Zero Point: Solving for it and what it is.

[Antonio Lao]

Furthermore, if the invariance given by

$$\vec{a} \cdot \vec{r} = c^2$$

is inserted into the simple formula for angular frequency

$$\vec{a} \cdot \vec{r} = \left( \frac{2 \omega}{\pi} \right)^2$$

 

[Epsilon Pi]

Are you not violating the uncertainty principle with a unitless radius of 1? At those levels you cannot suppose a particle with a definite radius, can you?
Regards
EP

therefore the above definition of angular frequency implies a unitless radius of 1.

 

[Epsilon Pi]

Remember it is not a wave, it is not a particle, it is an energetic system, whose state cannot be determined completely
Regards
RP

The circumference of a circle is $$2 \pi r$$, therefore the above definition of angular frequency implies a unitless radius of 1.

 

[Antonio Lao]

I think, in physics, the unitless radius is replace by the phase angle of a wave. a right triangle of sides 3-4-5 always has the same complementary angles regardless of how the sides are scaled as long as the ratio remains 3-4-5. The invariance of the phase angle is a scaling transformation invariance applicable to all similar right triangles at any given scale. And all trigonometric functions is the ratio of two sides of a particular right triangle. In a sense, it is applying the Pythagorean theorem over and over again. This theorem is the basis for the definition of a length and a distance and any higher dimensional metric.

 

[Epsilon Pi]

If I have in mind, at the background Euler relation, in a certain sense I can follow you, but then in it, phase angle is quite different from the radius or amplitude of that wave.
Regards
EP

I think, in physics, the unitless radius is replace by the phase angle of a wave...In a sense, it is applying the Pythagorean theorem over and over again. This theorem is the basis for the definition of a length and a distance and any higher dimensional metric.

 

[Antonio Lao]

The existence of a minimum triangular surface area of 1/2 corresponding to a unit square does not depend on the curvature of spacetime where the surface is embedded and clearly demonstrated in an Euclidean geometry using the parallel axiom.

 

[Antonio Lao]

The Euler's identities are given by

$$e^{+ i \theta} = cos \theta + i sin \theta$$

and

$$e^{- i \theta} = cos \theta - i sin \theta$$

A complex number z is given by

$$z = r \left( cos \theta + i sin \theta \right)$$

So Euler's identities is the same as when r=1 and

$$z = e^{ i \theta}$$

 

[Antonio Lao]

But the complex number is defined as z = x + iy. This is like adding apples to oranges where x is an apple and iy is an orange. Physically, I still failed to understand this but mathematically, I guess, anything is possible.