The Spectrum: Representation Through Multiple Disciplines

Everything can be thought of as a spectrum. Any adjective can be defined in varying levels, any debate has two sides, and any feeling can be felt at different amounts. In life, things are not one concrete entity, but rather they lie somewhere on a broader scale.

Even the number line itself is a “spectrum” and its form can be used to understand other concepts. With zero in the middle and negative and positive infinity on each end, we can view any two ends of a spectrum as “infinities” too, and zero represents the point of neutrality directly in between both extremities. 

It is often the case that we perceive one end of the spectrum as “good” and the other “bad.” For example, in the spectrum with the counterparts happiness and sadness, happiness often has a positive connotation and sadness a negative one. Yet being “too happy” is unhealthy just as much as being too sad is. Similarly, if we look at the ideas of love and hate, both hating and loving something too much can be equally harmful. 

This manifests clearly in opinions too. As humans, we form views about different topics and controversies. These views are often strong and so they fall close to one end of the spectrum, resulting in us discrediting the other side completely. Yet this is an inaccurate view of reality; by leaning too far on one side, we are limiting ourselves from understanding the full truth and setting ourselves up for failure. 

In any type of spectrum, reaching a point too far out on either end can be unhealthy and even dangerous. Although each end is expressed differently such as how being too happy looks different from being too sad, both points represent extreme states of imbalance where there is a lack of control.  

The word “spectrum” comes from the Latin words “specter” and “specere” which means to look at or view. “Specter” is defined as a ghostly apparition, a ghost itself, or simply an idea that people find frightening. From the same origin, “spectral” means having a ghostlike quality, seeming to vanish or disappear, or simply an idea that people find frightening. Perhaps it is no coincidence that the closer we approach the ends of the spectrum, the more dangerous and “scary” it can be. 

The two ends of a spectrum are also often perceived as opposites. For example, love and hate, hot and cold, positive and negative, and right and wrong are common antonym pairs. However, in their extreme states, these pairs are far more similar than they are different. Mathematically, this can be proved by looking at positive and negative infinity: although positive infinity and negative infinity have distinct mathematical identities, they both represent unboundedness, signifying that a sequence, function, or value is not limited by any finite number. They share many mathematical properties involving operations, such as how adding a finite number to infinity yields infinity, and adding a finite number to negative infinity results in negative infinity.

The true opposite of both extremities is the midpoint of the spectrum: the point of neutrality. The opposite of being too in love or in deep hatred is a state of indifference, the opposite of being excessively hot or cold is being at a perfect temperature, and the opposite of being so one-sided in a belief to the point of not being able to acknowledge the other side is being completely impartial to either. 

Thus, the spectrum is better represented as a circle with both of the extremities connecting to create one single state of “infinity.” On the opposite end of this circle lies this infinity’s true counterpart: the “zero” on the numberline.

The projectively extended real line is the extension of the number line by a point denoted ∞. It is thus the set R ∪ { ∞ } (where R is the set of the real numbers) with the standard arithmetic operations extended where possible. The added point is called the point at infinity, because it is considered as a neighbor of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded. 

The Riemann sphere is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point ∞  is near to very large numbers, just as the point 0 is near to very small numbers.

Unlike most mathematical models of numbers, the projectively extended real line and Riemann sphere allow for division by zero. The function f(x) = 1 / x  is not defined at x = 0 on the extended real number line where negative and positive infinity are treated as distinct points because division by  is not allowed. Yet when negative and positive infinity are treated as one point of infinity, the function becomes perfectly well behaved. Thus, a lot of complex analysis is done on the Riemann sphere rather than the two-dimensional plane. 

Just like some functions that are not defined on the number line become defined on the  projectively extended real line and Riemann sphere, our understanding of life becomes more accurate when we treat the spectrum as a circle. On this circle, we can see that it is healthiest to remain somewhere with substantial distance from the point of extremity. 

Yet the other place we do not want to be is directly at zero. Although we want to stay away from the infinity, we do want to reach its exact opposite either. Being confined at this point in the exact middle or neutral stance is also a form of extremity in and of itself. 

The Golden Mean is a theory proposed by Aristotle that all “virtues” lie in the middle between two extreme states: excess and deficiency. However, it lies not in the exact middle, but it will often be closer to one extreme than the other. For example, in the spectrum with the extremes of recklessness and cowardice, being in the middle might represent inaction. Instead, the point where courage would lie is closer to recklessness.

We can think of infinity as a concept that represents the absence of limits, and zero as a number that represents the absence of value. Like how negative and positive infinity share many of the same mathematical properties, so do infinity and zero. 

To live the healthiest life we can, we want to be far from the point of extremity and not directly on the zero either but at this fluctuating, dynamic in-between range.

View a representation of this idea through movement: https://drive.google.com/file/d/1wYt-GoLSlbxkzQnOKCXMGPBVv0jaUbp6/view?usp=sharing

To summarize:

• Everything can be thought of as a spectrum.

• It is often the case that we perceive one end of the spectrum as “good” and the other “bad.”

• Reaching a point too far out on either end can be unhealthy and even dangerous.

• The two ends are also often perceived as opposites. However, in their extreme states, these pairs are far more similar than they are different.

• The true opposite of both extremities is the midpoint which is also a type of extremity

• The spectrum is better represented as a circle

• The healthy choice is often to be at a dynamic point in between the extremeties