Math

 

All About The Bass

Because you know I'm all about that bass
'Bout that bass, no treble
I'm all about that bass, 'bout that bass, no treble
I'm all about that bass, 'bout that bass, no treble
I'm all about that bass, 'bout that bass (bass, bass, bass, bass)

Actually it’s all about the MATH!

Most people are familiar with Pythagoras’ Theorem, c=√(a²+b²), where c is the hypotenuse of a triangle with sides a and b.  This is actually only true for a flat, Euclidean, surface.  It is true because of the identity of a circle, cos²+sin²=1. For a triangle on a flat surface, the hypotenuse can also be expressed as

 cos(c)=cos(a)*cos(b).

For a triangle on a spherical surface, the formula for the hypotenuse of a right triangle is based on a similar principle, but it is

cos(c/R)=cos(a/R)*cos(b/R)  or c=R*cos^-1(cos(a/R)*cos(b/R))

where R is the radius of that spherical surface.  As R approaches infinity, it becomes identical to Pythagoras’ Theorem.

For a triangle on a hyperbolic surface, the formula for the hypotenuse is

cosh(c)=cosh(a)*cosh(b) or c=cosh^-1(cosh(a)*cosh(b)).

This is true because of the identity of a hyperbola, cosh²-sinh²=1.

What is true for any surface, regardless of its shape, is only that c²=a²+b². 

For the cylindrical coordinates (r, θ, z), of a point (x, y, z) in rectangular coordinates, it is true that x=r*cosθ, y=r*sinθ, z=z, and θ=tan^-1(y/x).  However the solution of r as √(x²+y²) is only true if x and y are on a flat surface. On any surface it is only true that r²=x²+y², and the solution for r depends on the shape of that surface. 

Additionally, hyperbolic trigonometric functions are defined in terms of exponentials such that

cosh(x)=½*(e^x+e^-x)

In this case, if the surface is hyperbolic, then the formula for the hypotenuse of a triangle is

c=cosh^-1(½*(e^a+e^-a)* ½*(e^b+e^-b))

From Euler's formula, e^ix=cos(x)+isin(x), it is also true that eⁱ⁰=-e^iπ.  Thus it is suggested that Pythagoras’ Theorem, taking the imaginary axis into consideration, is better simplified as  

c=ln(-cosh(a²+b²)±sinh(a²+b²))

rather than the conventional

c=√(a²+b²).

However it is also acknowdged that, when a and b are much less than infinity, just as Pythagoras’ Theorem can be used on the surface of the spherical Earth where its radius, R, is much greater than a or b,  Pythagoras’ Theorem is still a reasonable approximation. In fact there is virtualy no diffence between the two until a and b are more than 20% of the absolute/infinity.  The difference between the two does not become appreciable until after a and b are more than 7%[1] of the absolute/infinity.

Euler’s formula, e^ix=cos(x)+isin(x) can also be viewed as the rotation of Minkowski’s spacetime, reality, on an imaginary axis to create cylindrical coordinates. If, in Eulers’ formula, x is taken to be θ, that is  e^iθ=cos(θ)+ sin(θ)*i, this can then be viewed as a complex number in rectangular coordinates, a+b*i, where sin(θ) can be viewed as a rotation of the imaginary axis, i, by θ, and cos(θ) is the real portion of this complex number. However, while sin(x) is cyclical, and sin(0)=sin(π)=0,  the cosine, the real part of the complex number in rectangular coordinates, is out of phase with the sine such that cos(0)=1 but cos(π)=-1.  Thus it is proposed that if a surface passes through an origin of a two-sheeted hyperboloid such that sin(0)=sin(π)=0, that in one sheet of the two sheeted hyperboloid, cos=1, while in the other sheet cos=-1. This implies that if our universe, reality, is on one sheet of a two sheeted hyperboloid, that at its origin it must have undergone a phase change such that a hyperbola in one sheet, our universe, is a reflection of a hyperbola in the other sheet, and that one sheet is also a phase shift of π from the other sheet.

Setting the coefficient of the imaginary axis to 0, does not mean that the axis is eliminated, only that its coefficient is zero.

[1] Actually the mean of a choice of one, squared, multipled by π squared divided by three, is 0.5²π²/3 , which is equal to the variance, or 82.2%.