Trigonacci in the Real World

     The beginning of the fall semester has been productive. I presented Trigonacci at the LSAMP Conference at Kean University, touched upon it at Harvard University and gathered data on the presence of the Fibonacci terms in evergreen trees on campus. I am currently waiting for feedback from four experts in the field. As soon as I get their critiques and polish my paper, there will be developments of Trigonacci in three dimensions, as well as more research on its applications in the real world, such as photography, botany, etc.

Spirals Everywhere!

Found applications of Trigonacci at the Museum of Modern Art this past weekend!

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: