The Mathematics of Nature's Beauty

Everything is mathematics. It’s not just found in a textbook, or on a chalk board. It’s not limited to calculating your coupon savings at a grocery store, or even to the deeply complex world of quantum mechanics. Everything is mathematics… The universe, and everything in it is mathematics. The body that you live in everyday, the consciousness that lives in your mind, even the etheric soul that drives you, it’s all mathematics. Nature is nothing but numbers. And they are beautiful!

It all starts with 0 and 1. Yes and no; on and off; black and white; all and nothing. Zero and one begins it all. The electronic device that you’re reading this on is based on ones and zeroes; the leaf you see whipping in the autumn wind outside your window is based on zeroes and ones; this is where it all starts. The richness of the world is binary at heart, and all the beauty and diversity we see every day is but a set of combinations of the binary underworld. Let’s get started…

The Growth Function

The simplest form of arithmetic is 0 + 1 = 1. This is simply adding the simplest something to nothing. But, once we have one, we can add another one to it; hence, 1 + 1 = 2. Now, we have growth, the simplest form of growth, a doubling. If we follow this idea, then we can continue; 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16, etc. Here we have geometric growth based on doubling, represented by 2^x. The problem is, however, that this simple growth is based on set intervals; if we were to put the terms into a summation, then it becomes quickly obvious that the population of whatever we’re summing goes to infinity. Infinity is not fun to deal with, nor is it truly realistic in the real world.

So, how do we make it realistic? We must focus on a single time interval to allow for complex, or compound, growth. If you have a dollar, and you give it to me for investment knowing that you’ll get double your money after a year, then you will, indeed, have two dollars after twelve months. If I promise the doubling every year, then I get the following mathematical expression: ($1 + 100%)^# of years. But, what if I said that I’ll guarantee you a rate that will double your money after 12 months, but I’ll redo the calculation every six months? Or every month? Or every day? It would look like the following, after one year:

[$1 + (100%/2 periods)]^2 periods = $2.25

[$1 + (100%/12 periods)]^12 periods = $2.43

[$1 + (100%/365 periods)]^365 periods = $2.72

What we’re seeing in the above is splitting a single time interval into many periods of compounding, and the value is not going to infinity. In fact, it won’t even reach three! But this is real world behavior. Even if you begin with a single organism that can only split into two, the variable timing involved with this mitosis is what requires us to look at a singular time period compounded infinitely. Thus, taking the above to the extreme, we have:

lim n=>∞ [1 + (1/n)]^n = 2.7182818… = e

The value, e, is also known as Euler’s Number, and is the basis, as explained, of all natural growth in the universe, and is also the basis of logarithmic change. If you understand calculus, one of the more beautiful properties of e is the fact that it is its own derivative when used in the exponential function, e^x. In simpler terms, the value of the rate of change of the function at any point is equal to the value of the function at that point. This also means that e^x is also its own integral, or that the value of the area between the X-axis and the function, from -∞ to any point, is also the value of the function at that point. If you’re not a math nerd, perhaps the beauty of that may be lost on you.

The Circular Function

When it comes to geometry, envisioning the numbers 0, 1, and 2 are somewhat difficult. For me, I like to think of 0 as a line, whose definition is a one-dimensional specie of infinite length. The number one then becomes a ray, which starts at a point and goes on to infinity in one direction. Finally, two is represented by a line segment with a marked point midway between the endpoints. This provides a kind of angular definition as well, with one having an angle of 360˚ and two having an angle of 180˚. Thus, the simplest form of a polygon then requires three line segments, aka, a triangle. If we assume the line segments making up the triangle to be of equal length, we then end up with an equilateral triangle whose interior angles are 120˚.

Within the triangle, we can find the center (or centroid) and connect it to each vertex, bisecting the angles. If we assume the length of each bisector to be unity, or 1, then we can consider them to be radii of a unit circle for the remainder of this discussion. So, let’s find the perimeter of the triangle. We now must draw a bisector for an interior angle to the midpoint of one side. We know the length of radius, and we also know that the bisected interior angle is 60˚. Using our trigonometric ratio of sine, we know that the half-length of the side is sin(60˚) = (x/2)/1. This can than be rewritten as (√3)/2 = x/2, therefore x = √3. It also follows that the perimeter of the triangle is 3√3.

But we’re trying to equate this to the circumference of a circle, which is C = 2πr. As previously stated, r is unity, or 1. Let us now take the triangle’s perimeter, 3√3, and divide it by 2r for an equivalence to π; the result is 3*[(√3)/2]. If you look closely, you can see something in that value. A triangle has three sides, and if we attribute the value n as the number of sides, we can rewrite the above as n*[(√3)/2]. And we also know that (√3)/2 = sin(60˚); however, not all interior polygon angles are 60˚. Using n again, we can rewrite 60˚ as 180˚/n. What does this mean? At this point, not much, for the value of the π-equivalence above is 5.1962, which is a far cry higher than 3.1415. What if we increase the number of sides?

For n = 4,5,10,20,…,100, the following values for the π-equivalence using n[sin(180/n)] are calculated: 2.8284, 2.9389, 3.0902, 3.2188, …, 3.1411. The values are clearly approaching the actual value of π; so, if we run the π-equivalence to its limit, ∞, it does, indeed, result in the value of π itself.

The Fibonacci Sequence

Ok… So what? What does the above have to do with 0 and 1? On the surface, e and π really have nothing to do with 0 and 1, they are esoteric, termed “transcendental” numbers, and they are irrational and cannot be solved using algebraic solutions. But they are numbers of the natural world, as are 0 and 1, so there is a relationship to be found. To do this, we must begin with the Fibonacci Sequence. Starting with 0 and 1, we develop a sequence by adding them, 0 + 1 = 1. Now, we add the answer to the original 1 and get 1 + 1 = 2. Repeating this gives:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

If we assign n to represent each number in the sequence, then we can show it as follows:

n = 1, 2, 3, 4, 5, 6, … where

Sn = 1, 1, 2, 3, 5, 8, …

Now, let’s calculate Sn-1/Sn: 0/1, 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, …

And taking the limit, lim n=>∞ (Sn-1/Sn) = 1.61803…

This is, like the other two, an irrational number; but is it transcendental? The answer to that is ‘no’, and here’s why. While we know that 5/8 ≠ 8/13, at the limit of infinity we can assume that the last two fractions in the sequence are equal. Let’s assign variables so that we get the following relation: a/b = b/(a+b). If we cross-multiply, then we get a^2 + ab = b^2. To make it more algebraic, we will now say that a = 1 and b = x, resulting in 1 + x = x^2, and finally x^2 – x – 1 = 0. Not surprisingly, if we solve for x, the positive root is (1+√5)/2, or 1.61803… This is the definition of the Golden Ratio, φ.

What makes this number so fun? Well, you can see that φ + 1 = φ2, or, in other words, by adding one to the value of φ, you get its square. If you divide the equation above by φ, you get 1 + (1/φ) = φ, or 1/φ = φ – 1. So, by subtracting one from φ, you get its inverse! How about φ^3? By multiplying the above equation by φ, we get φ^2 + φ = φ^3. But we already have a value for φ^2, which is φ + 1, hence, φ^3 = φ + 1 + φ = 2φ + 1. Then there is the following:

(φ^2)^2 = (φ + 1)^2 => φ^4 = φ^2 + 2φ + 1 = (φ^2 + 1) + 2φ = φ + 2 + 2φ = 3φ + 2

=> φ^4 = φ^2 + 2φ + 1 = φ^2 +φ^3

I showed two separate answers for φ^4, but the first answer can be related back to the Fibonacci Sequence. Using the variable n, we can achieve the following general equation: φ^n = (Sn)φ + Sn-1. This is the Golden Power Rule, and it can be used to calculate the value of any number in the Fibonacci Sequence through the following derivation:

Let σ = (1 + √5)/2 and τ = (1 - √5)/2, the roots of the quadratic x^2 – x – 1 = 0. Using the Power Rule, we have two equations:

σ^n = (Sn)σ + Sn-1 and τ^n = (Sn)τ + Sn-1

Subtracting the above equations results in: σ^n - τ^n = (Sn)(σ – τ) + Sn-1 - Sn-1

σ^n - τ^n = (Sn)[(1 + √5)/2 - (1 - √5)/2] = (Sn)(√5)

Knowing that σ and τ are conjugate roots, with σ = φ, τ can be calculated to be (1 – φ):

Sn = [φ^n – (1 – φ)^n] / √5, and one can calculate the value of any number in Fibonacci Sequence.

But that still gives no relation to π or e. For that, we must think of φ in two dimensions. For each number in the Fibonacci Sequence, we make a square and put them together in a counterclockwise spiral. In each square we draw an arc that connects opposite vertices, and we end up with the picture above. The Fibonacci spiral is not a true spiral, but an estimation of one.

You will notice, though, that even this estimation shows growth and curvature; and that means that the real spiral, the Golden Spiral, will include π and e. Using the above picture, one can gain from it that the radius of a Golden Spiral will increase by a factor of φ for every 90˚ of turn, or π/2 radians. Without going through the proof, the equation for the radius of a logarithmic spiral is r = ae^bθ. Variable a is the initial radius of the spiral, which is 1 in this case; and b is the growth multiplier. We just stated that the growth of the Golden Spiral is φ for every 90˚, thus e^b(π/2) = φ. Solving for b, we get b = (ln φ) / (π/2). So, the final equation for the radius of the Golden Spiral is

r = e^(θ*ln φ)/(π/2)

A comparison of the Fibonacci Spiral (green) and the Golden Spiral (red) is shown below, with overlapping in yellow.

Where have you seen such spirals before? Well, let me tell you, if you think you haven’t seen one by now, then you didn’t know what you were looking at. Take a gander at where Golden Spirals can be found…

The beauty of nature begins with the binary, with zero and one. The complex is nothing more than a child of the simple. From zero and one we have built a pinecone, a sunflower, a hurricane, and a galaxy. The nature of mathematics is the foundation of the universe, and no matter what deity you believe rules the cosmos, none of them could have done anything without zero and one.