Many novel kinds of mathematics have been drawn in English crop pictures, for example “pi to ten digits” at Barbury Castle on June 1, 2008 (see www.dailymail.co.uk or www.telegraph.co.uk). That puzzle was not solved for over a week, while images of the crop picture spread widely over the Internet to millions of people around the world. Finally Mike Reed, a retired astrophysicist living in North Carolina, realized what it meant, then informed Linda Howe and myself. Here we will show four other mathematical puzzles which were drawn in crops during the summers of 2000, 2009 or 2012. No one has realized what they mean until now. They are all based on complex triangular geometries.
The first “triangle puzzle” to appear in crops was drawn near Silbury Hill in June of 2000. It came down in two separate phases on June 2 then June 4:
This puzzle relates the number of vertical levels in Pascal’s
Triangle, to the number of small overlapping triangles which may be
counted inside (see
jwilson.coe.uga.edu). The crop picture shows an example for
I am calling this “Pascal’s Triangle”, because we will return to that particular geometry in other slides below.
The next two “triangle puzzles” to be drawn in crops during 2009 or
2012 were more difficult to understand. In order to explain what
they mean, first we need to show an example of overlapping
triangular geometries. How many small triangles can you count,
within a large triangle that has been subdivided into
The answer is usually
Now everything is going to get a bit harder! A “triangular pyramid”
which appeared in crops near Chesterton Windmill on July 9, 2009 was
subdivided on each face as
Ideally for this example, we might expect to count 15 overlapping
triangles along each face, which is a value intermediate between 8
triangles for
Now one can count just 13 triangles on each face, instead of the expected 15 for ideal symmetry:
There are 12 small triangles which subdivide each face. Then we must add one large triangle for the whole face, to give a total of 12 +1 =13. This crop picture was drawn in a field near Chesterton Windmill, along with a small “key” motif which pointed towards the windmill. It showed six tiny lines on the top, which resemble a six-lined vertex from the triangular crop picture:
The “key” to understanding this puzzle is to realize that the
triangular crop picture would possess
The total number of small, overlapping triangles therefore equals 4 x 13 = 52, even though a lower face of the pyramid with 13 triangles remains “hidden”. Could there be some intent here to refer to “52 months” which separate July 2009 and December 2013, the latter date of which has been coded in other crop pictures? (see /fringe2013k) Those two coded numbers of “13” or “52” have distinct connotations in terms of ancient Mayan culture. There are 13 baktuns in one part of the Mayan Long Count calendar, while there are 52 panels on each face of the Pyramid of Quetzalcoatl in Chichen Itza:
Those 52 panels on each face of the Mayan pyramid represent “52 years” in their Sun-Venus calendar. Once again, might the Chesterton Windmill crop picture be referring to the “52 months” between July 2009 to December 2013? Will we perhaps see a large triangular UFO then?
A “large triangular UFO” was drawn in crops at West Kennett Long Barrow on August 13, 2013. It looks much like the crop picture drawn at Chesterton Windmill on July 9, 2009. Later in the same month of July 2009, another “triangular pyramid” as drawn in crops also resembles a triangular or pyramidal UFO.
We will show other interesting “triangle puzzles” below, but first we need to prepare the way with further explanatory information. Two different geometry diagrams from the summer of 2012 suggested “extra dimensions”. The first at Liddington Castle on July 1, 2012 (then July 21) showed the image of a cube passing through our 3-D space from other dimensions (see fringe2013b):
In their first diagram on July 1, we can see one corner of that cube “rising above” the plane of the crop field. Then in their second diagram on July 21, we can see a corner of that cube “falling below” the plane of the crop field. Later on August 26, 2012, they drew a very intricate 3-D cube which shows a similar change of perspective. We can apparently visualize the front corner of that cube by either “looking down” or “looking up” (see also hackpenhill3). How did the crop artists create such an amazing illusion? As shown in the next slide, they achieved this by changing the nature of visual perspective in its central parts:
Now quite remarkably, one of those geometry diagrams from the summer of 2012 also contains a “triangle puzzle”, like the example from Chesterton Windmill which we just studied above. We will be able to study this new triangle more easily if we “stretch” it away from the curved shape drawn in crops, back into a simple linear shape:
This new triangle is a clever variation away from the idealized case
of
Again we can count 25 small, overlapping triangles within each face. Then we need to add one more for the whole face to reach a total of 25 + 1 = 26. One small line near the centre of the triangle has been subtly omitted, or else we would find a total of 30 triangles rather than 26. Finally, in order to switch our linearized triangle back into its curved shape as was drawn in crops, we can apply a mathematical method known as “Steiner’s deltoid” (see /blog.zacharyabel.com):
The highly curved triangle which was drawn in crops appears “hyperbolic”, in order to symbolize a hyperbolic spatial geometry for “extra dimensions”.
During the day of July 1, 2012, we were inspecting a new crop picture near Stanton St. Bernard with Paul Jacobs and Stuart Dike. Then a call came in, to inform us that a new crop picture had just been found on top of Waden Hill, not far from Avebury or Silbury Hill. When we arrived there, the new crop picture was still pristine, because only three people were in it on the ground. One was Ross Holcombe, who was busy sketching a complex code which he had found. Six of its 36 triangular spaces showed standing tufts, while the other 30 spaces showed centres which were swirled just slightly off the ground. In the slide below, we have added six “red asterisks” to denote where six standing tufts of crop were located:
Once we saw an aerial photograph, we realized that the Waden Hill
crop picture shows “Pascal’s Triangle”, using the artistic style for
a “flower of life”. The total number of triangular spaces on each
level of the triangle equalled 1, 3, 5, 7, 9 or 11. Yet when studied
in the form of Pascal’s Triangle, only 1, 2, 3, 4, 5 or 6 spaces on
each level were available for coding. Those six standing tufts match
individual numbers of 1, 4, 5, 10, 5 or 1 from Pascal’s Triangle,
and add to a total of
Now the other crop picture which appeared at Liddington Castle on
July 1, 2012 also showed “
Around the outside of Waden Hill, we can see 2, 1 or 1 standing
tufts which add to of a triangular pyramid which
was drawn at Liddington? four faces
When we compare both crop pictures as seen from above, their similarities become striking:
Both crop pictures show large triangular pyramids, which have been coded on each face with either 13 or 26 small, overlapping triangles. When viewed in three dimensions, each triangular pyramid would possess four faces. We may summarize our current results as follows:
The triangle puzzle from Chesterton Windmill on July 9, 2009 shows 4 x 13 = 52 small, overlapping triangles. The triangle puzzle from Liddington Castle on July 1, 2012 shows 4 x 26 = 104 small, overlapping triangles. Both of those numbers “52” or “104” match the number of panels found on one or two faces of the Pyramid of Quetzalcoatl in Chichen Itza. Both crop pictures also resemble a “large triangular UFO”, which was drawn in crops at West Kennett Long Barrow on August 13, 2013. Who knows what will happen next? All of these spectacular field images must be leading somewhere. The past is prologue.
P.S. We would like to thank Marina Sassi for the “yellow windmill” image of a crop picture at Chesterton Windmill on July 9, 2009, and also Sarah Susanka for her careful analysis of the “cube” from Hackpen Hill on August 26, 2012 (see hackpenhill3 or Sarah_Susanka):
“As an architect, I draw window frames in perspective all of the
time. So it is very apparent to me that the way in which these lines
of see-through cubes have been drawn is actually ‘upside-down and
inside-out’, relative to what our eyes would expect to see at the
upper, outward-pointing corner of the cube. This tells me that we
are being shown an Indeed, the crop artists seem to come from a “mirror universe” which is upside-down relative to ours (see fringe2013k). If an architect and best-selling author such as Sarah can get it right, why not the scientists or academics in our schools? As Joni Mitchell sings in “Shine”, they have tunnel vision (see www.youtube.com). |