# Nurseryschool Numerology

1. Dec 20, 2003

### marcus

Numerology 1A

Decided it wasnt nice to say "nurseryschool" numerology and would try calling it beginning or introductory or numerology "one-A"

numerology, that hated word. a magnet for crackpots.
a pitfall that even good people stumble into
something at the historical root of science: "All is Number" said Pythagoras.
To be remembered at all times: Numerology is Not Physics.

in recent times maybe the worst examples of numerology are the people who give formulas for 1/137.036...
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but Beginner's Numerology is actually a casual backdoor into science.
we should consider doing some.
People come to PF who will turn and run if they see a differential equation but are willing to be curious about the fundamental numbers in nature.

the fundamental universal numbers are, in a way, nicknames for some of the physical laws

a beginner may not remember that a spheres volume or area is
V=(4/3)pi R3
or
A=4pi R2

but may be willing to consider the idea that pi is a pervasive ratio in nature, at least approximately, and fascinating.

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So likewise with this number alph which is a deep proportion in nature operating in all sorts of contexts thruout the universe---this number 1/137.036...----for which there is no formula, so that it can only be found experimentally, unlike pi.

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the deep proportions, these pervasive numbers, are features of our beloved universe and they are in a way Sexy.
that is the point about Numerology.
It is a pedagogical opportunity.
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because it is easier for a beginner to entertain the vague notion that pi (or alpha) is interesting and in everything around us
than it is for him or her to comprehend and know about all the
separate formulas and phenomena

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alpha is in the size of atoms and the light they glow with and in the fusion at the core of the sun, but if you begin by talking about these things you might lose listeners who think it is too hard to understand

so suppose you start telling the story differently: Once upon a time there were Seven Numbers that made all nature....and....
(what do you suppose those seven numbers were?)

or if not seven then some bunch
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and then there are universal quantities like the speed of light which are not numbers
that also belongs in Nurseryschool Numerology. Basic quantities like the speed of light are not numbers.

they are generalized Things. and you can compare other things to them---the speed the earth travels in orbit can be compared to light (and is one tenthousandth)
---the speed sound travels in the stratosphere can be compared to light (and is one millionth)

the speed of light is a basic quantity the same for people in Andromeda as for people in Milkyway---the quantity is universal but has no universal number attached to it.
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I am proposing something pedagogical to think about.
In the suggested line of discussion you would learn about a few universal quantities (like the speed of light) and a few universal numbers (like pi but not pi)-----without too much emphasis on the pride-and-joy crown-jewell formulas called Laws of Physics that are the venue where they make their famed appearances and perform their daredevil feats.

"How does that strike you Tyson?"
"If I may say so, with a dull thud..."

which numbers and quantities would you personally nominate for
inclusion

Last edited: Dec 26, 2003
2. Dec 20, 2003

### marcus

Hmmm (replies to his own question)....what about 1836?

does everybody recognize this one or does it need an introduction?
any other ideas?

3. Dec 26, 2003

### marcus

so far no interest in this thread
I can erase it, I guess, or continue solo.
I think the Planck units are interesting and that it is
interesting to see what features of nature you can estimate
(in terms of those units) using just a few numbers as input

the point is that if you use arbitrary units like in the metric
system then a lot of other quantities, that you have to remember, come into the picture
so that things get more complicated

in planck units most of the basic constants one needs are one
the speed of light is one
the boltzmann temperature coefficient is one
planck's hbar constant is one
the newtonian gravitational constant is one
the electron charge is one

so you dont need to remember the metric values for those things----they are just one----if you work in natural or planck units

that means that besides the whole numbers, there are just a few other numbers needed (like 1/137 and 1836, and pi) in order to calculate quite a bunch of stuff about nature

4. Dec 26, 2003

### marcus

But I shouldnt forget to say that if any feature of nature is specific to this planet or to this solar system, then numerical data about it will be needed as a separate input

the natural units make a bunch of UNIVERSAL quantities and conversion factors come out to be one, or other simple things like 4pi/3 or pi-squared

but for instances, typical gravity at the earth's surface is not universal
and the average atomic weight of a molecule in the earth's atmosphere is not universal
so if we are going to calculate some things like air pressure and the speed of sound in the earth's atmosphere we need a few other numbers specific to the earth

the surprising thing is how much is calculable (to a good approximation) from so little input
this kind of economy of inputs is not exactly science itself
(physical models do much more) but is like what physical models
do----gives a rough notion. that's why I'm putting up a possible thread on it in case anyone wants to try

BTW we do need a number for the compton wavelength of the proton----it is a handle on the size of the proton and also the inverse mass, but in natural units.
Since it is stated in planck lengths (a very small length unit) the number for the proton wavelength will be very large:
13 quintillion.
Or 13.01 quintillion, but 13 is close enough for most things I would calculate.
quintillion is E18 or 1018
which may seem very large but that is just how it is.

Last edited: Dec 26, 2003
5. Dec 26, 2003

### meteor

Don't forget that some famous scientists were also attracted to the study of numerology, notoriously Eddington and Dirac. Eddington was convinced that the fine structure constant should be exactly 1/137, but was demonstrated later that this is not true. Dirac proposed the Dirac number, 1040, that according to this page:
www.edge.org/3rd_culture/lloyd2/lloyd2_p3.html
is the ratio between the size of the universe and the classical size of the electron. Unfortunately, I'm unable to find the value of the classical size of the electron. I've searched in the web and there's no way. So, I can't confirm if Dirac's affirmation is true
You can also have Euler's formula: ei*pi+1=0. I think that there's some hidden information in Euler's formula, is a too much simple for to be a casuality

Edited: The classical radius of the electron is 2.82*10-15m. So the classical size must be 5.64*10-15m. So, Dirac supposed that the size of the universe must be 5.64*1025 m.
I've discovered the meaning of 1836. 1/1836 is the ratio between the electron mass and the proton mass

Last edited: Dec 27, 2003
6. Dec 27, 2003

### marcus

Right and the current experimental value for it is
1836.15266...

I hope it is clear I'm not a real numerologist. I dont want that ratio to be exactly 1836 or to be anything except an experimentally determined ratio.

Same with 1/137.036... I dont want there to be a mathematical formula for it, or for it to be exactly 1/137 as you say Eddington wished it would be.

What I'm discussing is a small set of key numbers from which a lot of other features of the universe can be calculated. The prime example of this is the Standard Model-----which has IIRC a couple of dozen numbers (like 1/137.036...) from which vast amounts of other stuff can be calculated, if you know how!

What I would like, I guess, is a "toy version" of the standard model.
Instead of a couple of dozen basic numbers I would like there to be three or four or a half-dozen. And no great effort devoted to precision.

So for example,
you mentioned the electron classical radius.
We should be able to calculate that (in planck units) from our basic numbers. I will try in the next post.

7. Dec 27, 2003

### marcus

calculating Bohr radius, compton wvlngth, classical r

[edit: I put the formulas in LaTex so they look better. BTW Meteor it is a disgrace, as you say, to have two versions of planck's constant. the first one he published, in 1899 curiously enough, was hbar, but almost no one knows about the 1899 paper. The symbol he used was the letter a, in that paper. In 1900 he came back to it, but multiplied by 2pi and called it h. Original sin began with Adam and our disgrace began with Planck.]

Let us start with three numbers given only approximately (to 3 or 4 significant figures)
1/137
1836
13E18

these are the finestructure constant, the proton/electron mass ratio (as you mentioned) and the proton compton (in planck)
or if preferred think of 13E18 as the ratio of planck unit mass to the proton mass.

the bohr radius turns out to be $$137*1836*13E18$$

the electron compton turns out to be $$1836*13E18$$

the electron's classical radius is $$\frac{1836*13E18}{137}$$

I guess we should be able to get electron cyclotron frequency (in ratio to magnetic field strength) and also Thomson scattering cross-section, and a bunch of stuff like that. The main wavelengths of the hydrogen spectrum---the atom's "colors"---should be easily calculated from these numbers

Also the masses of the proton and electron----expressed in planck mass units----are easy.

The proton mass:
$$\frac{1}{13E18}$$

The electron mass:
$$\frac{1}{1836*13E18}$$

The Rydberg energy----the ionization energy of the hydrogen atom (expressed in planck energy units)-----should be easy.

what makes all these things easy to calculate is that we express the answer in natural units. so few inputs and conversion factors are involved

So far we are using only 3 numbers. What other number or numbers should we add to the basic set in order to expand the scope of what can be calculated?

Last edited: Dec 29, 2003
8. Dec 28, 2003

### meteor

I'm not comfortable with natural units, I only know how operate when you only set c=1, and then time is expressed in meters, but I'm assuming you set c=G=hbar=k=e=1. How do you get to the cipher of 13E18 for the proton Compton wavelenght? In SI units the proton Compton wavelength is 1.32*10-15m. If I only set c=1 it gives me h/m = 3.6*10-7 m.

Last edited: Dec 28, 2003
9. Dec 28, 2003

### marcus

that is right. there are other possible choices but this is
not too unusual. because hbar is used it is more convenient to use
the reduced wavelength (lambda with a slash through it, "lambda-bar")

just so we can refer to a common listing of the fundamental physical constants, here is what I am using:

http://physics.nist.gov/cuu/Constants/

The NIST constants website gives both versions of the proton compton.
Its figure for lambda is
1.3214...E-15 meters
Its corresponding figure for lambda-bar is that divided by 2pi
2.103089...E-16

The NIST site is just one of several sources for the CODATA constants (you may have them in whatever handbook). The values as of 1998 were published in "Physics Today" August 2000). Now the NIST posts the values as of 2002. The 2002 recommended CODATA value for the Planck length is 1.61624...E-35 meter. None of this precision matters for what I have in mind here, of course.

Dividing 2.103089E-19 by 1.61624E-35 gives 13.01E18.
Not being worried about precision, and wanting a number that's easy to remember, I take that to be 13E18

This may seem like a stupid game to be playing. It amounts to not very much more than simply taking the conventional form of the planck units (e.g. from NIST website) literally and naively. And seeing how the world looks seen through those glasses.

Last edited: Dec 28, 2003
10. Dec 28, 2003

### meteor

I think is a little disgrace to have 2 different values for the Planck length, the other value is 4E-35 m.
http://scienceworld.wolfram.com/physics/PlanckLength.html
I don't know which of the two values is more used between the physics community, but we can adopt the value that you give, 1.61E-35 m
I think that I understand the major part of your posts, I will try to do some calculations tomorrow, I don't have time now

Last edited: Dec 28, 2003
11. Dec 28, 2003

### meteor

A propos, Marcus, what do you think about a possible variation in the value of the fine structure constant? For example, in this model
http://www.spaceref.com/news/viewpr.html?pid=13101
the change in the value of the fine structure constant is due to the variation of quintessence

12. Dec 29, 2003

### marcus

Hello Meteor,
thanks for staying with me here. it is very out near the edge and a bit lonely to be talking about the fundamental constants

(I think it is important to ask why they are the values they are---why is 1/137 equal to whatever it is. I mean 1/137.036...but omit some digits. Eventually new physics may come from finding explanations for the values of some basic numbers)

This Physical Review D article is cited in the article you refer to:

"Anchordoqui and Goldberg's journal article, "Time Variations of the Fine Structure Constant Driven by Quintessence," available at http://arXiv.org/abs/hep-ph/0306084. "

I can not say anything enlightening about this but I will tell you my raw impression. Looking at a natural nuclear reactor seemed to indicate that alpha DID NOT CHANGE for 2 billion years. Now the people who want alpha to change must look for nonlinearity. They are just fiddling with the story so that the data can appear to show that it did change a long time ago, even tho it did not change in the last 2 billion years. For me, this is too speculative, and also quintessence is too speculative. But I would be honored if you would discuss things like changing alpha in this thread.

For me, the fundamental constants are a pedagogical tool because they are the deep inner proportions of nature which even non-physicists can grasp---a way that essential physics can communicate to a wider audience (I hope). But I do not want to impose my limited approach to fundamental constants on you or anyone else! Therefore please do not be discouraged by my somewhat narrowly defined interest.

Last edited: Dec 29, 2003
13. Dec 29, 2003

### marcus

how to explain what is alpha

if one adopts the perspective of the units inherent in nature
then how does one describe what basic numbers are
like 1836, 1/137, 13 quintillion

1836 is easy, just the ratio of proton mass to electron mass
and 13E18 is the ratio of planck mass to proton mass

so one can say that $$\frac{1}{13E18}$$
is simply the proton's mass expressed in the natural unit.

This is fairly intuitive and widely understandable, but what is 1/137?

One way to think of it is that it is the Coulomb's constant expressed in natural units.

If one says c = G = hbar = e = 1
then the value of the ordinary Coulomb's constant turns out to be 1/137.

I hope that many people find the Coulomb's constant a practical understandable concept. To tell the force between two charges, one multiplies their product by the Coulomb's constant and divides by the square of the distance between them. What simpler indicator of the strength of electrical attraction/repulsion could there be?

I think other descriptions of what 1/137 is probably require more sophistication on the listener's part. But do you have one that might be more informative or work better?

14. Jan 2, 2004

### meteor

Can someone have an answer to this?: Is possible to set the value of 1 to more that 5 constants?. For, example, instead of setting c=G=k=hbar=e=1, is possible, say, to assign the value 1 to six or seven constants? This would be really a saving of calculations!

Last edited: Jan 2, 2004
15. Jan 2, 2004

### marcus

in case other people read this we should first say that with some constants there is no choice what they must be, since algebraically dependent on these 5.

for example the Stefan-Boltzmann radiation constant of the "fourth power law" is by definition

$$\frac {\pi^2}{60}*\frac{k^4}{hbar^3*c^2}$$

so the Stefan-Boltzmann can not be just anything it must be

$$\frac {\pi^2}{60}$$

16. Jan 2, 2004

### marcus

Indeed there is a great simplification of constants. Because even with just these 5 set equal to unity one has other constants that, because they are algebraically derived from these, take on simple values, like 1/2, or 2, or 2pi, instead of values one must look up in the handbook.

The Josephson constant is 2e/h, or if you prefer a different convention, 2e/hbar. So it is a simple number.

the von Klitzing constant is h/e^2, so likewise.

And groups of constants which normally have different values take on the same value. The electron mass and the electron rest energy and the reciprocal Compton wavelength (wavenumber) are all the same number.

But you have asked a more interesting question, which is, in the algebra of physical quantities what is the rank. What is the number of generators of this algebra?

Now I have said the simple stuff, in case, for completeness. So we can talk about this. How many independent types of physical quantity are there, which are not algebraic combinations of the others? To me this is a very interesting question. there may be types of physical quantity we dont know of.

why dont you take a shot at this, Meteor?

area and volume and density and pressure and current and voltage do not count because they are just algebraic combinations of basic ones we already have.
can you think of any type of physical quantity that is not buildable out of mass, length, time, charge, temperature?

Last edited: Jan 2, 2004
17. Jan 3, 2004

### meteor

The only constants that are not dependant of other quantities like mass, charge etc. that I can think right now are individual constants, those specifics for a given element, for example, the Curie point, is specific of each element, and is the temperature at which a ferromagnetic material pass to be paramagnetic, and this Curie point I think that is not related to other properties of the material. The same applies for example to the Neel temperature, the temperature at which a antiferromagnetic or ferrimagnetic material pass to be paramagnetic

Edited: To correct an error

Last edited: Jan 3, 2004
18. Jan 3, 2004

### marcus

Meteor, just to simplify our conversation let's ignore "color charge" and QCD and strong force and weak force and all the deep kinds of physical quantity that we dont see in everyday life.

Let's pretend we just have gravity and EM force based on ordinary electric charge.

Now would you agree that all the types of physical quantity you can imagine----length, area, volume, mass density, temperature, energy, heat capacity, pressure, current, capacitance, charge density, etc.----that all those types can be built up out of just five types?

I think you probably would agree. All types of quantity can be built of mass, length, time, electric charge, temperature.

The "curie point" you are talking about is just a temperature. I am not talking now about physical constants, but simply about TYPES.
volume is a type, force is a type, etc.

If all quantities are algebraically generated by 5 types then one can only have a free choice of 5 scales.
and so one can force at most 5 independent fundamental constants to be unity.
whatever other constants then TURN OUT to be unity will not be under our control but will depend on them being algebraic combinations of the others.

And then beyond this there are other types of quantity implicit in the standard model or in QCD or latent in still-undiscovered theory. these too could be set to unity.

Anyway, that is how I see the situation at the moment. Does it make sense to you or have I missed something?

19. Jan 3, 2004

Staff Emeritus
On your list, I am dubious about charge and temperature. Temperature is just a form of kinetic energy, and can be expressed in MLT just as all energy can. I'll accept charge for the moment; it's usually expressed in terms of charge on an electron. But I think it can be rendered into MLT via quantum electrodynamics.

20. Jan 3, 2004

### marcus

it is somewhat like discussing whether birds should be categorized as dinosaurs or not. associated with a system of units is a catalog of types of physical quantity and a way of writing the constants.

if you like to measure temperature and energy on the same identical scale you are welcome to
and then Boltzmann's constant is dimensionless for you
while someone else is free to measure temp and energy on two separate scales (like kelvin and joule) and then for them Boltzmann's constant is dimensionful
and has dimensions "joule per kelvin"
we are talking largely human taste and convention
and the main thing (but not the only thing) is to be consistent

since there is no one right way, it is maybe not too interesting a discussion

but I think Meteor and I see 5 independent types of quantity
mass, length, time, temperature, electric charge
and what I am wondering is, can anyone point out a SIXTH or seventh to me?

Unless one gets into the bowells of chromodynamics I dont think there is any other. All the quantities I am familiar with can be built up from those 5 (and you may choose to build them up from 4, or 3, and are welcome to if you so desire!)

Meteor's question is how many fundamental constants can you force to be unity. I say that if your customs allow for 5 independent TYPES of physical quantity then you can force at most 5 (algebraically independent) fundamental quantities to be unity. Or if 4, then 4, if you want. Does that sound right?