Pi

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 e^ix which should be positive but is negative, cos(π) + sin(π)*i= -1. It should be restated in a hyperbolic domain as e^ix=cosh(x)‑sinh(x)i and then with a rotation of π when traversing domains, it would be e^ix=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 e^ix=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 e^ix=cos(x)+sin(x)i in one sheet and e^ix=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.

 

 

 

 

 

 

 

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(−∞,∞)

·        The graph of y=cosh(x

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·        The graph of y=sinh (x

·        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 e^ix which should be positive but is cos(π) + sin(π)*i= -1. It should be restated in a hyperbolic domain as e^xit=cosh(x)‑sinh(x)i=and with a rotation of π it would be e^ix=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 e^ix=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 e^ix=cos(x)+sin(x)i in one sheet and e^ix=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.

 

 

 

 

 

 

 

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