# Exploring Euler’s Formula as a Framework for the Universe

Written on

Euler’s formula connects imaginary numbers, natural logarithms, trigonometric functions, and the relationship between a circle’s diameter and circumference, all within a single mathematical expression. I propose that this formula also captures the dynamics of potential and kinetic energy in an ideal oscillating system, offering a valuable framework for understanding the theoretical expansion and contraction of the universe.

The equation is as follows:

If mathematics isn't your strong suit, allow me to clarify the symbols briefly…

<i>e</i>, known as Euler's number, serves as the base for the natural logarithm. Like pi, <i>e</i> is an irrational number with a non-terminating decimal expansion. Its initial digits are 2.718, and it is represented by the following series:

One effective way to conceptualize <i>e</i> is through the lens of compound interest…

Imagine I lend you a dollar with a promise of 100% interest at the year’s end. If the interest is calculated once, you would expect to have $2 at year’s end.

If the interest is applied biannually, the rate would be 50%. After six months, you’d have $1.50, and at year-end, the total would rise to $2.25.

Increasing the frequency of interest compounding leads to similar outcomes. Quarterly compounding yields approximately $2.44, monthly about $2.61, and infinitely frequent compounding approaches <i>e</i>:

$2.718281828…

While there’s much more to discuss regarding <i>e</i> and its characteristics, this overview provides a foundational understanding.

— — —

The next symbol needing clarification is <i>i</i>.

<i>i</i> represents the square root of -1.

<i>i x i</i> = -1

Typically, we don’t associate negative numbers with square roots, but imaginary numbers allow us to envision their existence. If we picture the real number line extending from negative infinity through zero to positive infinity, we can also visualize an imaginary axis perpendicular to this real axis. Complex numbers, which consist of both real and imaginary components, can be plotted on a coordinate plane, with the imaginary part functioning as the “y” axis.

## Kinetic and Potential Energy

Kinetic energy (KE) refers to the energy an object possesses due to its motion, mathematically expressed as e = (1/2)mv².

Potential energy (PE) is the energy an object has based on its position, though other factors can also influence it. PE can vary depending on the specific system being analyzed. For instance, the potential energy of a falling object is defined as PE = mgh, with g representing gravitational acceleration and h signifying the object's height. For a spring, it is defined as PE = (1/2)kx², where k denotes the spring constant and x represents displacement from equilibrium.

While these mathematical definitions of KE and PE hold true within specific mechanical systems, they are not universally applicable. For our discussion, it suffices to understand KE and PE as abstract concepts.

Key takeaways include:

**KE is the energy of motion.****PE is the energy of position (or the potential to induce motion).****Energy values are expressed as e = mass * velocity².****The total energy of KE + PE in any system is conserved.**

As potential energy is utilized, it converts into kinetic energy. Conversely, kinetic energy, such as that of a pendulum, transforms back into potential energy as it swings past equilibrium, demonstrating energy conservation: it merely changes forms rather than disappearing.

In ordinary systems defined by our arbitrary constraints, energy is inevitably lost through friction, air resistance, heat dissipation, and countless factors, leading to a decline in the system’s total energy.

But what if we envision an **ideal** oscillating system, one so efficient that no energy can escape? How would we graphically represent its energy balance?

Euler’s formula offers a groundbreaking and insightful solution.

## Graphing Energy with Euler’s Formula

Entering Euler’s formula into graphing software like Wolfram Alpha yields a graph resembling this:

Here, cosine (blue/real) and sine (red/imaginary) values appear on the same x-y plane. While this representation is convenient, it is not entirely accurate. Multiplying by <i>i</i> rotates the value 90 degrees from the real axis to the imaginary axis. A more accurate graph would incorporate a third axis for the imaginary dimension, positioning the sine wave perpendicular to the cosine wave.

Pardon my rudimentary graphical skills; I make no claims to be a designer. However, the three-dimensionality and helical nature of the graph should be evident.

The green wave represents our cosine x in the real (x,y) plane, while the purple wave illustrates i*sin(x) in the imaginary (x,z) plane.

The correlation with an oscillating system is straightforward. Let Cos(x) symbolize potential energy and i*Sin(x) represent kinetic energy, with x corresponding to time.

When a perfect spring is compressed and released (x=0), it initially possesses maximum potential energy and no kinetic energy. This potential energy quickly transitions to kinetic energy, peaking at equilibrium (x = ?/2) as potential energy reaches zero. Momentum propels the spring beyond equilibrium, generating a negative force that diminishes kinetic energy until it stops at maximum compression (x = ?). The negative potential energy then propels the spring back to the center, repeating the cycle. In an ideal oscillating system, there’s negligible difference between these “beginning” and “ending” moments.

An observant reader may realize that sin(x) and cos(x) do not consistently sum to the same value for any given x. For example, sin(0) + cos(0) = 1, while sin(?/4) + cos(?/4) = 1.414. The sum of KE and PE must remain consistent for energy conservation to hold, suggesting a potential issue that arises from misplacing both graphs on the same axis. This problem resolves when we recognize i*sin(x) as situated on the imaginary plane. Instead of simply adding sin(x) and cos(x), we must perform vector addition, measuring the distance between the two waves at any x value.

The red lines illustrate our vector addition. Regardless of the direction, the magnitude remains constant at 1, as the calculation resembles finding the hypotenuse “a” using the Pythagorean theorem, a²=b² + c². The sum of the squares of sine or cosine always equals one, implying that energy is conserved for our metaphor.

This energy summary method is intriguing because, while preserving a consistent magnitude, it also captures the dynamic nature of energy balance between kinetic and potential forms. In the gif below, the length of the green bar symbolizes total energy at any moment, while its direction reflects the character of that energy.

## The Universe as an Ideal Oscillating System

The infinite or finite nature of the universe remains uncertain. Its shape and dimensionality are also unknown. On a grand scale, we can definitively state that galaxies are receding from one another, with this rate of separation increasing.

Now we venture into pure speculation and thought experiments, making broad assumptions to explore their implications.

From observing galaxies moving away, physicists generally agree that the universe is expanding, suggesting that matter was once densely packed in a singularity, which subsequently underwent a “Big Bang” to create the observable universe.

While we can't conclusively assert that all energy was once concentrated at a single point, we can logically deduce from observations that visible matter was once far more condensed. We can treat the Big Bang as an ideal limit, representing a logical extreme. While history may not have unfolded in precisely that manner, we can be confident it occurred within those logical boundaries.

In an ideal oscillating system, the moment of the Big Bang parallels our maximally compressed spring or suspended pendulum. At this hypothetical singularity, the universe's kinetic energy is zero due to complete stillness, while potential energy is maximized, resulting in immense outward pressure from all visible matter. This pressure must instantaneously propel the condensed matter outward, converting potential energy into kinetic energy. The system's expansion accelerates continuously as long as positive potential energy exerts outward force.

This raises fundamental questions regarding the essence of matter, mass, gravity, energy, and space. When considering matter compressed into a singular point, what about space? Does the universe exist outside that point as an infinite void? Or is space itself included in that singularity? Is the outward movement of galaxies also indicative of space's expansion? And what constitutes this “outward force”? What drives this separation?

Most physicists attribute the accelerating expansion of the universe to “dark matter,” a term coined because it remains unseen, yet it seemingly accounts for about 85% of the universe's total mass.

Introducing an entirely new form of matter complicates a straightforward issue. Since Einstein’s General Relativity, we’ve understood that space possesses physical properties and mediates gravity. If space-time is responsible for attracting distant objects, it’s not surprising that it also accounts for their repulsion. The perceived “missing mass” vanishes when we consider that mass might arise from space's distortion. Instead of merely curving space-time, as proposed in General Relativity, mass could better be defined as the curvature itself.

However, I digress. Redefining mass requires a more extensive discussion. In an upcoming article, I will delve deeper into this new mass definition, its viability, and its far-reaching implications for our understanding of the universe.

For now, whether we consider this unseen pressure as stemming from “dark matter” or space itself, we can ascertain two facts:

- It constitutes approximately 85% of the universe's mass.
- Energy is necessary to generate outward force.

Given #1 and applying it to our Euler's formula graph, we can crudely estimate our position in the timeline. If 85% corresponds to PE, we could simply state: “we are at 85% PE and 15% KE.” With basic calculations—taking the square root of .85 and .15—we could pinpoint ourselves at .921 + .387i, or roughly ?/8 on the x-axis.

Of course, it’s not that straightforward. We cannot assume that dark energy correlates exactly with PE, nor can we take the 85% figure as precise, especially as we refine our understanding of mass. Nevertheless, it’s captivating to consider that Euler’s formula could provide a rough predictive model for the future. By estimating our position x, we might gauge the time until the “equilibrium point” at ?/2, the “Big Crunch” at ?, or the next “Big Bang” at 2?. It’s equally fascinating to perceive time as cyclical and space as a form of energy, suggesting that reality can compress, expand, and stretch beyond its equilibrium, much like a spring.