Euclidean Natural Units

[FrankM]

A truly intelligent species would derive its fundamental base units mathematically. Our current set of scientific base units are serially defined, starting with the definition of a time duration then defining a length, etc.

 

The relationships between the lengths and the angle that form an isosocles right triangle are illustrated using dimensionless numbers, that is, they do not have any unit designators. These pure geometric relationships can be exploited by assigning wavelengths and frequencies as the parameters of a right triangle. A set of base units can be mutually defined using pure geometric relationships, and they can be scaled to practical values.

 

http://vip.ocsnet.net/~ancient/EuclideanUnits.pdf

[Turtle]

A truly intelligent species would derive its fundamental base units mathematically. Our current set of scientific base units are serially defined, starting with the definition of a time duration then defining a length, etc.

 

The relationships between the lengths and the angle that form an isosocles right triangle are illustrated using dimensionless numbers, that is, they do not have any unit designators. These pure geometric relationships can be exploited by assigning wavelengths and frequencies as the parameters of a right triangle. A set of base units can be mutually defined using pure geometric relationships, and they can be scaled to practical values.

 

http://vip.ocsnet.net/~ancient/EuclideanUnits.pdf

:shrug: :) :D :( :wave: :wave: :wave:

Frank, you rock! If we tell another 98 monkeys, then everyone can rock too.

Tau to re-read & re-flect.:)

[Turtle]

... These pure geometric relationships can be exploited by assigning wavelengths and frequencies as the parameters of a right triangle. A set of base units can be mutually defined using pure geometric relationships, and they can be scaled to practical values.

 

http://vip.ocsnet.net/~ancient/EuclideanUnits.pdf

 

Frequency is defined as the number of repetitions of some event that occurs in a unit of time, time duration; we currently use the duration of the second that is defined in SI. Frequency can be expressed using its basic angular form. When the mathematics of waves were developed, the motion of a wave was given the equivalence of a point rotating on a circle and the angular position was measured in radians. A complete wave encompassed a full circle, 6.2831 or 2π radians. Angular frequency is how many radians occur in a specified time duration. The common equation for angular frequency is ω = 2πf, where f is the frequency.

 

I need some time to wrap my head around this Frank. How long ago did you write this? What inspired it? Two pie or not two pie, that is the question. :cup: I highly recommend readers take the time to read this succinct & insightfull bit of geometry that Frank has proffered. :cup:

[CraigD]

The following is my synopsis of Makinson’s (FrankM’s) paper “Euclidean Natural Units”

 

The Euclidean Natural Unit (ENU) system of measurement of length and time has length unit [math]L_E[/math] and time unit [math]\tau[/math] (“tau”).

 

[math]L_E[/math] is defined as the wavelength of the hydrogen line a well-known physical constant arising from the wavelength of the photon emitted when the spin of the electron and proton in a hydrogen atom change from the same direction to the opposite direction.

 

[math]\tau[/math] is defined as [math]\frac{\sqrt{2} * 2 * \pi * 100}{c}[/math], where c is the speed of light, a well-known physical constant.

 

So, in summary, the ENU system uses a characteristic of the hydrogen atom to define its length unit, and the speed of light to define its time unit.

 

The SI system of measurement has units of time s (“seconds”) and length m (“meters”).

 

s is defined as the duration required for 9192631770 periods of the frequency of the photon associated with a caesium-133 atom making the same transition as desribed above for the hydrogen atom, a well-measured physical constant.

 

m is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 s, a well-measured physical constant.

 

Comparing the 2 systems, I find distinct equivalences, pros, and cons with each

• Pro: Both systems are based on well-known physical constants
  
• Con: These physical constant are determined by experimentation, so, as the accuracy and precision of the measurement increases, the true value of the units of both the SI and ENU system also change.
  
• The SI system is based on the arbitrary integer constants 9192631770 and 299792458.
  • Con: These are invented number – they have no natural significance.
    
  • Pro: They are precise - they do not require approximation to represent as a rational number.

  [*]The ENU system is based on the real number constant [math]\sqrt{2} * 2 * \pi[/math]

  • Pro:This number is a naturally occurring constant – it is a consequence of number theory and Euclidean geometry
    
  • Con: It is an irrational – it cannot be exactly represented as a rational number. It must be approximated to be represented as a rational number.

Approximating [math]\sqrt{2} * 2 * \pi * 10^8[/math] by truncating (not rounding) it to 5 significant decimal digits gives a value of 888.57. This is the value used in the paper. The ENU length and time units, then, are approximately related to the SI system by:

1 [math]L_E[/math] = 0.2106 m

1 [math]\tau[/math] = 6.242079712358874619*10^-7 s

 

In conclusion, As a practical system of measurement, I don’t think the ENU system offers any advantage over the SI system. To be practically used in a digital calculator, an approximation must be used. For different users to be get the same results from the same arbitrary-precision calculator, they must agree on an arbitrary, fixed precision of this approximation. Deciding on this “standard” precision requires the selection of an integer as arbitrary as the SI system's 9192631770 and 299792458.

 

As an “extraterrestrial intelligence test,” I think it has value. An ET listening to a faint SETI radio signal containing a diagram of a conspicuous astronomical distance, such as the distance between the Milky Way and Andromeda, and its measure in some length unit, is arguable more likely to expect it to use something like ENU units than SI units.

 

I have a couple of technical and editorial criticisms/suggestions:

• The inclusion of powers of 10 in ENU is mathematically arbitrary. 10 is not a naturally occurring physical or mathematical constant – it is just a common numeral system base, likely due to the biological coincidence that normal human beings have 10 fingers.
  
• As 2 is the lowest possible numeral base, binary should be used. For convenience, based 10 (decimal) numbers may be used, but multiplication by any arbitrary constant for “convience” should use a power of 2, and some mention of avoidance of arbitrary uses of 10 should be included in the paper.
  

FrankM, please let me know if I’ve misunderstood a factual detail of your paper. In particular, in equation 9, the term [math]L_U[/math] appears. I didn’t recognize it, and, taking it as a typographical mistake, assumed it to be [math]L_E[/math]. Please let me know if this is correct, and if not, what the term [math]L_U[/math] means.

 

I found this an impressive and interesting paper. My complements to the author. :thumbs_up

[FrankM]

I revised my article to include the "direct derivation" mathematics. I did this before I received and read Craig's response. In the revision, I believe I corrected some inconsistencies in the way I had presented the material, a couple were noted by Craig. The revision uses the same name.

 

http://vip.ocsnet.net/~ancient/EuclideanUnits.pdf

 

I have a number of responses to Craig's analysis, and I appreciate the time he has taken to examine the mathematics.

 

To be practically used in a digital calculator, an approximation must be used. For different users to be get the same results from the same arbitrary-precision calculator, they must agree on an arbitrary, fixed precision of this approximation. Deciding on this “standard” precision requires the selection of an integer as arbitrary as the SI system's 9192631770 and 299792458.

Every digital calculator I have has a fixed precision, it depending upon the number of digits that they display. My truncated calculator values for 2Pi and the square root of two have significantly higher precision than the "defined" value assigned to the SI speed of light, thus any calculations using the SI SOL has limited precision, but not because of my calculator. I suspect this will be true with every current scientific calculator on the market. I am still using a Ti-36. Those using newer model calculators will have greater numeric display length, but their precision using calculations that include the SI SOL value has the same precision limit. One of the computer programs I use can give a much greater precision, but I limit it to a reasonable sized display or print out length when calculating "pure" ENU units. In ENU units, one only needs to indicate the level of precision, how many decimal points one wants to use as a "standard". Every scientific institution could produce the ENU ESOL value to whatever precision they need.

 

I do not recommend that the ENU system be used for commercial and common use, scientific only.

---

# Pro: Both systems are based on well-known physical constants

# Con: These physical constant are determined by experimentation, so, as the accuracy and precision of the measurement increases, the true value of the units of both the SI and ENU system also change.

The SI defined value for the SOL is based upon an astronomical unit of measure. The duration of the second is based upon a average value for the Ephemeris second, and then it was determined how many Caesium transition counts would fit within that duration. The publication that tells how this was done is available on the leapseconds site.

 

http://www.leapsecond.com/history/

http://www.leapsecond.com/history/1958-PhysRev-v1-n3-Markowitz-Hall-Essen-Parry.pdf

 

The SI value for the SOL does not have a relationship to a known physical science or mathematical constant.

 

The true value of ENU units do not change, their absolute value is limited by our computational capability, they are mathematical values not measured values.

---

Approximating by truncating (not rounding) it to 5 significant decimal digits gives a value of 888.57. This is the value used in the paper. The ENU length and time units, then, are approximately related to the SI system by:

1 = 0.2106 m

1 = 6.242079712358874619*10^-7 s

The ENU length should be 0.21106.... m

 

The ENU time unit duration relative to that of the second is determined by the ratio of the cosecant value at 26.25400 degrees as compared to that at 45 degrees. The earth second is about 1.5985180.... longer than Tau.

---

The numerical convergence of the ESOL value with that of the ENU frequency for hydrogen has significance. If the eV and other physical science values all started with the ENU values, there would be quite a few other convergences taking place.

 

As an “extraterrestrial intelligence test,” I think it has value. An ET listening to a faint SETI radio signal containing a diagram of a conspicuous astronomical distance, such as the distance between the Milky Way and Andromeda, and its measure in some length unit, is arguable more likely to expect it to use something like ENU units than SI units.

Your SETI comment is appropriate. I passed on the ENU "hydrogen" frequency to an individual that has contact with the SETI people on a regular basis. He said he would pass it on but whether he did or didn't I do not know. I mentioned that an intelligent species would likely use that numeric value rather than what we use (1420.405 MHz). If they were considering logical offsets, an offset of 3.1415 based upon the smaller number 888.5765 would have a significantly great spectrum shift than offsetting from the larger number. I believe most of their receiver bandwidths would fail to cover the spectrum area a 3.1415 offset would produce.

 

Turtle,

How long ago did you write this?

Actually, I identified the primary dimension (47.713 cm at 26.25400 deg) over five years ago. I knew the significance of the numbers immediately, but it took me awhile to develop a coherent explanation that other people can understand. I have been submitting various iterations of the current report to many, many people, institutions, and a few publishers over the years, but this is the first use of that particular title. On one forum I actually presented the concept as "ET Time". I managed to get the concept (under another name) published in an on-line scientific journal many years back, but it is no longer active. My last attempt for publication, earlier this year, was rejected by my "peers", it steps on the sacred cow SI. I might try to resubmit my current version and see if it would be considered using its new title. The previous title was "Defining the Basic Constants: A Mathematical Method".

 

I have additional information extracted from dimensions related to the above dimension set, but it would extend the article making it much more complicated. This other information is why I used the tens multiples, and I sure don't want to explain that here. I haven't tried to convert the ENU

numeric values to a different base number system. Feel free.

[FrankM]

I need some help in defining an algorithm that will identify the angle and length of the vertical leg that corresponds to values expressed in SI units. I know what the angle and length are by an iteration process but it appears this can be done by integration. The section in the EuclideanUnits.pdf article, "Constant Wavelength - Constant Frequency" establishes the conditions.

 

The equation y = z / csc(a) would identify the length of y if we knew the angle and the length of z. We know the length of z in SI units, 47.713 cm and it will convert to a frequency with the numeric value 628.31 (10^6) at one particular angle.(See first form of equation (2)). This gives a function that has two variables, y and a, and this function will identify y and the angle when the length of y and the angle result in the value of z, when expressed as a frequency, equals 628.31 (10^6). The value of c, the numeric value for the speed of light in SI units, will be valid for only one angle. This angle is where the duration of Tau equals the duration of the second.

 

There must be an elegant integral that will allow the solution to be made directly, rather than the cumbersome method I used.

[FrankM]

I put together a shorter version of the article, eliminating the mathematics that details the various relationships. Now it can be printed on one sheet of paper.

 

http://vip.ocsnet.net/~ancient/EuclideanUnits-Basics.pdf

 

The existence of a natural unit of length is a peculiarity common both to hyperbolic and elliptic geometries, and differentiates them from Euclidean geometry.

I wasn't aware of the above statement and it has been around a long time. I found it in the 1911 version of the Encyclopedia Brittanica, which is on-line.

 

http://www.1911encyclopedia.org/Geometry

 

I found a few other sites that quoted or paraphrased that statement. The Euclidean natural length defined by applying wavelengths and frequencies as parameters of a right triangle invalidates that long standing statement. How long will it take for others to learn that the wavelength of the hyperfine emission of neutral hydrogen is the Euclidean natural unit of length?