EIGHTH BRIEF:  Asymptotic Variance (Approached Linearity)

“Linear” means straight or flat. “Nonlinear” means curved or exponential variance.

For all practical purposes the earth is flat (except for mountains, hills and valleys). Of course it’s not really flat (at least most people don’t think it is). So, why does the surface of the earth appear to be flat (linear), when it really isn’t?

The surface appears to be flat because we are relatively very small and in close proximity to the curvature.  The smaller we are relative to the size of the curvature and the closer we are to it, the more linear the curve becomes. This is a phenomenon known as perspective variance and is the cause of what artists refer to as the “vanishing point” in paintings. Mathematically it’s called asymptotic variance.

For example, suppose you are in a spacecraft traveling from the moon to the earth. At first, the surface of the earth appears obviously curved, in a spherical shape. As you get closer to the earth however, the curve becomes larger and more linear relative to you. When you reach the earth, the surface has become flat. Then again, not really flat, just so close to flat that you can’t tell it’s not flat.

This size and distance relationship to a curvature we may refer to as “approached linearity” or relative linearity. Approached linearity means we can get extremely close to a linear state, but can never actually achieve linearity. This is asymptotic variance, and this type of curve is defined by an equation referred to as an asymptote (and is related to the nonterminating decimal discussed in the next Brief).

The simplest example of an asymptote is the equation y = 1/x as it varies relative to the x-axis when plotted on an x-y graph. As x increases, the equation y = 1/x decreases (approaches zero), and thus the subscribed curve approaches the x-axis. We say: the equation is asymptotic to the x-axis.

There are two significant points about asymptotic variance, as it pertains to linearity:

1.   As x increases, the curve subscribed by y = 1/x becomes closer and closer to the x-axis, but the curve will never actually reach the x-axis. No matter how large x becomes, the subscribed curve can only approach the x-axis.

2.  Since the x-axis is linear (a straight line), as x increases, the curvature of the equation y = 1/x approaches linearity (approaches a straight line). Here again however, the equation can never become totally linear. Although it may become so close to linear that the curvature is undetectable, it will always be curved to some extent.

An interesting side note: the most essential conservation law of physics says, “A mass cannot displace another mass.” This is an example of asymptotic variance. Masses within the space continuum can only approach displacement but can never reach actual displacement. As masses approach each other, the variance between them is asymptotic. They can get extremely close, but they cannot actually displace each other. When masses are forced together (by gravity for example) the masses start turning into energy that is released in the opposite direction of the force. This appears to be an attempt to force a nonlinear condition into a state of linearity.