Can't Help Myself
Sugar pie, honey bunch
You know that I love you
I can't help myself
I love you and nobody else.
But what
about Sugar Pi?
I love pie, but I love Pi, π, even more. A wave has a repeating form that
can be described by its amplitude, its wavelength, and its phase and that typically involves Pi.
Sinusoidal wave functions, such as the sine and cosine, have several distinct characteristics:
- They are periodic functions with a period of 2 π.
- The domain of each function is (-∞, ∞) and the range is [-1,1]
- The graph of y=sin(x)=sin(-x) is symmetric about the origin because it is an odd function.
- The graph of y=cos(x)=-cos(-x) is symmetric about the y-axis because it is an even function.
- A cosine is a sine that has been phase shifted by π/2, one quarter of its period.
It is thus not surprising that hyperbolic sinusoidal wave functions, such as the hyperbolic sine, sinh, and the hyperbolic cosine, cosh, have similar characteristics.
- They are periodic functions with a period of 2 π i.
- The domain of the hyperbolic sine, sinh, is (-∞, ∞) and the range is [-1,1] and while the domain of the hyperbolic cosine, cosh, is also (-∞, ∞) its range is [1, ∞].
- The graph of y=cosh(x)=cosh(-x) is symmetric about the origin because it is an odd function.
- The graph of y=sinh(x)=-sinh(-x) is symmetric about the y-axis because it is an even function.
- A hyperboic cosine is a hyperbolic sine that has been phase shifted by π/2 i, one quarter of its period.
A hyperbola is a function that does not change signs because of the sign of its input. A negative or a positive input always yields a positive output OR a negative or a positive input always yields a negative output. An ellipse can change signs. A negative input can yield a negative or a positive output AND a positive input can yield a negative or a positive output. A hyperbola can change signs, if it is rotated by π radians. (which is equivalent to a rotation of π/2 radians AND a reflection). Euler's Formula is true in an elliptical domain in all cases and in a hyperbolic domain if the phase is less than π/2. It is not true for a hyperbolic domain with a phase greater than π/2. For example, for a complex number that is x+0i, if the phase, rotation of the imaginary axis is π, then Euler’s formula is eix which should be positive but is negative, cos(π) + sin(π)*i= -1. It should be restated in a hyperbolic domain as eix=cosh(x)‑sinh(x)i and then with a rotation of π when traversing domains, it would be eix=1 in both domains.
It is
proposed that the entire universe consists of a two sheeted hyperboloid, while in one sheet it is true that eix=cos(x)+sin(x)i
due to the elliptical identity cos2+sin2=1 while in the other
sheet the hyperbolic identity cosh2- sinh2=1 applies. The observable
universe is one sheet of this hyperboloid, the sheet in which the hyperbolic identity
applies. The two sheets of the hyperboloid connect at the origin. Thus a 2-D Minkowski
space becomes a 3-D two-sheeted hyperboloid when an imaginary axis is added to the two axes
of space and time, and a hyperbolic surface
is rotated by 2π on this imaginary axis. Then the two cones, one
of which is inverted, become two connecting sheets of a two sheeted hyperboloid.
A hyperbolic surface which passes through
the imaginary axis at π and the origin would satisfy eix=cos(x)+sin(x)i
in one sheet and eix=cosh(x)-sinh(x)i in
the other sheet. For any equation to be valid in both sheets, that equation would
require a rotation by π when it passes between the two sheets at
the origin. And that rotation is important and why I love Pi.
· The graph of
·
A hyperbolic cosine is a hyperbolic
sine that has been phase shifted by πi/2, one quarter of its
period.
A hyperbola
is a function that does not change signs. A negative or positive input always yields
a positive outputs OR a negative or positive input always yields a negative output. An ellipse can change signs. A negative input can yield negative
or positive output and a positive input can yields a negative or positive output.
A hyperbola can change signs, if it is rotated by π radians. (which is equivalent
to a rotation of π/2 radians AND a reflection). Eulers’s formula is true
in an elliptical domain in all cases and in a hyperbolic dominance if the phase
is less than π/2. It is not true for a hyperbolic domain with a phase
greater than π/2. For example, for a complex number that is x+0i,
if the phase, rotation of the imaginary axis is π, then Euler’s formula
is eix which should be positive but is cos(π) + sin(π)*i= -1.
It should be restated in a hyperbolic domain as exit=cosh(x)‑sinh(x)i=and
with a rotation of π it would be eix=1 in either domain.
It is
proposed that the unobservable universe consists of a tow sheeted hyperboloid, when
in one sheet it is true that eix=cos(x)+sin(x)i
due to the elliptical identity cos2+sin2=1 while in the other
sheet the hyperbolic identity cosh2-sinh2=1 applies. Th observable
universe is one sheet of this hyperboloid, the sheet where the hyperbolic identity
applies. The two sheets of the hyperboloid connect at the origin. Thus a 2-D Minkowski
space become a 3-D hyperboloid when an imaginary axis is added to the two axis
of space and time, and hyperbolic surface
is rotated by 2π on this imaginary axis, and the two cones, one
of which is inverted, becomes two connecting sheets of a two sheeted hyperboloid.
Then a hyperbolic surface which passes through
the imaginary axis at π, and the origin would satisfy eix=cos(x)+sin(x)i
in one sheet and eix=cosh(x)-sinh(x)i in
the other sheet. For any equation to be valid in both sheets, that equation would
require a rotation by π when it passes between the two sheets at
the origin. And that rotation is important and why I love Pi.