Cyclic Frequency - Cyclic Angular Frequency

[Harmonic]

It is a minor point, but I find the current descriptive definition for frequency to be a bit lacking. We are taught that an event or a specific period can be represented as a cycle. If it is a repetititve cycle it can be expressed as how many cycles occur in a specified duration, and the duration didn't have to be that of the SI second. Classically, the term frequency is defined as,

f = 1/T

and the numeric result is typically dimensioned as s ^-1, and we know that this refers to "per second". If the "1" had been replaced with "P", for period, it would be universal, where "P" could be one or more than one. Everything is okay if one understand that cycle, a dimensionless value, is a phantom descriptor. Going back a ways, frequency had been annotated as cycles per second (cps), which made sense, as one cycle represented one period of an event, and the dimensionless value was noted. One can easily call this "cyclic frequency".

Then they redefined the unit descriptor for frequency as Hertz. I know this isn't refering to a rent-a-car company, but for the populaton at large this term has to be explained, where cycles per second differentiated it from a commercial term. John Hertz is better known than Heinrich Hertz.

The reason I introduced the term "cyclic frequency" is the definition for angular rate, which is generally referred to as "angular frequency",

Omega = 2Pi*f

and the result is dimensioned as s ^-1. This descriptively means radians per second, but since radians are dimensionless it isn't reflected in the dimension unit associated with the numeric value. Others are aware of the lack of a descriptor and often add rad per sec to be unambiguous. The Greek letter Omega alerts us that the meaning is different from the letter "f", although both are referred to as "frequency".

2Pi represents one full radian cycle. When one radian cycle is defined in relationship to a unt of time, it then has a rate. I wouldn't object if it was defined as "cyclic radian frequency, or "cyclic angular frequency".

If others haven't noticed, there is exactly one numeric value where "cyclic frequency" and "cyclic angular frequency" will return the same numeric wavelength value when their values are calculated using the equation that defines the inverse proportional relationship between wavelenth and frequency. It is not a trivial convergence.

[Epsilon=One]

...I find the current descriptive definition for frequency to be a bit lacking.You are correct and for good reason.

We are taught that an event or a specific period can be represented as a cycle. If it is a repetititve cycle it can be expressed as how many cycles occur in a specified duration, and the duration didn't have to be that of the SI second. Classically, the term frequency is defined as,

f = 1/TThis is where the difficulty begins. As with all Standard Model dimensions, "time" is poorly defined.

In fact, time is most likely the least understood of said dimensions. You are correct that events, and time is an event, can be represented "as a cycle." The difficulty occurs in defining a "specified duration" that does not join the other circularly defined orthogonal dimensions. It is of little import whether the duration is a "second" or not; as the “second” and all currently defined "clocks" are all fundamentally ambiguous.

…and the numeric result is typically dimensioned as s ^-1, and we know that this refers to "per second". If the "1" had been replaced with "P", for period, it would be universal, where "P" could be one or more than one.Defining “P” as “1” or vice versa leads to the same problems of provability that mathematics encounters: How do you define “One”? And, the answer is much the same: The problem must be solved by relying upon fundamental concepts of Nature, which provide rational answers to both problems with the same geometry of motion.

Everything is okay if one understand that cycle, a dimensionless value, is a phantom descriptor. Going back a ways, frequency had been annotated as cycles per second (cps), which made sense, as one cycle represented one period of an event, and the dimensionless value was noted. One can easily call this "cyclic frequency".A cycle (I prefer “pulse”) is neither “a dimensionless value” or a “phantom descriptor”; for it must be that from a fundamental pulse that both a constant “unit of time” and the illusive “One” (www.CQthus.com/PT/PoO) of mathematics must be derived. In fact surprisingly, they are more than related, they are the same phenomenon, which I refer to as εpsilon; or, heuristically, as the Elliptical Constant (www.CQthus.com/PT/EC).

Then they redefined the unit descriptor for frequency as Hertz. …The balance of your comments are moot; as, they incorporate “time” that isn’t defined (not unusual as the Standard Models also illustrate); or, you erroneously introduce dimensionless concepts and the ambiguous “1.”

…there is exactly one numeric value where "cyclic frequency" and "cyclic angular frequency" will return the same numeric wavelength value when their values are calculated using the equation that defines the inverse proportional relationship between wavelenth and frequency. It is not a trivial convergence.I agree!

[Harmonic]

This is where the difficulty begins. As with all Standard Model dimensions, "time" is poorly defined.

In fact, time is most likely the least understood of said dimensions. You are correct that events, and time is an event, can be represented "as a cycle." The difficulty occurs in defining a "specified duration" that does not join the other circularly defined orthogonal dimensions. It is of little import whether the duration is a "second" or not; as the “second” and all currently defined "clocks" are all fundamentally ambiguous.
I would agree that time is not defined in relationship to the real physical world. In the scientific community there is a need to have a standard reference for a "duration" and the SI second does this poorly,which is not surprising considering its origin. The real duration is based upon the ephemeris second and this was then tied to so many transitions of a particular Cesium isotope to give it a stable reference. The duration of the SI second does not have a real relationship to any mathematical or physical science constant except by definition.

It is possible to mutually define a "unit of time" using fundamental concepts of nature and the mathematics of basic geometry. One needs to understand the relationships that exist between wavelength and frequency and how these two natural attributes can be applied to a basic geometric form. The geometric relationships create a "unit of time" when wavelength and frequency are expressed in their "primitive" form.

Do you have a problem with the ambiguous "1" when it is applied to define a basic geometric form, the 45 degree right triangle? Isn't "1" in the geometric application a mathematical constant?

Would you accept the premise that it is possible to identify a fundamental numeric value in Nature that has a value of "1", such as your "pulse"?

[Epsilon=One]

I would agree that time is not defined in relationship to the real physical world. In the scientific community there is a need to have a standard reference for a "duration" and the SI second does this poorly,which is not surprising considering its origin. The real duration is based upon the ephemeris second and this was then tied to so many transitions of a particular Cesium isotope to give it a stable reference. The duration of the SI second does not have a real relationship to any mathematical or physical science constant except by definition.Quite astute. I agree.

It is possible to mutually define a "unit of time" using fundamental concepts of nature and the mathematics of basic geometry.Yes.

One needs to understand the relationships that exist between wavelength and frequency and how these two natural attributes can be applied to a basic geometric form.This is true; though I would contend that wavelength defines frequency (or vice versa) if the basic geometric form is at the fundamental limit and it is Natural; ie: defined by seminal motion (www.CQthus.com/PT/SM).

The geometric relationships create a "unit of time" when wavelength and frequency are expressed in their "primitive" form.Yes. And, the relationship is not a ratio; it is a difference.

Do you have a problem with the ambiguous "1" when it is applied to define a basic geometric form, the 45 degree right triangle? Isn't "1" in the geometric application a mathematical constant?I have no problem as long as you are aware that such a "1" is ambiguous.

No; said "1" is not a geometric mathematical constant. To be so it must be the difference between two geometric parts; such that it remains constant regardless of the size of the two geometric parts or the geometric construct when all parts remain the same algebraic ratio to one another.

Would you accept the premise that it is possible to identify a fundamental numeric value in Nature that has a value of "1", such as your "pulse"?Yes.

[Harmonic]

....The geometric relationships create a "unit of time" when wavelength and frequency are expressed in their "primitive" form.
Yes. And, the relationship is not a ratio; it is a difference.
When the "primitive units" for wavelengths and frequencies are used in a geometric form, the "duration" of the "unit of time" is not a difference nor a ratio, it is an absolute value, at least as absolute as we know how to define the duration mathematically.

I should have been more explicit and stated that wavelength and frequency should be expressed in their "primitive units"', a "unit wavelength" and a "unit frequency".

A "unit wavelength" suggests that is has a value of "one", and as far as I can ascertain this a valid conclusion. The "unit wavelength" does not have a unit descriptor, at least not in the sense where we use something like feet, inches, meters or centimeters, it is simply a "unit length".

A "unit frequency" is not as obvious, as it is not intuitive, and it is not "one". At the moment, the only way I know how it can be supported mathematically is within the framework of the geometric relationships between wavelength and frequency. An old engineer knew this value a long time ago.

…there is exactly one numeric value where "cyclic frequency" and "cyclic angular frequency" will return the same numeric wavelength value when their values are calculated using the equation that defines the inverse proportional relationship between wavelenth and frequency. It is not a trivial convergence.
In SI units this can be demonstrated by dividing the "cyclic frequency" value 6.2831... into the value for the speed-of-light, a numeric value of 47713.... will be the result. For a "cyclic angular frequency", when the multiplier is 1 cycle, you have the same 6.2831 value and this will give the same numeric result. Any tens multiple or division of the 2Pi value will always return the same numeric value, differing only by decimal placement.

The current method of defining frequency and angular frequency creates a mathematical conflict, because the "f" in the angular frequency definition, 2Pi*f, infers it is the same value for regular frequency, f=1/T. When a "unit frequency" of 2Pi is used in the geometric relationships, some unexpected characteristics are revealed.

Given two right triangles each with a 45 degree angle, denote one as representing wavelength and assign a value of 1 to each of the legs. Denote the other triangle to represent frequency and assign the value of 2Pi to each of the legs. The hypotenuse of the wavelength triangle will be the sqrt of 2, and that of the frequency triangle the (sqrt of 2) times (2Pi). These values should be left in their mathematical forms rather than expanding them to the numeric values.

One can mathematically show that these triangle pairs are mutual, as one reflects the inverse proportional relationship of the other. The hypotenuse of one is the inverse relationship of either of the legs of the other.

The constant of proportionality is obtained when the product of the leg of one and the hypotenuse of the other are both equal. The constant of proportionality is valid only when both sets are equal, which also indicates the triangle pair are mutually related.

One needs to understand the relationships that exist between wavelength and frequency and how these two natural attributes can be applied to a basic geometric form.
You replied,
This is true; though I would contend that wavelength defines frequency (or vice versa) if the basic geometric form is at the fundamental limit and it is Natural; ie: defined by seminal motion.
The "primitive units" can be scaled to practical units, and it is as you stated, frequency defines wavelength in this case. At 45 degrees, a right triangle is at a particular "fundamental limit", it is the null point where both legs are equal, a "Natural" form.

Would the scientific community accept a mathematically defined value for the speed-of-light when the value is derived using both mathematical and physical science constants?

[Epsilon=One]

When the "primitive units" for wavelengths and frequencies are used in a geometric form, the "duration" of the "unit of time" is not a difference nor a ratio, it is an absolute value, at least as absolute as we know how to define the duration mathematically.This comment I believe is perceptive beyond your intent.

Whatever, I believe we are in near complete agreement after a quick review of all your comments. Possibly, a little semantic tweaking; my perusal was rushed; however, I am generally pleased with your perception of what seems difficult for many theoretical physicists.

I am in the process of moving and am very limited in time. I will try to comment in more detail in a few days. Thanks for your input.

[Harmonic]

This comment I believe is perceptive beyond your intent.

My intent is very precise and limited to a mathematical construct that is understandable by even high school physics students, if they are taught anything about geometry, algebra, trigonometry and waves nowadays.

An end product of the mathematical construct is illustrated in the TrianglePair-SI.pdf article.

http://www.vip.ocsnet.net/~ancient/TrianglePair-SI.pdf

The old scientists (non-Western) that introduced (used) the concept did not explain the fundamentals, they expected those that "read" their material to know them. I have laid out and annotated the triangles with familiar SI units, something they did not use. Fortunately, I knew enough of the fundamentals to understand the mathematical and fundamental basis what the triangles meant.

The concept is easier to understand when it is presented graphically than in pure mathematical form.

Wavelengths and frequencies are inversely proportional to each other. The triangle pairs illustrate their inverse relationships in a geometric form. The constant of proportionality is calculated by the product of the hypotenuse of one triangle and the vertical leg of the other triangle. When the cross products of the two triangles are equal they are mutually related, one is the inverse of the other.

One can easily progress to the "basic geometric form" from what is presented. One element of the constant of proportionality is a function of the angle, that is the key. Everything one needs to know about the triangle pair is taught in Physics101, but no one is taught to represent the elements in geometric form.

When put into the basic geometric form you will have three fundamental "base units" defined mathematically, one which is probably related to your "pulse".

[Epsilon=One]

My intent is very precise and limited to a mathematical construct that is understandable by even high school physics students, if they are taught anything about geometry, algebra, trigonometry and waves nowadays.This quote and your further comments leads me to conclude; and, I again repeat: "This comment I believe is perceptive beyond your intent."

I don't believe that you completely understand the fundamental mathematical concepts that you allude to; or, their importance to an understanding of all physical phenomena.

You are very close to what string theory (ST) has sought for nearly forty years; and what Kurt Gödel mistakenly theorized was unprovable.

[Harmonic]

I am sure I do not understand everything that is related to the mathematical construct in the TrianglePair-SI.pdf article, but I am aware that it allows those who use it to have a "base system of units" that are physically and mathematically related to the physical universe, unlike the SI base units.

For those that haven't figured how to go to the "basic practical mathematical construct" I will provide most of it in the article Backwards.pdf. I had to take that direction originally, and at the end it doesn't identify what I call "primitive units", which are the fundamental units which define the wavelength frequency triangle pair in its most primitive construct.

http://www.vip.ocsnet.net/~ancient/Backwards.pdf

I do know that the 888.5765...(10^6) frequency was used as the reference for their fundamental atomic level unit of energy. We define the eV separate from frequency. In SI units, we equate 5.9 (10^-6) eV as being the energy level represented by a frequency of 1420.4 MHz. The energy level represented by a frequency with the numeric value of 888.5765...(10^6) defined their base unit of energy, which would be somewhat equivalent to what we refer to as 1 eV. Think about that, all electric potential values would be related to the energy level of the H1 precession emission.

[nina060609]

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