From our knowledge about the Fibonacci sequence, the golden ratio, and their respective characteristics, we can build to higher levels of mathematics. In the Explanation section, we found that φ2 = φ + 1, which can be rewritten as φ2 − φ − 1 = 0. This is the characteristic equation for the Fibonacci sequence, and it can be represented as x2 − x − 1 = 0 (Bahşi, p. 2). From the characteristic equation, we can derive the Binet formula, Fn = (φn − (-φ)-n) ⁄ √5, and solve for the roots, where the positive root is the golden ratio φ, and the negative root is 1⁄φ, or 1 − φ.
In 1993 two mathematicians, Stakhov and Tkachenko, introduced hyperbolic Fibonacci functions. Hyperbolic functions are similar to trigonometric functions, however the key difference between the two is that hyperbolic functions are defined based on a hyperbola while trigonometric functions are based on the unit circle. Encountering a hyperbolic function will look like a trigonometric function with an h attached to the end like sinh(x) = (ex − e-x) ⁄ 2. Hyperbolic Fibonacci functions are a special type of hyperbolic functions using the characteristic equation instead of e, and looks like sFs(x) = (φx − φ-x) ⁄ √5 (Bahşi, p. 2). This led Stakhov and another mathematician, Rozin, to define symmetrical hyperbolic Fibonacci functions (Bahşi, p. 2). In their study of this new class of functions, they found that the graphs are similar to normal hyperbolic functions and that they have similar properties (Bahşi, p. 3). We use hyperbolic functions to help us describe the formation of satellite rings around planets and the shape of a curve, usually created by something hanging between two points (Hyperbolic Functions). Since symmetric hyperbolic Fibonacci functions have similar properties, we can continue to find applications of the Fibonacci sequence and the golden ratio in objects that require a hyperbolic description.
Another extension of the Fibonacci sequence is applying the characteristic equation to higher dimensions.
If we were to have a k dimensional object, then the characteristic equation would be
xk − x(k − 1) − x(k − 2) − . . . − x − 1 = 0
which resembles our original characteristic equation (Dutta, p. 5).
When k=2, we have the Fibonacci sequence and the golden ratio of 1.618.
With k=3, we have a sequence of tribonacci numbers, and when k=4 it is tetranacci and so forth according to k (Dutta, p. 6).
Finding the roots of the k characteristic equation will similarly rely on a constant like φ, known as k-nacci constants (Dutta, p. 6).
The constants for k=2, 3, and 4 are approximately 1.6180, 1.8393, and 1.9276 respectively (Dutta, p. 6).
Therefore, they proved that as k increases towards infinity, the k-nacci constant will asymptotically approach 2 (Dutta, p. 6).