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*(ex+e-x)
In this case, if the surface is hyperbolic, then the formula
for the hypotenuse of a triangle is
c=cosh-1(½*(ea+e-a)*
½*(eb+e-b))
From Euler's formula, eix=cos(x)+isin(x),
it is also true that ei0=-eiπ. 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, eix=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 eiθ=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%.
