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…