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. 

Sunday, August 2, 2015

Introducing the Polytonic Ratios

The following is a summary of the starting point for my crop circle research. I realized early into my research in 2010 that someone else had already noticed the patterns I was seeing in the crop circle designs. His name was Professor Gerald Hawkins, and he was a pioneer in the study of the geometric properties that appeared in these designs. As an astronomer, however, he should have left the theoretical mathematics to mathematicians, and his "crop circle theorems" are nothing more than the application of a well known formula I will explain below. That being said, I am very grateful for the work of Professor Hawkins, because not many credible scientists even take the subject seriously.

Circles Inside, Circles Outside

A regular polygon is a 2-dimensional closed shape, consisting only of straight lines, that is both equilateral and equiangular. This means all the sides have the same length, and all angles have the same measure.  A regular polygon can either be convex, with all sides “bulging outward,” or shaped like a star (with a mixture of concave and convex sides).  The most common examples of convex polygons are the triangle and square, and the most common examples of star polygons are the pentagram and hexagram.

Every regular polygon can be both inscribed and circumscribed by a circle. The circle that is tangent to all sides of the polygon is said to inscribe the polygon. This unique circle is called the incircle, and its radius is referred to as the inradius (r) or apothem of the polygon. 

circumscribed triangle with inradius r
Incircle with inradius “r”

Likewise, a circle can be drawn around the polygon, passing through all of its vertices, and this circle is said to circumscribe the polygon. This is referred to as the circumcircle, with a radius called the circumradius (R).

Circumcircle with circumradius “R”
Circumcircle with circumradius “R”
The formula for the circumradius (R) of any polygon (“n-gon”) can be expressed in terms of its inradius (r) and number of sides (n) as follows 

Suppose we want to compare the size of the circle that nestles the interior of the shape with the circle that caresses the outside.
Inradius r and Circumradius R

Dividing both sides of the equation by r gives us a generalization for the ratio of circumradius/inradius:
This formula tells us the ratio of circumradius/inradius is constant for each regular polygon, and based on a periodic function (cosine) that is used to describe light and sound waves. Equivalent ratios are produced by looking at circumference or diameter, because they are both proportional to the radius, which means they grow at the same rate.

The ratio of areas of the circumcircle and incircle form a second set of ratios that is the square of the first. The formula to calculate these values is the square of the circumradius/inradius formula above.
I have named the ratios defined by these two formulas as the polytonic ratios, and polygons surrounded by circles as encapsulated polygons. Each shape has two interval ratios that are unique to that polygon and independent of scale. The values for both the linear and the square polytonic ratios are given in the table below. The red bold ratios are the ones referenced in Gerald Hawkins’ Theorems.

Table 1 – Linear and Square Polytonic Ratios
Looking at the table of polytonic ratios, it is easy to see why Professor Hawkins picked out the values of 2, 4, and 4/3. I know the trigonometry is scary for some, so let's look at some pictures. This formula gives us a way to figure out which polygon (if any) fits between any two given circles.

Table 2 - The First 12 Encapsulated Polygons

A Look Ahead...

As you might guess, many of the above rings compose key elements in the design of the crop circles. Rings can be combined with each other, producing a multiplicative effect of the polytonic ratios which resembles the way musical diatonic ratios are combined.There is much to be explored here, but for now I will leave you with a polytonic representation of the perfect fifth formed with two hexagons and a heptagon. It's actually not a perfect fifth ratio (1.50), but it's close (1.48)

A Polytonic Perfect Fifth

Monday, January 5, 2015

Circles as Musical (Diatonic) Ratios

A Short Lesson in Music Theory

Pythagoras was the first one credited with realizing that a string could be cut at certain ratios to produce harmonically pleasing sounds. This idea undoubtedly predates Pythagoras, as it is the basis for all musical theory, and he likely encountered it in Egypt or one of the many places he traveled in his youth. Pythagoras constructed his scale using only perfect fifths (3/2) and octaves (2/1). Mathematically speaking, it means he used only powers of 2 and 3 to generate the interval ratios.

The Pythagorean tuning system produces perfectly consonant fifths, but presents several issues for musicians. For example, enharmonic notes such as A and Gare considered to be different notes with distinct frequencies. In the equal temper tuning we use today, enharmonic pairs are treated as the same note because they have the same interval ratios and frequencies. 

The real issue with this system is that the circle of fifths does not close. That means that as you progress through the notes of the scale, you won't end up on the same note an octave above where you started. Some adjustment needed to be made in order to close the circle, and that adjustment was to lop off the last interval and make it smaller then the rest. Thus, the Pythagorean tuning system consists of 11 intervals of perfectly just fifths, and 1 slightly shorter, badly-tuned interval known as the wolf interval. This wolf interval is the direct result of forcing the circle of fifths, which is actually spiral in nature, to be a circle.

The next evolution in musical theory was to introduce other prime numbers into the construction of musical intervals. In addition to using powers of 2 and 3, as in the Pythagorean tuning, the number 5 was introduced in a system we now refer to as a 5-limit just tuning. Pythagorean tuning is sometimes referred to as 3-limit, and other tuning systems such as 7-limit and 11-limit also exist, but are less common. All of them seek to represent harmonic ratios using the smallest integers possible.

There are some people today who argue that the 3-limit tuning, despite its dissonant wolf interval, is superior to the 5-limit tuning system, so let’s compare them a little:

  • 3-limit has consonant fourths and fifths, but dissonant major and minor thirds
  • 5-limit has consonant major and minor thirds, but some dissonant fifths
  • 3-limit tuning lives on today mostly in the form of 2-dimensional isomorphic keyboards. More will be said about these later, but for now it will be noted that the way to eliminate the Wolf interval is to add a spatial dimension to our musical instruments.  
  • 5-limit tuning became popular in the late Middle Ages because chords and triads are based on three notes, which means major and minor thirds need to be consonant in order to achieve a certain complexity in the music. Chords are the basis for most of the music we are familiar with today.
The 5-limit tuning was used until the late 1500s, when equal temper tuning was discovered and a method for calculating 12 equal musical intervals was developed. Today, in the West, we use a twelve-tone equal temperament tuning system, or 12-TET. Each note in the chromatic scale is the same distance apart, and that distance is 2^(1/12). It is my opinion that by moving away from pure ratios and into a logarithmic-based tuning system, we have done our culture a disservice. This will be the subject of a future blog post.

For now, I want to focus on 5-limit tuning, and explore it a bit further… The following table shows what the notes in the chromatic scale look like. Diatonic ratios are generally considered to be only those intervals found in a major scale, although some places on the web you will see a much broader use, or overuse of the term. Diatonic ratios are in red in the table below. The key of C has been chosen for the Note column because it is the one I am most familiar with.

5-Limit Tuning System

5-limit tuning also has its own issues, and its own wolf intervals. As shown in the table above, the notes F# and Gare extremely close to each other, but not equal in ratio/pitch. By seeking to use just intonation and produce pure harmonic ratios, equal interval sizes are sacrificed. This means that each of the interval types, except for the octaves, has three or even four different sizes. 5-limit tuning uses a combination of justly tuned fifths and non-just fifths of several sizes to close the circle of fifths. It's a complex and imperfect system.


Musical Ratios in 2-Dimensions - Circles

Up to this point we've been looking at musical ratios in terms of strings or lines, which are theoretically 1-Dimensional. Now imagine you were to fix each string at one end and spin them around to trace out a circle. The relative radii (or diameters) of the circles would be in the same ratio as the strings.This provides a 2-Dimensional model of musical intervals that in many ways bridge the gap between music and geometry. 

5-Limit Chromatic Scale Using Circle Radii

Note the slim interval between A4 (1.406, F#) and d5 (1.422, G). In an equal temperament system, these two notes are equivalent, and the spacing of the circles would be logarithmically equal.

It turns out there are a lot of ways to represent music using 2-Dimensional figures, but circles are the most natural element for this visual representation of sound. Because circles are defined by one number (radius), they can be thought of as a 2-D embodiment of number itself. For more information, see my previous blog post Circles as Numerical Sequences.

The chromatic scale represented by the circle radii above shows all the notes stacked on top of each other. Each of the 7 major diatonic notes are shown individually in the table below. It demonstrates the relative sizes of the intervals.

 5-Limit Diatonic Ratios Represented as Circle Radii

By using circles to represent musical ratios, we can:
  1. Express ratios in terms of linear measurements like radius, diameter or circumference
  2. Express ratios in terms of square measurements like area
  3. Change the positions of the circles to create an interplay of ratios using linear and square measurements
Expressing ratios in terms of radius and area are fairly straightforward, but once you add another variable representing the positions of each circle, a whole new level of complexity develops. The figure below demonstrates the relationship between the octave, fourth and fifth more elegantly than any 1-D representation ever could...

Octave, Fourth and Fifth - The Circular Dance

The red circles are in a perfect fourth ratio (4/3) and the blue circles are in a perfect fifth (3/2) ratio. There are also three pairs of circles in octave ratios (2/1, 4/2, 6/3)  and three other perfect fifth pairs (3/1, 6/1, 6/4), some spanning more than one octave.

I've seen designs like this appear in crop fields around the world, most notably in the UK. It appears in many forms, but has the same underlying structure. For example, the following design appeared near Liddington Castle in the UK in June of 2010.

Liddington Castle crop circle near Swindon, UK - June 2nd, 2010.

Monday, December 22, 2014

Crop Circle Hoaxers Can Suck my Ovaries

I don’t have balls. I’ve never wanted them. I suspect most of the crop circle hoaxers have them, along with an inversely proportional, large ego. 

There is a group of people who live mostly in Wiltshire England, and every year they create their own crop circle designs. Most of the time the farmer is the victim in what is essentially an act of vandalism. There are also rare instances when a farmer or company will commission a design. The unwilling farmers, however, face damaged crop fields, as well as additional damage caused by people wanting to come see the design. Researchers like me face a corruption and muddying of the data.

When I first started studying the geometry of crop circles, I wasted a lot of time studying hoaxed crop circles. A lot of time. A lot of wasted time. I want to personally thank the hoaxers for wasting so much of my precious research time, which I carved out of my daily life. It took me years to figure out the difference between the hoaxed crop circles and the genuine ones. Now I don’t waste as much time because I’ve developed a few geometric litmus tests that can be used to spot the fakes and exclude them from the data pool. 

So perhaps you understand my dilemma of whether to reveal what I’ve learned in my nearly 5 years of crop circle research. If I teach the hoaxers what I know, they can create better hoaxes and waste more of my time. I am reluctant to publish specifics, but at the same time I am angry at myself for self-censoring. Which brings me back to the point of this post: Crop Circle Hoaxers Can Suck my Ovaries

I don’t think they are bad people, but they are misguided in the belief that they are not harming their community. I suspect it is their disproportionally large egos that have lead them astray. I don’t think it is about money. I don’t see how one profits from faking crop circles. Some hoaxers believe their circles are part of a dialog with the circle makers, and I say if you are that compelled, find a place where you have permission, or choose a different medium for your creative expression.

Here are a few general problems I’ve seen with the geometry of hoaxed crop circle designs:

  • Nonexistent proportioning system. Complete lack of geometric order, sometimes interspersed with random elements from sacred geometry
  • Inconsistent proportioning. The system of proportion varies depending on which part of the design you look at.
  • Tram lines are used in the construction of the design elements rather than integrated into the overall geometric design.

So my message to the hoaxers is as follows:
Knock it off; find a more productive hobby. You are muddying the data and wasting the time of people who are trying to study the phenomenon from a scientific/mathematical point of view. Stop harming those you should be helping within your own community.