pi, but does it? .... or is this the "Enigma of
(Focus on the circle squaring properties of the right triangle
within Karotumus: angles = 62.402.., 27.597.., 90 degrees.)
This new concept of Pi will intrigue geometers as well as crop
circle aficionados. Please add the following comment after the
sentence "Focus on the circle squaring properties ...":
Consider also that the Pythagorean Theorem confirms these
(1/2 square root of Pi = 0.88622692545275801364908374167057...)
a = 463.25137517610424292137983379471... (top horizontal
b = 886.22692545275801364908374167057... (left vertical
c = 1000 units (hypotenuse; radius of large magenta circle)
a˛ + b˛ = c˛ = 999999.99999999999999999999999999...
Here's the latest research on the Manton
Drove crop circle's geometry as it relates to the Barbury
Castle's Pi crop circle (June, 2008).
I was curious about the rounding of Pi
(from 3.1415926535... to 3.141592654...) in the Pi crop circle
since I believed that a number is not rounded if an ellipsis is
used to continue that number. But Bert Janssen's conjecture on
those "three little circles" in the Pi crop circle provided
great insight: that ellipsis may also convey circle squaring
Interestingly, the circle makers clearly
demonstrate their geometry expertise by choosing this tenth
digit of Pi: not only does this tenth digit (3) properly round
to 4, but the relative diameters of the "three little circles"
(the ellipsis) within that 36-degree angle appear to
Today, preliminary "reverse
engineering" of the Barbury Castle's Pi crop circle's "three
little circles" around the circle-squaring 62.403.. degree angle
(from the Karotumus research) produces a fascinating
correlation: the tangent between the two smaller circles is very
close to the 62.403.. degree angle! (see above BCPi image).
Perhaps, this Pi crop circle's subtle clue
for squaring a circle might soon be confirmed and be followed by
a geometric solution for proving the square (via the
non-isosceles right triangle which includes the 62.403..
A great day for triangular Pi !
: The 4/3/2 relative diameters are actually the
three circles' tangent-to-tangent chord lengths.
This Symmetry design, further
exploration of the Manton Drove crop circle, appears to
provide important clues for "squaring the circle". Focus
on the two large, green, scalene triangles and consider
the diameters of the small circles, especially when
these circles are analyzed in groups of two (one small +
one slightly larger). But the horizontal centre (not
shown) of the green parallelogram may contain the more
To celebrate the final day of the 13th baktun (o oo
oo o o oo), here's a design that colourfully
articulates the significant lines in a squared
Humour: When asked to summarize his articulation of
the geometry of a squared circle, Pythagoras just
winked and blurted "My Pi, My Pi".
To celebrate the beginning of
the 14th baktun, here's a design that
colourfully articulates the symmetry of a
squared circle. This design, rich with "new era"
symbolism, proffers that circles and their
squares have impressive alignment with universe
significance. The full lengths of the light blue
lines on this 2000-units-diameter circle are
equal to the square root of 2 * 1000 (=
A personal interpretation of
this new era symbolism:
"What is important in life is
not what you can do or have done, but always
what you believe that you can do ... for your
life's work often complements what you
The squared circle symbolizes
that which is believed to be impossible and the
snowflake symbolizes the uniqueness of every
human being. The design is coloured to emphasize
the inscribed parallelogram, that both squares
the circle and symbolizes the unseen, parallel
spiritual realities of our material universe.
A 14th baktun, first day perspective on the
challenge of squared circles. This design
symbolizes the expected discovery of a
solution to this Greek challenge during the
next 143999 days of the 14th baktun.
"In 2013 we taste the
cheese, moving closer."
These two squared circles
(light blue; the larger created first)
display the essence of squared circle
geometry and suggest that only three points
are required to square the circle. In the
large, green scalene triangle, the left
diagonal line (essence of a square) and the
bottom horizontal line (essence of a circle)
reveal the square roots of 2 and of Pi,
respectively. Does this design display the
elusive and transcendent geometric
associations that prove the circle's square?
(large circle's diameter = 2000 units;
smaller circle's radius = 1/2 square root of
2 * 1000)
Interestingly, this design's geometry
appears to confirm a thesis about proving
the square of a circle:
When half of the side
length of the larger square (1/2 the square
root of Pi) equals half of the
diagonal length of the smaller square (1/2
the square root of Pi), both circles are
squared. This equal condition exists only
when both circles are correctly squared. And
Pi does not constrain this geometric proof
of a squared circle since both line lengths
are equal, despite the number of decimal
digits in Pi ... if the thesis is correct.
Zdenka Pávková's June
2012 analysis of the Manton Drove crop
circle reinforced my belief that one side of
a circle's inscribed square would be
required to "anchor" a geometric proof. The
smaller light blue circle in this design
successfully incorporates one side of an
inscribed square (full square not
The precision of this
circle squaring method is directly related
to the exact length of the left diagonal
side of the scalene triangle (this side
represents the square root of 2) and to the
intersection of a point on the circle's
circumference with a point on the length of
the golden "T" pin (this side represents the
square root of Pi). This squaring method
begins with the golden circle, then a chord
(one side of an inscribed square), then the
"T" pin that effectively creates the scalene
Interestingly, the other
two squared circles seem to be clues to
geometrically proving the square of the
golden circle: each diagonal side of the
green scalene triangle effectively squares a
circle when the other side of the triangle
is the diameter of that circle.
The small red circle
helps create the symbolism of an exclamation
mark, the red triangle an allusion to
trigonometry that also squares this circle.
Correction: The "T"
pin concept as presented in the recent
design is not correct - the scalene
triangle cannot be created this way.
"Close, but no cigar!" (of US origin
from the mid-20th century when
fairground stalls gave cigars as
prizes). But research still shows that
the scalene triangle - when drawn
precisely - squares the circle and its
diagonal sides facilitate the
drawing/squaring of two additional
Presenting the essential scalene
triangle of squared circle geometry,
suggesting that only three points
are required to square the circle:
- pivot point is at the top left of
- golden circle (diameter =
2,000,000 units) is given.
- green line at 45 degrees (length
unknown) is given.
- horizontal line at 0 degrees
(pivot line, length unknown) is
- shortest side of the scalene
triangle = side of an inscribed
- light blue arc is a portion of a
circle (radius = golden circle's
- dark blue arc is a portion of a
circle (radius = golden circle's
- rotate golden circle and its chord
counter clockwise to create scalene
- many lines shown cannot be drawn
until the rotation is stopped.
- knowing when to stop the rotation
is the challenge!
The dark blue arc
(circle's diameter = 4 mil. units)
is the "pivotal" arc (the more
important geometric association with
the pivot point at the top left of
the diagram) because the large, dark
blue, Pythagorean triangle confirms
the "square of the circle": the
golden circle (diameter = 2 mil.
units) is squared by the small red
Pythagorean triangle at the top of
circles, golden and dark blue, are
squared by the same Pythagorean
triangle (same angles): the area of
the square of the dark blue circle
is exactly four times the area of
the square of the golden circle. And
the golden "I" highlights two equal
line lengths: length of square's
diagonal (midpoint to midpoint) =
length separating the circle's two
chords (midpoint to midpoint).
This Scalene Progression design
finalizes this study and
commemorates Pi Day 2013 (now 10
trillion digits of Pi).
The design reveals that only 3
points are required to "square the
circle" (left side of green triangle
and of horizontal side reflect the
square roots of 2 and of Pi; large
circle's diameter = 1 million
units). The diameters are
decremented by half the square root
of 2: lengths from largest to
smallest = 1000000.0, 707106.781,
The red triangle provides a new "no
Pi" trigonometry formula for the
area of the circle. With a right
triangle having an hypotenuse as the
radius of a circle and the cosine
angle (near the circle's center) =
degrees, the trigonometry formula
Area = ((cos
) x diameter) squared
That this cosine angle might
complement Pi digit-for-digit in a
perfectly squared circle ... and
that this Pythagorean triangle
exists in every squared circle ...
hints that Pi Day has always had a
certain Pythagorean ambience.
Perhaps, Pythagoras even
contemplated this right triangle's
always-present, geometric sibling: a
circle-squaring scalene triangle.
The lines in this Scalene
Progression design directly or
effectively square the four
This geometry and the supporting
trigonometry hint that squaring the
circle may be less "impossible" than
finding the limit of Pi digits.
After all, the three points of a
certain scalene triangle seem to be
a sufficient number on a circle to
determine its square (although 8
would be the maximum required since
a circle's square rests upon 8
points ... only).
Which leads us to this Pi Day
perspective: Some Pi are square,
some are round, but no Pi is