# Euler's Formula: A Unique Perspective on the Universe's Dynamics
Written on
Chapter 1: Understanding Euler's Formula
Euler's formula intricately weaves together imaginary numbers, the natural logarithm, trigonometric functions, and the constant ratio found in circles, all encapsulated within a single equation. I propose that this equation can also serve to illustrate the concepts of potential and kinetic energy in an ideal oscillating system, thereby offering a compelling framework for contemplating the universe's hypothetical expansion and contraction.
The equation is expressed as follows:
If mathematics isn't your forte, let's break down the symbols quickly.
The letter e, known as Euler's number, serves as the foundation for the natural logarithm. Much like pi, e is an irrational number, meaning its decimal representation goes on infinitely. The initial digits are 2.718, and it can be defined through a specific series.
A straightforward way to grasp e is through the concept of compound interest.
Imagine I lend you a dollar and promise to pay you 100% interest after one year. If I compute and credit the interest just once at the end of the year, you would expect to have $2. However, if the interest is calculated semi-annually, each half-year would yield a 50% interest rate. After six months, you would receive $0.50, bringing your total to $1.50. At the year's end, you'd earn 50% interest on the new total, resulting in $2.25.
The more frequently we calculate interest, the closer we get to e. For example, quarterly compounding gives you approximately $2.44, monthly gives about $2.61, and if we compound continuously—essentially paying you interest every moment—the total approaches e at approximately $2.718281828…
Now, let’s clarify the next symbol: i.
The letter i represents the square root of -1.
Thus, i × i = -1.
While negative numbers typically don't have square roots in the realm of real numbers, the use of imaginary numbers allows us to explore this concept. If we visualize the real number line extending from negative infinity through zero to positive infinity, we can imagine the imaginary axis as a perpendicular line to this real axis. Complex numbers, which consist of both real and imaginary parts, can be plotted similarly to points on a (x,y) coordinate plane, where the "y" axis represents the imaginary dimension.
Kinetic and Potential Energy
Kinetic energy (KE) refers to the energy an object possesses due to its motion. In classical mechanics, it is defined as ( KE = frac{1}{2} mv^2 ).
Potential energy (PE) refers to the energy an object holds due to its position and can be defined differently based on various scenarios. For instance, for a falling object, the formula is ( PE = mgh ), where g represents gravitational acceleration and h is the object's height. In the case of a spring, it is expressed as ( PE = frac{1}{2} kx^2 ), where k is a constant representing the spring's stiffness, and x is the displacement from its equilibrium position.
While these mathematical expressions for KE and PE are useful for specific mechanical systems, they should be understood as abstract concepts.
Key takeaways are:
- KE is the energy associated with motion.
- PE is the energy linked to position or the potential to create motion.
- Energy values are expressed as ( e = text{mass} times text{velocity}^2 ).
- The total energy, comprising KE and PE, remains conserved within any system.
As potential energy is utilized, it converts into kinetic energy. Conversely, kinetic energy transitions to potential energy, as seen in a pendulum that slows due to gravity while passing its equilibrium point. This process illustrates the conservation of energy: it never truly disappears, only changes form.
In typical systems, defined by our own constraints, energy inevitably dissipates through friction, air resistance, heat loss, and various other factors, leading to a gradual decline in the total energy within the system.
Now, imagine an ideal oscillating system so effective that no energy can escape. How would we graph its energy balance?
Euler's formula provides a fascinating solution.
Chapter 2: Visualizing Energy with Euler's Formula
If you input Euler's formula into a graphing tool like Wolfram Alpha, the resulting graph will typically illustrate the cosine and sine waves overlaid on the same plane. However, this representation may not fully capture the concept. When we multiply a value by i, the imaginary unit, we effectively rotate that value 90 degrees from the real axis to the imaginary axis. A more accurate depiction would introduce a third axis, showcasing the sine wave in the imaginary dimension, perpendicular to the cosine wave.
Please forgive my rudimentary graphical representation. Although I’m no design expert, I hope the three-dimensional and helical structure of the graph is clear.
The green wave represents our cosine function in the real (x,y) plane, while the purple wave illustrates ( i cdot sin(x) ) in the imaginary (x,z) plane.
This relationship to an oscillating system is quite straightforward. Let ( cos(x) ) signify potential energy and ( i cdot sin(x) ) represent kinetic energy, with x denoting time.
When we compress a perfect spring and release it (at ( x=0 )), it possesses maximum potential energy and no kinetic energy. This potential energy quickly translates into kinetic energy, reaching its peak at the equilibrium point (( x = frac{pi}{2} )), while potential energy dwindles to zero. The momentum carries the spring past the equilibrium, generating a negative force that reduces kinetic energy until the spring halts at its maximum compression. The process then reverses, accelerating the spring until it reaches equilibrium again at ( x = frac{3pi}{2} ), where the force direction shifts, diminishing kinetic energy until the spring stops at its compressed point—marking the cycle's beginning once more at ( x = 0 ) or ( x = 2pi ).
In this ideal system, the initial and final moments are indistinguishable.
An observant reader may note that the values of ( sin(x) ) and ( cos(x) ) do not yield a consistent total at any given ( x ). For instance, ( sin(0) + cos(0) = 1 ), while ( sin(frac{pi}{4}) + cos(frac{pi}{4}) = 1.414 ). To adhere to the conservation of energy, the sum of KE and PE should remain constant, raising a potential dilemma. However, this discrepancy arises from misplacing both graphs on the same axis. The issue dissipates when we consider ( i cdot sin(x) ) existing on the imaginary plane. Instead of merely adding ( sin(x) + cos(x) ), we must perform vector addition, which essentially measures the distance between the two waves at any given ( x ) value.
The red lines illustrate this vector addition. Regardless of direction, the absolute magnitude remains 1, as this calculation is akin to finding the hypotenuse using the Pythagorean theorem: ( a^2 = b^2 + c^2 ). At any point, the sum of the squares of sine and cosine will always equal one, affirming that, for our metaphor, energy is conserved.
This method of summarizing energy is intriguing because it retains a consistent magnitude while simultaneously portraying the energy's varying nature in terms of its kinetic and potential components. In the accompanying visualization, we can interpret the green bar's length as the total energy at any moment and its direction as the nature of that energy.
Chapter 3: The Universe as an Ideal Oscillating System
We remain uncertain whether the universe is infinite or finite and do not fully understand its shape or dimensionality. However, on a macro scale, we know with certainty that galaxies are receding from one another, and the rate of this expansion is accelerating.
At this juncture, we enter the realm of speculation and thought experiments, making broad assumptions to explore their implications.
Based on the observation that galaxies are moving away from each other at increasing rates, physicists generally agree that the universe is expanding. This implies that matter was once much more densely packed—a singularity—before the "Big Bang" propelled it outward to create the observable universe we see today.
While we cannot definitively state that all energy was once condensed into a single point, we can logically deduce from our observations that visible matter was once significantly more concentrated than it is today. The Big Bang serves as an idealized limit, a conceptual extreme of reality. While the history of the universe may not have unfolded precisely in this manner, it certainly occurred within the constraints of this logical framework.
In an ideal oscillating system, the moment of the Big Bang mirrors the scenario of a maximally compressed spring or a pendulum held at its peak. At this hypothetical singularity, the universe would possess zero kinetic energy, as no motion exists. However, it would have an incredibly high potential energy due to the superposition of all visible matter exerting outward pressure. This pressure must instantaneously convert to kinetic energy, propelling the matter outward as the system accelerates, fueled by the remaining potential energy.
At this stage, we must contemplate fundamental questions regarding matter, mass, gravity, energy, and space. When we discuss matter being compressed into a singularity, what about space? Does the universe exist outside that point, like an infinite void, or is space itself also compressed into that singularity? Does the expansion of galaxies also signify the expansion of space itself? And what do we mean by "outward force"? What drives this separation?
Most physicists attribute the accelerating expansion of the universe to "dark matter"—a term coined for this unseen mass, which seems to account for up to 85% of the universe's total mass.
Introducing a new type of matter may complicate matters unnecessarily. Since Einstein's General Relativity, we have understood that space possesses physical attributes and mediates gravitational forces. If we accept that spacetime is responsible for attracting distant objects, it is reasonable to consider it may also be responsible for their repulsion. The apparent "missing mass" phenomenon vanishes when we entertain the notion that mass is a function of space distortion. Rather than merely influencing spacetime curvature, as proposed in General Relativity, mass may be better understood as the curvature itself.
I digress; redefining mass warrants a more thorough examination. In a future article, I will delve deeper into this new mass definition, its viability, and the numerous implications it could have for our comprehension of the universe.
For now, whether we attribute this unseen pressure to "dark matter" or the properties of space, we can agree on two points:
- It constitutes roughly 85% of the universe's mass.
- Energy must be expended to create the outward force.
Given this knowledge and applying it to our Euler's formula graph, we might crudely approximate our temporal position in the universe's history. If the 85% correlates directly to potential energy (PE), we could simply state: "we are at a position of 85% PE and 15% KE." With some calculations—specifically, the square roots of these values—we could determine our position as ( 0.921 + 0.387i ), roughly equivalent to ( frac{pi}{8} ) on the x-axis.
Of course, the reality is not so straightforward. We cannot assume that dark energy corresponds perfectly to PE, nor can we take for granted the accuracy of the 85% estimate, especially when we plan to refine our definition of mass. Nevertheless, it is an intriguing notion that, through Euler's formula, we might develop a model to predict the future, even in a rough sense. By estimating our x-position, we could anticipate when we might reach the "equilibrium point" at ( frac{pi}{2} ), the "Big Crunch" at ( pi ), or the next "Big Bang" at ( 2pi ). This notion of time as a cyclical entity and space as a form of energy is captivating—reality itself can compress, expand, and stretch beyond its equilibrium point, much like a spring.