Trigonacci Grows Up

Most formulas I have developed so far circle around the circular trigonometric functions, sine and cosine.

The dependency is evident. It is time for our trigonacci to come of age and find a representation for itself that does not use sine and cosine, but only the angle “x” and the dimension “n”. This can be done by re-opening our Calculus II book to the chapter on power series. The power series representation of sine and cosine are as follows:

By substituting the trigonometric functions with the power series, we get the following independent trigonacci formulas in the form of a power series. Since the "n" is the same nth term for both the series and the Fibonacci sequence (represented as dimensions in spiral coordinates), and since the limits of the power series coincide with the domain of trigonacci functions, the formula as a whole can be brought under sigma.

Expanding the Trigonacci Family

“Mathematics is more dramatic and exciting than most people think, and - more importantly - today’s mathematics is far closer to the flexibility of life than it is to the rigidity of Euclid.” To live up to these words by Ian Stewart, we must broaden the family of spiral functions. For a more flexible use and application of trigonacci, identities equivalent to those in regular or hyperbolic trigonometry have been made. While some identities like the odd/even ones do not apply as negative numbers are not included in the domain, others such as reciprocal and ratio identities, double-angle identities and half-angle identities have been reshaped to fit the truths of how trigonacci functions work.

Sources: Stewart, Ian. Life's Other Secret: The New Mathematics of the Living World. New York: John Wiley's Sons., n.d. Print.

Properties of Trigonacci

Below you will find an incomplete list of general properties that characterize Trigonacci and spiral coordinates. Let’s take a look at them more carefully:

• The domain in which angles and dimensions can occur is only amongst positive real numbers. Unlike other coordinate systems, this one does not accept negative values, as it portrays an ever-expanding spiral that is portrayed often in nature.
• Similar to regular trigonometry, the functions of sine and cosine serve as building blocks for other functions or identities (coming up in the next post).
• The functions of spiral-tangent, spiral-cotangent, spiral-secant and spiral-cosecant.
• The ratio of the Fibonacci sequence is known to be the golden ratio, approximately 1.618. However, this number reveals itself more accurately as the terms get larger and larger. Since larger angles are accommodated by larger dimensions, and since the golden ratio is used in Binet’s formula, the values of the trigonacci functions become more accurate as the angles increase.
• See previous posts for a geometric proof of the Pythagorean trigonacci identity.
• Since every circle is responsible for a different set of angles, -which, if rearranged into the Fibonacci spiral get larger and larger- every circle is now called a dimension.

Additional Functions

Now that we have the two defining trigonometric  functions to work with, we can start generating more. Since we are still working with right angle triangles, the functions of spiral tangent (tansx), spiral cotangent (cotsx), spiral secant (secsx), and spiral cosecant (cscsx), can be found by using their properties in regular trigonometry, as seen below. You may notice that the functions of tangent and cotangent are equal to their spiral counterparts, while those of secant and cosecant, similar to sine and cosine, are tagged along by the Binet's nth term formula.

Our trigonacci family is now complete!

Trigonacci's Pythagorean Identity

 Does the famous identity still apply?

Prediction: No, because sinsx and cossx are no longer bounded between 1 and -1. My hypothesis is, however, that instead, the sum of their squares would be equal to the nth term squared. Logically, this makes sense and goes hand in hand with how the Pythagorean identity is proven. In the unit circle, the radius is one unit, the horizontal axis represents cosx and the vertical axis represents sinx. So any right triangle formed with a unit vector as the hypotenuse would satisfy the Pythagorean identity above.

On the other hand, in spiral coordinates, the radius is the nth term, the horizontal axis is cossx and the vertical axis is sinsx.

In this case, we accept the conjecture:

How Trigonacci Came to Be

After studying the previous model of a system of coordinates, I realized that it was unnecessary to complicate it by switching the axis. Instead, let us examine the coordinate system without switching the axis, while keeping the circles of the Fibonacci spiral stacked on top of each other. We obtain the following.

Here, sinsx and cossx represent spiral coordinates (no, it was not a typo). Equivalent to how hyperbolic trig functions are named, the “s” in sinsx and cossx stand for “spiral”. My thought process before coming to this graph was to use rectangular coordinates, since this graph resembles a unit circle. That way we would get rid of units since they do not apply to the Fibonacci spiral, and the graph would be the same. However, we would not get trigonometric functions if we used a rectangular coordinate system. So I scratched that out of my notes. Looking back at the graph above, I realized that this is definitely not the graph of a unit circle, or polar coordinates. This is a whole different graph. Each quarter term here will be called a dimension, and they are represented by the following angles:

Dimension 1: 0 degrees to 90 degrees

Dimension 2: 90 degrees to 180 degrees

Dimension 3: 180 degrees to 270 degrees

Dimension 4: 270 degrees to 360 degrees

Dimension 5: 360 degrees to 450 degrees

Dimension 6: 450 degrees to 540 degrees

Dimension 7: 540 degrees to 630 degrees

Dimension 8: 630 degrees to 720 degrees…

And so on. Angles that repeat themselves at the limits of each dimension can be portrayed geometrically by the spiral itself, at the point where the previous circle is tangent to the following circle. In the picture, the black dots show where the circles are tangent to each other at 90 degrees, 180 degrees, 270 degrees and so on.

One would ask, what is the difference between spiral coordinates and polar coordinates since they look similar.

1- The axis change. Instead of “r” and “theta” we have “sinsx” and “cossx”.

2- The angles do not recur. For example, 20 degrees is not equivalent to 380 degrees.

Meanwhile, there are fundamental differences between spiral coordinates and the unit circle as well:

1- We are not using a unit, lengths change according to the Fibonacci sequence.

2- Angles do not recur. Every subsequent quarter circle is bigger.

3- Quadrants “take turns” every fourth term.

In pursuit of the relation between spiral trigonometric functions (or “Trigonacci functions”) and regular trigonometric functions, quarter terms were looked into in more depth.

Placing these quarter circles onto spiral coordinates, sinsx goes in place of Xn and cossx in place of the horizontal side of the triangle. So in the formula  , we replace the unproven nth term formula with Binet’s formula (see previous post for the proof that the two are equivalent) and substitute Xn with sinsx. Now, we arrive to the formulas for the first spiral trigonometric functions:

Where x is the angle and n is the number of the term (i.e. the dimension).

Now that we have the formulas of Trigonacci, let’s calculate some well known angles:

A New Approach

These last couple of days I have looked at the Fibonacci spiral more in depth by taking it apart and separating the quarter circles that make it up.  

I wrote their circumferences in terms of one another to then be able to write their trigonometric functions in terms of one another, all the while using Binet's formula from past posts. When trying to put it back together, though, this is what I got.

In this image, the angle I am trying to work with appears all over the place, and will continue to do so as the spiral continues. So instead, what I did was stack the circles on top of each other while switching the coordinate axis from sinx to cosx and vice versa, after every circle. The result came as follows:

Having this new view of the Fibonacci spiral and using the properties of similar triangles, as well as Thales’ theorem (also known as the intercept theorem), will hopefully open a doorway to more discoveries (or inventions, as you wish).

Books, books, and more books

This is going to be an interesting week.

Binet's Formula

After countless endeavors of searching for an nth term formula, the answer came in John A. Adam’s book: “Mathematics in Nature: Modeling Patterns in the Natural World”. On page 226, Adam reveals Binet’s formula, which connects terms in Fibonacci’s sequence to the golden mean: . Is this, however, equivalent to the formula we were trying to prove? Let’s find out.

Because the equations turned out to be true, both formulas are indeed the same, and so here I end the pursuit of an nth term formula with success.

In Pursuit of an Nth Term Formula

So there already exists an nth term formula for the Fibonacci sequence… but it sucks. It’s more of a definition than a practical formula.

Let’s say I wanted to find the 14938593th Fibonacci number. Using this formula, I would have to first find the 14938592nd and the 14938591st Fibonacci number before I could find their successor. Doesn’t that sound fun?

In order to play with the Fibonacci sequence in trigonometric functions, we’d need a more pragmatic formula for the nth term of the sequence. The following was presented to me:

Although this formula looks tempting, I have yet to find a peer-reviewed source that says it is legitimate. So I undertook the task of proving or disproving it:

So far, so good. Stay tuned for the second half of the proof!

The Golden Ratio

“Mathematics is to nature as Sherlock Holmes is to evidence.” (Adam 3) This is especially the case when studying the golden ratio. The golden ratio, also known as the golden mean or the divine proportion, is a number found often in nature, sometimes even referred to as “God’s fingerprint”. It is approximately 1.618034. The golden ratio has been used in art since antiquity; it was used to build the famous Egyptian pyramids, as well as the Stonehenge. We know that artists like Leonardo da Vinci and Albrecht Durer used it in their paintings, and that it is used in constructing musical instruments.

In nature, we see it expressed often in the Fibonacci spirals that characterize the behavior of plants and animals. What does Fibonacci got to do with the golden ratio? Well, if you divide numbers in the Fibonacci sequence with their predecessor, you will come up with numbers that get closer and closer to the golden ratio.

Below you can see my attempt to find an equation for the Fibonacci spiral using the golden ratio , and the graph of it. Sadly, after finishing it I realized that Google had beat me to it, and already had similar graphs.

Source: 

Adam, John A. Mathematics in Nature. Princeton: Princeton UP, 2011. Print.