Cordinate Transformation

 

Reflections

Through the mirror of my mind
Time after time
I see reflections of you and me
Reflections of
The way life used to be
Reflections of
The love you took from me

But can you reflect an absolute?

Mathematically you can translate a function with respect to an x-axis as g(x)=f(x-µ), where µ is the new location.  You can rotate a function with respect to an x-axis as g(x)=sin(θ)*f(x) where θ is the angle between the rotated and the original x-axis.  You can reflect a function with respect to an x-axis as g(x)=-f(x). But if x is the relationship to an absolute then that can restrict the ability to translate, rotate, or reflect a function of x, f(x). Translation, rotation, and reflection together can be called cordinate transformation

The simplest one to explain is the translation.  There is an absolute zero temperature.  It is measured in degrees Kelvin.  There is a 0° Kelvin but by definition you can not have a temperature below 0 degrees Kelvin.  It is more convenient for us to use a translated temperature scale where the origin, zero, has been translated by 273.15 degrees Kelvin, to the freezing point of water.  You can observe a temperature of -273° Celsius but you can’t observe a temperature of -274° Celsius because that would be below absolute zero.

Rotating a function in one direction is equivalent to rotating a function in the opposite direction by 2*π because the period of the sine function is 2π.  This has an impact on Euler’s Formula e^ix=cos(x)+isin(x) which is a rotation of the imaginary axis by x degrees.  If x is the relationship to the absolute, then x can not be less than zero because you can’t rotate the imaginary axis by something less than absolute zero.  That is why e^ix cannot be defined as a real number, or else it would create the paradox which would require both that cos(-x)= cos (x) and cos(-x) = -cos(x). 

This same restriction applies to the reflection.  If the origin of zero has been translated to µ, then you can reflect that function between µ and 0, but you can not reflect that function below 0 or else it would be a reflection below absolute zero.

Since a random number has the parameters µ and σ, and the variance, σ², defines the range of that random number, a random number must be µ>3σ or else the random number could be below absolute zero.