Generalising the Golden Logic

In order to transition from examining the Fibonacci sequence, we must expand the logic describing the creation of the arithmetic patterns commonly found in nature, by generalising fibonacci logic. This expansion in complexity, it will later be argued, is the fundamental mechanism in the evolution of the Universe from the beginning of time forward.

The Fibonacci Sequence is an ordered sequence of numbers, starting with a 0 and a 1, where each value is equal to the sum of the previous two numbers in the sequence.

The Generalised Fibonacci Sequences, often called Gibonacci Sequences, are a series of sequences, where instead of adding the previous 2 (two) terms together to get the next, we add some variable p previous numbers together to get the next number. The pth Gibonacci Sequence is defined as follows:

As before, we start with 0 and 1, but now each successive term in the pth Gibonacci sequence is equal to the sum of the previous p numbers in the sequence.

The Fibonacci sequence [Golden Ratio, Golden Spiral] is merely the first glimpse at the underlying logic of the physical universe. In order to capture an idea of Why the Fibonacci sequence appears everywhere in nature, we must generalise the Fibonacci logic in the following manner:

The Fibonacci sequence is a sequence of numbers, starting with a 0 followed by a 1, where each successive term is produced from the sum of the two previous Fibonacci numbers.

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...]

Mathematically we express the Fibonacci sequence, f(n) as follows:

f(0) = 0

f(1) = 1

f(n) = f(n-1) + f(n-2)

In order to generalise this logic, we will add a new variable and instead of defining each Fibonacci number as the sum of the previous two numbers, we will instead define the Generalised Fibonacci sequences [Gibonacci] as follows:

For some integer number p, the pth Generalised Fibonacci Sequence starts with a number of 0s [p-2 0s by the convention of this site] and a 1, and each successive term in the sequence is equal to the sum of the previous p numbers in the sequence.

In mathematical terms, the statement above produces the following functional representation, for any two positive integers p and n:

for p>1, 0 < n < p-1:  

f(p, n) = 0

for p>1, n = p-1:  

f(p, n) = 1

for n >= p:  

f(p, n) = f(p, n-1) + f(p, n-2) + f(p, n-3) + ... + f(p, p-n-2) + f(p, p-n-1) + f(p, n)

If you are confused at this point, please visit the next page:

Examining the numbers may be far easier to grasp, instead of mathematical functions.

Again, this says that the pth Gibonacci Sequence begins with p-2 zeros and a 1, and each successive term is equal to the sum of the previous p numbers in the sequence.

The generalisation that occurs as a result of adding the second dimension to the input of our function produces some fascinating and elegant results.

Next we look at the numbers that the series of generalized fibonacci sequences produces to understand the simple mathematical concept at work.

 

Thermodynamics, Number Theory and The Goilden Ratio
Creation, Evolution and the Golden Rule
Theory of Order
Why Fibonacci and Gibonacci sequences appear everywhere in nature,
and how simple combinatoric math can describe how a Universe with simple beginnings evolved into a complex form