Tuesday, March 19, 2013

Tibetan Monks and the Geometry of Levitating Stones

In addition to the ancient accounts of large stones being levitated or “walked into place,” there are also a few modern accounts that provide evidence for such a phenomenon. One of the most compelling cases is that of Henry Kjellson, a Swedish engineer who in his 1961 book The Lost Techniques detailed the visit of his friend Dr. Jarl to a Tibetan monastery. While at the monastery, Dr. Jarl witnessed Tibetan monks levitating large stones from the ground up to a cliff approximately 250m high.

This account of stone levitation is remarkable because of the scientific observations and measurements that were documented by Jarl (or Kjellson using a pseudonym). The detailed measurements allow for an analysis of geometric proportions used by the monks. The following excerpt comes from Kjeller:

In the middle of the meadow, about 250 meters from the cliff, was a polished slab of rock with a bowl like cavity in the centre. The bowl had a diameter of one meter and a depth of 15 centimeters. A block of stone was maneuvered into this cavity by Yak oxen. The block was one meter wide and one and one-half meters long. Then 19 musical instruments were set in an arc of 90 degrees at a distance of 63 meters from the stone slab. The radius of 63 meters was measured out accurately. The musical instruments consisted of 13 drums and six trumpets. (Ragdons).

Eight drums had a cross-section of one meter, and a length of one and one-half meters. Four drums were medium size with a cross-section of 0.7 meter and a length of one meter. The only small drum had a cross-section of 0.2 meters and a length of 0.3 meters. All the trumpets were the same size. They had a length of 3.12 meters and an opening of 0.3 meters. The big drums and all the trumpets were fixed on mounts which could be adjusted with staffs in the direction of the slab of stone. 

The big drums were made of 3mm thick sheet iron, and had a weight of 150 kg. They were built in five sections. All the drums were open at one end, while the other end had a bottom of metal, on which the monks beat with big leather clubs. Behind each instrument was a row of monks. The situation is demonstrated in the following diagram: 

When the stone was in position the monk behind the small drum gave a signal to start the concert. The small drum had a very sharp sound, and could be heard even with the other instruments making a terrible din. All the monks were singing and chanting a prayer, slowly increasing the tempo of this unbelievable noise. During the first four minutes nothing happened, then as the speed of the drumming, and the noise, increased, the big stone block started to rock and sway, and suddenly it took off into the air with an increasing speed in the direction of the platform in front of the cave hole 250 meters high. After three minutes of ascent it landed on the platform.

Continuously they brought new blocks to the meadow, and the monks using this method, transported 5 to 6 blocks per hour on a parabolic flight track approximately 500 meters long and 250 meters high. From time to time a stone split, and the monks moved the split stones away.

Personally, I find the account very compelling because there are a lot of details, as well as a group of measurements that can be analyzed regardless of whether the whole thing is fiction or not. The geometry can speak for itself. From the side view, the ground and cliff form two sides of nested right triangles as shown below. The green arc represents the flight path of the stone blocks.

The distance from the instruments to the stone (63m) is about 1/4 the distance from the stone to the cliff base (250m). Consider the base of the cliff as the center of two circles with the two triangle legs as radii. The circle diameters as well as the triangle legs are in the ratio of 5:4 as shown visually below.

These two circles in a linear ratio of 5:4 are seen repeatedly in crop circles and megalithic monuments around the world. One significance of this set of circles is that they provide an easy approximation for squaring the circle. In the above figure, the red square has an area equal (within 0.2%) to that of the green circle. I have often wondered why squaring the circle would play such a prominent role in crop circles, ancient architecture and monuments. Recently, I have come to realize that the angles encoded within this geometric structure may play a key role in non-linear resonance because they are an embodiment of the golden ratio, phi (ɸ). This is Nature’s fractal.
The angle formed by the musicians, point of levitation, and base of the cliff (51.38°) is very close to the angle seen in a vertical cross-section of the Great Pyramid of Giza (51.83°). Cultural myths allege that the stones used to build this pyramid were levitated into place, so the geometric similarity is interesting to note.
The vertical cross-section of the Great Pyramid of Giza appears to be composed of two triangles known as Kepler triangles. A Kepler triangle is the only right triangle with sides in a geometric progression according to the golden ratio (1, sqrt(ɸ), ɸ). Geometrically speaking, Kepler’s triangle is the embodiment of the golden ratio in more ways than the “golden triangle” some readers may be familiar with.

The question remains about the geometric intent of the Tibetan Monks, assuming the story is true. Were the monks using a 5:4 ratio to approximate the geometric progression of the golden ratio? A comparison of the triangles in question show how close they all are.
The connection between Kepler triangles and resonance will the subject of a future blog post.

Drum and Horn Proportions

The description tells us that there were 13 drums and 6 Ragdons, which are the long, skinny trumpets still in use at Buddhist monasteries today. The drums came in 3 sizes, and it is assumed that Kjellson’s measurement for “cross-section” represents the diameter of the head of the drum. Then we have the following drum sizes:

Large (L)
1.0 m
1.5 m
Medium (M)
.7 m
1.0 m
Small (S)
.2 m
.3 m

Upon further inspection, the proportions of the drums appear to be based on powers of 2 and 3, within a small margin of error. The number of drums is also based on powers of 2.

Large (L)
Medium (M)
Small (S)

The medium drum is 2/3 the size of the large drum (in diameter and length), and the small drum is 8/27 times smaller than the medium drum. Based on these ratios, the small drum would be significantly higher in pitch than the other two drums. It seems logical that the small drum was the one leading the beat, since it could be heard above all the other instruments according to Jarl’s account. 

The ragdon trumpet produces a very low frequency sound, one I would describe as somewhat “farty.” It’s similar in frequency to a tuba, but more piercing, and with less “umpah.” With a length of 3.12m and an opening of 0.3m, the length/width ratio is about 10.4.  If I was sticking with the powers of 2 and 3, that would be close to 34/23 = 81/8 ≈ 10.125. 

From a music theory point of view, it makes perfect sense to use the ratio 3/2 for proportioning instruments, because it represents one of the most important musical intervals, the perfect fifth (P5). The musical system supposedly developed by Pythagoras (but likely borrowed from the Babylonians) relied entirely on powers of 2 and 3. This 3-limit tuning system was the dominant music theory used up until the Renaissance where it was supplanted by an equal-temper tuning system that allowed for easier key changes. 


I tend to believe that this account of Tibetan monks levitating stones is genuine. It is a lot of detail to fake, and the geometry is just too familiar. I see this Kepler triangle, this weird 51-ish° angle everywhere, in conjunction with geometric progressions of the golden ratio. The central angle of a septagon, the 7-sided regular polygon that plays a prominent role in sacred geometry, is 51.42°. Perhaps if it was named “sonic geometry” instead of “sacred geometry” more people would take it seriously and start analyzing the fractal nature of sound.

Saturday, March 9, 2013

Are Crop Circles “Real”?

When I was a teenager I remember being curious about crop circles, but in the days before the World Wide Web, information was limited for those topics deemed “new age.” I was going to college in 1991 when Doug Bower and Dave Chorley received a huge amount of publicity by claiming they had created hundreds of crop circles from 1978-1991. I remember seeing demonstrations of how they used ropes and planks to stomp down the crop, and I thought “well, I suppose that explains it.” Back then I trusted the media to provide unbiased information, but now I just feel duped.

Image courtesy of Lucy Pringle, ©2008 (http://www.lucypringle.co.uk/

I never gave crop circles a whole lot of thought after that, until 2010, while surfing the internet, I stumbled across a picture taken in June 2008 of a crop circle at Barbury Castle, Wiltshire, UK. The stepped, spiral design is a geometric representation of π to 10 digits. The first time I saw the crop design, I felt like a bell had been rung in my psyche. Intuitively, I knew that the ratcheted circle was proportioned and placed very carefully in the field, and a very insistent voice in my head kept telling me “look deeper into it…” And this is how my geometric study of the crop circles began.

Are All Crop Circles Man-Made?

This is the inevitable question which confronted me again in 2010 for the second time. Almost 20 years after the Bower and Chorley hoax, I decided to do my own investigation. What I found astounded me so much that I was compelled to go to England in June 2012 to see the crop circles for myself.

I’m a mathematician, but I also consider myself a scientist because I place great weight on empirical evidence and the scientific process. I was shocked I didn’t know about this sooner, because there is significant empirical evidence to support the idea that some crop circles are not man made. Our best and brightest scientists should be studying this phenomenon, but instead, those who study the subject seriously are mocked by their peers. It’s an appalling injustice.

Here is some of the empirical evidence that supports the idea of non-man-made crop circles:
  • There are many eyewitness accounts of crop circles being spontaneously formed. The book Crop Circles – History, Research and Theories by the Norwegian researcher Eva-Marie BrekkestØ contains an entire chapter, 40 pages of different types of eyewitness reports. Some reports have more detail than others, but many of the accounts from all over the world are eerily similar to each other.
  • There are several accounts by pilots and other witnesses who have flown over a field where there is no design, and a short time later, fly over the same spot and a crop circle has appeared. Some of the designs appear in broad daylight, seemingly instantaneously.
  • Michigan biophysicist W.C. Levengood compiled an extensive amount of data in the study of plant abnormalities found in downed crop (http://www.bltresearch.com/plantab.php). More than 250 crop formations from multiple countries, over a 10-year period were examined.
    The study found the following in downed crop but not the crop left standing (control):
    • Abnormal enlargement of cell wall pits in bract tissue
    • Enlarged plant stem nodes (both laterally and longitudinally).
    • Bending of the stem nodes or base of the stem. Crop bent by planks or mechanical means usually snaps at the base, especially later in the season when plant stems become more brittle.  Crop bent at the nodes across a large area cannot be accomplished by mechanical means either.
    • Expulsion cavities in one or more of the nodes, most likely caused by microwave radiation that turns internal moisture into steam. These “exploded nodes” are not found in man-made crop circles. 
    • Stunted, malformed seeds and germination effects. These effects “are thought to be caused by exposure of the plants to a complex atmospheric plasma energy system which is emitting heat (probably microwaves) in association with unusual electrical pulses and strong magnetic fields.”
  • Many instances of strange phenomenon have been documented by people entering crop circles. This includes the malfunction of electronic devices such as cameras, as well as emotional and physiological effects on the human body. Lucy Pringle has been working for many years to compile this empirical evidence. She has documented much of it on her website (http://www.lucypringle.co.uk/). 
  •  Some of the designs contain crop that has been woven stalk by stalk, alternating from side to side. I have also seen pictures of a basket weave, where groups of crop run perpendicular over or under another group that was standing next to it. This type of weaving would be impossible to fake on such a large scale. The hoaxed ones are usually made of concentric, plank-width circles, which breaks the stem of the crop at a 90 degree angle to the ground. 
  •  As I saw for myself in June 2012, some of the circles that appear on muddy nights have no trace of mud on top of the downed crop, and seem to float above the mud. I was one of the first people to enter a crop circle that appeared on a rainy night during one of the rainiest Junes on record. The ground was like clay soup, and my boots were caked in it when I finally reached the downed crop. The wheat was bent, but not all the way to the ground, and there was no mud anywhere on the crop. As I stepped onto the downed wheat, it was impossible to not mash it into the ground, and after our group left, the new design was caked in mud and completely flattened.
Stanton St. Bernard, Phase II (June 17 2012)
  • Some of the designs are rotated and scaled to intersect with tram lines and other geographical landmarks, while still maintaining complex geometric rules consistent with the other (“genuine”) designs. These intersection points mark locations of geometric importance within the design. It would be nearly impossible to do this without a very accurate survey of the land, including tramlines, and incredible accuracy in determining the centers of the circles on the ground. 
  •  Crop circles from different countries separated by miles share the same complex geometric rules. There would need to be a well-organized group of people all around the world who have agreed to base their designs on the same geometric principles, and who are coordinating with each other.

So, ARE all crop circles man-made? You decide!

This blog intends to explore the above ideas in more detail, with emphasis placed on sound mathematics and scientific principles. There are so many connections to be explored, because crop circles often appear near the megalithic monuments of the world, and seem to share the same affinity for irrational and transcendental numbers such as square roots or the Golden Ratio.

Saturday, March 2, 2013

MegaGeometry.com is now live!

New! Graphic
Howdy, the site is now live! Please bear with us while we get everything up and running. You can reach the webmaster at amber@megageometry.com