Monday, November 14, 2016

Megalithic Spirals of Malta, Puma Punku Cross, & the Fibonacci Sequence

When I was in elementary school, I stumbled across the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …) for the first time in one of those logic puzzle / mathematical curiosity books written for kids. The sequence was described in terms of fictional breeding rabbits, which was interesting at the time, but in retrospect not the best way to apply the sequence because in reality, rabbits don’t breed according to that model (and inbreeding is gross)!

In middle and high school, I was introduced to the Fibonacci sequence in a cursory manner, nothing really in depth or meaningful. At Humboldt State University, as a math major, I saw the power of the Fibonacci sequence, and more importantly, the Golden Section, for the first time. I remember thinking, why is this not a big part of the curriculum in high school? Well, a few years later, when I was a high school math teacher, I got the answer to my question… The curriculum is a mile wide and an inch deep; nothing of value or consequence can be covered in depth, because there is too much to cover.

Regardless of what I read in books or was taught in school, it is clear to me now that Fibonacci was not the first one to figure out this sequence, any more than Pythagoras “discovered” the Pythagorean Theorem. Many of the big mathematical concepts were either taught by, or reverse engineered from the more advanced cultures that came before. Megalithic sites like Stonehenge, the Pyramids of Egypt and Mexico, and the ruins on Malta use complex geometric concepts that we are just now catching up with. They seemed to not only be aware of the Golden Ratio, but able to apply the mathematical concepts to sound amplification and propagation.

I have a tendency to hoard my research, but I am going to try to be better and release it in small bits, imperfect and unconnected though they may be. Here are a few examples of Fibonacci sequences that clearly predate the man credited with its discovery.

Malta - Spirals of Tarxien Temple (~3150 BC)

On the small island of Malta, off the coast of Greece, are some of the oldest and most puzzling megalithic ruins on the planet. There is a subterranean structure (Hypogeum) carved out of rock, and on the second level of this structure is a small chamber known as the Oracle Room. Anyone speaking from this room can be heard throughout the Hypogeum, and it has some strange acoustic properties that I don’t think have been explained fully.

Until I can travel to Malta (Bucket List!) and check out / measure the Oracle room / Hypogeum for myself, I have to look for other clues to study the geometry of the site. If I am correct in my theory, the Universe is scale-invariant, which means that the very small looks identical to the very large. I believe megalithic people were aware of this fact, and that their carvings of smaller objects use the same geometric ideas as their largest constructions.

There appear to be spiral carvings all around the temples on the island, and these carvings can shed some light onto the mathematics used in the overall construction of the site. I’ve looked at one of the carvings so far, and guess what I found…

Spiral Carving from Tarxien Temple, Malta

There is a lot going on in this carving, but its basic structure is recursive and encodes the numerical sequence 3, 5, 8, 13, 21. There it is, about 4200 years before Fibonacci was born.

Puma Punku Cross, Bolivia (~600 AD)

Last May I was fortunate enough to travel to Machu Picchu and on the way I dragged my travel partner on a crazy side trip to Bolivia to see Tiawanaku and Puma Punku (Thanks, Tracy!). I’m really glad that I did, because I felt a very strong connection to the place.

The Tiawanaku and Puma Punku sites are about a quarter mile apart, but as I stood on the back side of Puma Punku looking over the valley behind it, I slipped back in time for a few moments, and became aware of a clay wall in front of me that extended to my left and far behind me. An inner voice said “it goes all the way around,” and in my mind’s eye, I saw a layout of the surrounding area and a wall encircling both Tiawanaku and Puma Punku. They are not separate sites, but part of the same complex. The clay wall has been partially excavated, but I don’t know if they’ve figured out it goes all the way around yet (I don’t think so).

The carved crosses and H-blocks of Puma Punku are not as old as the spirals on Malta, but the Bolivian carvings are amazing. I think they may not be carvings at all. They are all uniform in size, and the interior cuts would have been very difficult, even with modern technology. It is more logical that they were poured into molds, which has startling implications about what people knew 1500+ years ago…

Regardless, here is a cross to ponder. The dates on these crosses are iffy, maybe 600AD? The Fibonacci sequence dates to about 1200 AD.
Stone Cross from Puma Punku, Bolivia

There it is again – the sequence 3, 5, 8, 13…

The tricky thing with putting numbers on these structures is that there may be alternate scalings, but in this case I have chosen to use integers because they seem like the best fit. There may be alternate scalings that use powers or multiples of the Golden ratio. Regardless, both the Malta spirals and the Bolivian cross are proportioned based on the Golden section. The existence of Fibonacci numbers is likely but not required to establish a Golden relationship. The Golden section is present in both carvings, regardless of scale.

Next up, Lucas Sequences…

The Fibonacci sequence is actually part of a larger set of sequences, known as Lucas sequences. Without going into too much mathematical detail, Lucas sequences are constant-recursive integer sequences. That means the sequences contain only integers, and recursion is used to generate new elements.

In the case of a Fibonacci-like sequence, we add the last two elements to get the next. My favorite Fibonacci-like sequence is 3, 4, 7, 11, 18, 29,… In the case of another important sequence known as the Pell sequence, each Pell number is the sum of twice the previous Pell number and the number before that (1, 2, 5, 12, 29,…). Just as the Fibonacci sequence is related to the Golden ratio, the Pell sequence is related to the Silver ratio, another very important and often ignored proportioning system.

I use the words “proportioning system” because that is the remarkable quality of all of these Lucas sequences… They provide a way to recursively divide space while maintaining constant proportions.  If you were going to create a simulation using space-time as a medium, such a mechanism would be very useful…

Wednesday, February 10, 2016

Nolan’s Cross on Oak Island – Tree of Life, or NOT?

So I’ve been sucked into the History Channel’s show The Curse of Oak Island because it combines two things I love – the hunt for treasure, and geometry. More than a year ago, I hypothesized that Oak Island had been shaped by people in order to conform to certain geometric principles. I have since found significant geometric evidence that suggests the whole eastern part of the island has been modified in the past, and was thrilled to discover in season 3 that Fred Nolan believes this as well, but for different reasons. 

The logical starting point for the study of Oak Island’s geometry is Nolan’s Cross, which consists of 6 large stones placed on the island in a cross formation. Thanks to Fred Nolan’s early surveys, we have good measurements that give us the actual layout of the cross. A Norwegian man named Petter Amundsen proposed slightly different measurements, but based on my analysis, I believe Nolan’s measurements are more accurate. 

Below shows the cross with Nolan’s measurements, as well as the “distance matrix” I used to analyze them. The idea is that each distance is divided by every other distance to discover the relationship between all of them as a set. I have used this method for years to study crop circles and other megalithic monuments around the world, and have found that it is only necessary to consider the quotients > 1.

Figure 1 - Fred Nolan's cross measurements and the Distance Matrix used for analysis

It is much easier to understand the process by looking at it visually. The diagram on the left shows the measurements from Nolan, and the one on the right shows the simplification suggested by the division matrix. I call this simplified diagram a “relational model,” and I find it useful to visualize distances as circles. 

Figure 2 - Actual cross measurements vs. Relational Model

Note that the arms of Nolan’s cross are in a 6/5 ratio, meaning the long arm is about 1.2 times longer than the short arm of the cross.
  
Petter Amundsen appeared on the first season of Curse of Oak Island to explain the geometry of the cross. His theory, in a nutshell, is that Nolan’s Cross is part of a Tree of Life geometry, and that a cipher buried in Shakespeare texts points to the treasure being hidden under what he calls the “mercy point” on this tree. So, let’s compare the geometry of the Tree of Life with the relational model of Nolan’s cross and see if they match.

Figure 3 - Tree of Life construction and relative distances

The Tree of Life has a (long arm)/(short arm) ratio of 4/sqrt(3), which means the long arm is about 2.31 times as long as the short arm.
   
As previously mentioned, the (long arm)/(short arm) ratio of Nolan’s cross is 6/5, or 1.2 times as long. Geometrically speaking, the arms of Nolan’s cross are proportionally different than the arms of the Tree of Life. Armundsen tried to fix this problem by adding another data point to make Nolan’s cross longer, but even this does not fix the proportions.

This mathematical mismatch is shown below as the two diagrams are scaled and superimposed (with the Tree of Life in red, Nolan’s cross in black). The first diagram shows how Armundsen viewed it, and the gray point at the bottom is his proposed new point on the cross. The resulting 8/5 ratio is still out of proportion with the 4/sqrt(3) ratio, which basically just means the short arm of Nolan’s cross is still too long, even with the modification. In the first diagram, you can also see that the central stone of Nolan’s cross does not actually correspond to a point on the Tree of Life. The second diagram shows what happens proportionally when the short arms are scaled to fit the Tree of Life – none of the other points line up. 

Figure 4 - Nolan's Cross superimposed on Tree of Life

In summary, Nolan’s cross is not geometrically compatible with the Tree of Life. There is some similarity in the placement of points along the long axis, but if the builders intended a Tree of Life, I believe they would have used the correct proportions. Geometry was obviously important to those who “constructed” Oak Island. That being said, I do believe that Armundsen was probably correct about the placement of the extra stone at the bottom of the cross, but only because it fits in with the larger geometric figures that define the shape of the island.

Before I reveal how the shape of the island was changed, I want to take a closer look at Nolan’s cross, because it is critical to understanding the overall geometry of the island. Nolan’s cross is like the island’s legend, because it provides both scale and direction.  The integers 1, 2, 3, 4, 5, and 6 are encoded into the cross, but it is done so in a mathematically elegant way. 
This is what I see when I look at Nolan’s cross…

Figure 5 -Circular interpretation of Nolan's Cross

Each circle serves a purpose, and there is a lot going on mathematically, considering there are only 6 points. This seems familiar to me, and I need to look back into my crop circle research to see if this same pattern has turned up somewhere before. The beauty of mathematics is that it allows us to empirically compare two designs.

Let’s switch gears, and look at the rectangles that compose the cross.  I am still not finished with my analysis here, because it involves the larger geometry of the island, but I have noticed one interesting property that is indicative of megalithic monuments, and architecture in general. The outer proportion of the cross is repeated on the inside, in a non-trivial manner. In the diagram below, the shaded blue rectangle is geometrically similar to the outer blue rectangle around the cross, which means the inner proportions reflect the outer. This idea can be found in Mayan/Incan cultures as well as Templar geometry. 
Figure 6 - Inner and outer proportions of Nolan's Cross

In summary, I believe that Nolan’s cross is not related to the Tree of Life, but provides 1) scale and 2) direction relating to the larger geometric construction of the island. The scale, based on the distance matrix in Figure 1, is 1:145ft, which gives us a way to analyze the island in terms of pure number, as well as actual distance on the ground.  When I talk about direction, I believe the cross is rotated 30 degrees from a North/South orientation, but I need to confirm this. Also, the arms of the cross provide an AXIS along which the centers of the circles that define the island are placed. The whole island screams GEOMETRY.

My next blog will show the larger geometry of Oak Island, but this is very much a work in progress, and I do not have the GPS coordinates for the stones, or other key markers, otherwise I could move a lot faster on this. In addition to Oak Island being altered, I believe the long skinny island right next to it, as well as Birch Island have been modified as well.