## T E T R O L O G Y

## or the basic base for dimensions, quantities and units

The reader who is familiar with these concepts

can skip the introduction

and can jump to the section unit dimension.

From there remarkable things happen.

Quantities

All measurable properties are called quantities.

Examples of quantities are speed, age, acidity and volume.

The speed of a car at some point is

14 m/s = 50 km/h = 0.042 × the speed of sound.

So the number depends heavily on the unit off.

In general, aquantity= anumber× aunit.

Often one leaves the × but not always. See the example above.

Very generally for an arbitrary quantity it is written A = {A} [A].

Here are

A = the symbol for the quantity

{A} = the measured number and

[A] = the corresponding unit.

Dimensions

Here always a dimension is a physical dimension and

nothing like the third dimensionin which we live.

The quantities length, width, height, depth, distance, perimeter,

diameter and radius have something in common,

The are all lengths. They belong to the dimension length.

The unit of measure used in this dimension belongs is the meter

and derived from it the mm, cm, dm and km.

The velocity of light, the speed of sound, the rapidity of a car,

the units knots, m/s and km/h, they all belong to the dimension speed.

The quantities content, space, volume, displacement and the

units liter, gallon and m3 they each belong to the dimension volume.

Physical work, energy, heat thermo and the units Btu, cal, kWh and joule

they each belong to the dimension energy.

The quantities (start and end) capital (net and gross) profit and loss

and interest as well as the units, dollar, euro and other currencies

they each belong to the dimension capital.

This dimension does not play a role in physics.

to multiply and to divide with quantities and dimensions

An example

to multipy dimensions: speed × time = length to multipy quantities: average speed × duration of time = distance a numerical example: average 50 km/h × 2,4 h = 120 km

Gelijkwaardig hiermee

to divide dimensions: length divided by time = speed to divide quantities: distance / duration of time = average speed a numerical example: 120 km / 2,4 h = average 50 km/h. Dimensionless quantities

Some quantities do not havea unit.

Some examples are: the refractive index of glass to air,

magnification, effectiveness or efficiency,

kinds and relative quantities and friction coefficient.

It is said that these quantities are dimensionless.

An example

If a radio receiver will be able to do its job,

it needs electric power as input.

What you want from the device is power as sound.

This is the output.

Now applies to these quantities the following formula:

output = input × effectiveness

On the left hand is a number of watts.

On the right hand you see an equal number of watts.

Input and output are both of the dimension power.

always it was said, that the effectiveness is dimensionless.

## The unit-dimension

Here is a definition of a new dimension, the 'unit-dimension'.

All quantities that belonged to the dimensionless quantities

are members of the unit-dimension.

Why? Check again the previous example:

output = input × effectiveness

dimension power = power dimension × unit-dimension.

Actually, the unit-dimension is identical to the natural number 1,

hence the name.

The quasi unit of the unit-dimension

Again we look at output = input × effectiveness.

A random numerical example:

80 watts = 0.4 ??? × 200 watts

At the place of the ??? nothing has to stand there.

Nevertheless I place the quasi unit Re.

Re is an abbreviation of real number. So 80 watt = 0.40 Re × 200 watt.

It will be understood that the Re is identical with the number 1.

So is true: the effectiveness = 0.40 Re.

(This is also the same as 40%.)

Re is a 'pretend unit', which indicates that we deal with

a quantity of the unit-dimension.

Re demonstrates explicitely, that we are aware of it and

that we did not forget the unit.

Moreover, it is possible to use the known prefixes.

Thus MRe for mega, a million without unit.

A= {A} [A]

We saw this expression above.

Also we see hereAasks for the unit [A].

WhenAbelongs to the unit-dimension

[A] = Re can be the and satisfying answer.

Percent, permille and ppm

The permille = ‰ can be replaced by mRe

the percent = % by cRe.

the next quantities belong to the unit-dimension

- all the measurable and calculable quantities without units;
- all quasi units, as the radian (rad) and the steradian (sr).

They only remind you that the angle or solid angle is expressed

in a real number and not in degrees, with or without minutes and seconds;- the percent (%), the permille (‰) and the ppm;
- all the real numbers.
Basic dimensions in science

The traditional seven basic dimensions are

lengte, time, mass, current, temperature, number of moles and brightness.

I added the unit-dimension.

## Three superfluous dimensions

It is a unpleasant thought, that three of them are redundant.

They are the dimensions of moles, temperature and brightness.

That's awful, because physicists have the task to describe

nature as precisely as possible and

simultaneously as simple as possible.

Scientists are expected to obey these rules.

They need to take a break from historical noise. They fail here.

In the following sections, a brief explanation.

The basic dimensions number of mole and temperature

In the theme 'Thermo' the redundancy of both is clearly explained.

Highly motivated the basic dimension temperature is replaced by

the derived dimension thermo and the mole by yotto

Yotto belongs to the unit-dimension.

The basic dimension brightnessis reduced to

the derivated dimension power in a responsible way.

This is covered in the theme 'Photonmetry'.

Only four basic dimensions

The only sensible basic physical dimensions are

only the following four:

the dimensions of length, time, mass and current.

For mathematical reasons the unit-dimension is not

mentioned as a basic dimension.

Please see the explanation in the next sections

about the so-called tetroes.

## Tetrology

Historical developments

Assume we are completely free in the choice

of basic dimensions associated with basic variables

and basic units today.

Would we come to the same choice as historically happened,

ie length, time, mass and current?

Predominantly a historical development is not a systematic or rational choice.

An objective

A goal is to invent a basis of dimensions only

by means of perfectly rational arguments.

How the best base can reached?

This base would be construed as the most ideal one.

the ultimate goal is to replace the current base

into the perfect base, the basic base.

The continuation of this site is a search

for the scenario to find it.

The bunch of demands for the ideal set of basic dimensions

- Nature should be described by te smallest number of

basic dimensions being as completely as possible;

without the unit-dimension the number is four.- The basic dimensions must be independent of each other.
- It must be possible to define all the existing

derived dimensions only by means of (repeated)

multiplication and (repeated)division of

basic dimensions.- A dimension only has a meaning when at

least one quantity belongs to it.- A basic dimension must be connected

to a physical constant or to a

combination of several physical constants.The demands for basic quantities

- You must be able to define derived quantities

by means of basic ones.- You must be able to build a coherent system

of units by means of the basic quantities.The demands for basic units

- each basic dimension has its own basic unit.
- You must be able to define derived units

by means of basic ones.- basic units must be reproducible and
- practically useful.
- basic units must be independent of each other.
- basic units must be independent of qualities

of any element or any chemical compound.Additional requirements

When a derived dimension is expressed in basic dimensions

brokenexponentsmay not occur.

The same applies to quantities and units.

The logarithm of any quantity is not defined.

Exp(any quantity) = e^{(any quantity)}is not defined.

When derived dimensions are expressed in basic dimensions

broken exponents are not allowed.

The same for quantities and units.

Physical constants

The demand of reproducibility of leads quickly

to use physical constants.The demand Choise can be made out of

All kind of combinations as

Physical constant connected dimension velocity of light in vacuum speed the spin of the proton angular momentum the electric charge of the proton electric charge the dielektric constant of the vacuum capacity/length the rest mass of some stable elementary particles mass the rest energy of some stable elementary particles energy

*) only if thermo has been accepted. See later that theme.

Physical constant bijbehorende dimension the magnetic permeability of the vacuum self induction/length Wien's constant energy × length *) Faraday's (new) constant charge *) the bohr-radius of the H-atom length the rydbergconstante for H unit-dimension/lengte the bohrmagneton current × oppervlakte the atomic massa-unit mass the chemical massa-unit *) mass ... ...

Because of the requirement of independency of chemicals,

all sorts of physical constants are missing:

fixed energy jumps in atoms and molecules;

various fixed temperatures:

the curie temperature, melt point, triple thermo or

boiling point of all kinds of substances.

The gas constantR, the Avogadro numberN_{A}and the Boltzmann constantk_{B}

have been omitted deliberately after the introduction of thermo.br>

Doubt exists about the gravity constantG.

Is it really constant or not??

That is whyGis missing in the list above.

For the same reason Planck's constants have been left.

They are connecting toG.

In practice physical constants are used already

defining the so-called SI-units.

However my proposals are providingly further.

Unwanted dependency

As we saw we need only four basic dimensions.

More physical constants exist. So

this indicates dependency. Look. dimension energy = dimension massa × (dimension snelheid)^{2}.

hcε_{o}= 68,5181 ×e^{2},

or as a dimension relation:

angular momentum × speed × (capacity per length) =

= eenheiddimensie × (charge)^{2}.

Combinations of physical constants have been allowed.

An example

energy = massa × (speed)^{2}.

So energy can be fixed at the mass of an elektron ×c^{2}.

Of coursecis the velocity of light in vacuum.

The product of two (physical) constants is a (physical) constant again.

Tetroes

To decide what base of dimensions is the best

a new help tool is introduced: the tetro.

In an example we use the traditional base {mass, length, time, current}

only because of familiarity but

not because of the necessary final choice.

We express the dimension of force in the aforementioned basic dimensions:

dimension force = dimension mass × dimension length / (dimension time)^{2}.

Look at the exponents:

the mass has exponent 1, the length has exponent 1,

the time has exponent -2 and the missing current has exponent 0.

Define the tetro of the dimension force as the mathematical fourvector

[1,1,-2,0]. This is the force tetro.

The mass tetro = [1,0,0,0]. This is one of the four basic tetroes.

The three other basic ones are [0,1,0,0] as the length tetro,

[0,0,1,0] as the time tetro and [0,0,0,1] as the current tetro.

It reminds linear algebra and it is!

Every general-physical-dimension-tetro is a linear combination

of the four basic tetroes you saw. The mathematicians say:

these four basic tetroes 'span' a linear space, the tetro space.

Also you can take the inverse way:

[0,1, -2,0] must be the acceleration tetro.

Every physical dimension can be represented as a

unique point in the four-dimensional tetro space.

Most points are not connected to a physical dimension.

for instance: [1, 2 ,3 ,4] has no physical meaning at all.

The zero tetro [0,0,0,0]belongs to the tetro space and

has been connected to the unit-dimension.

Because of dependency [0,0,0,0] does not

belong to the base mathematically seen.

Properties of tetroes

There is a one-to-one relationship between a physical

quantity and its tetro at a chosen base.

To a multiplying of two dimensions belongs

adding of two tetroes:

mass × acceleration = force

1,0,0,0] + [0,1,-2,0] = [1,1,-2,0]

To a division of two dimensions belongs

subtraction of two tetroes:

output / input = rendement

power dimension / power dimension = unit dimension

[1,2,-3,0] - [1,2,-3,0] = [0,0,0,0]

Lattice points

Because we do not allow broken exponents,

always the four numbers in each tetro are integers.

That is why you only find the tetroes in the lattice

points of a four dimensional lattice.

Only about sixty point have been occupied.

That is the number of defined physical dimensions.

The distribution of tetroes in the tetro space

By definition the modulus is the distance of a tetro to

[0, 0, 0, 0] in the tetro space. Never it is negative.

The modulus of the tetro [a,b,c,d] equals √(a^{2}+ b^{2}+c^{2}+ d^{2}).

The longest modulus of all the possible tetroes

defines the radius of a hyper sphere with centre [0, 0, 0, 0]

in which you find all the tetroes.

We define it as the tetro sphere.

The great tetro sumis the vector sum of all the sixty tetroes.

(Adding three tetroes goes like that:

[0,2,3,1] + [-2,0,0,2] + [1,2,0,-1] = [-1,4,3,2])

The smaller the modulus of the great tetro sum

the closer the great tetro sum is to [0,0,0,0]

the more symmetrically the tetroes dwell around [0,0,0,0].

The smallest tetro sphere

For the first time we will use tetroes functionally.

we define the best base of dimension the one

with a tetro sphere that has the smallest radius.

Then the average values of the exponents,

the coordinates of the tetroes, as small as possible.

then the relations between the dimensions are as easy as possible.

Dan zijn the waarden van the exponenten gemiddeld zo klein mogelijk.

the relaties tussen the dimensions zijn dan zo eenvoudig mogelijk.

An other option is, that the average modulus

of all tetroes must be as small as possible.

Now the best rational choise for a dimension base can be made.

the fundamentele energieconstante

We hebben nog een vrije keus:

the keus van the natuurconstante die bij energie behoort.

We hebben the rustenergie van enige stabiele

subatomaire deeltjes ter beschikking.

the vraag reduceert dan tot: welk elementaire deeltje kiezen we?

Omdat the elektron the voorkomende vrije deeltje is

waaraan gemeten kan worden,

is the een praktische keus om the elektron

als uitverkorene te beschouwen.

the fundamentele natuurconstante, the rustenergie van the elektron,

wordt gesymboliseerd met symboolu.

Preliminary summaryIt has been proved to be possible to express all the physical dimensions

in only the dimensions, corresponding to four independent base dimensions

for completeness the unit-dimension must be added to the collection.

All the physical dimensions can be connected to a physical constant.

I investigated a number of possible bases

(Lower you find the results.)

Very surprising the base {mass, length, current, time} had the

smallest great tetro sum and as well the smallest total modulus.

So preliminary this was the best base!

basic dimension is associated to length spin _{electron}/ (mass_{electron}×c)mass mass _{electron}current charge _{proton}× massa_{electron}×c^{2}/ spin_{electron}time spin _{electron}/ (massa_{electron}×c^{2})

Armed with this knowledge the next table can be composed.

It has been sorted on the exponents.

dimension lengte massa stroomsterkte tijd permittivity -3 -1 2 4 particle density -3 0 0 0 electric charge density -3 0 1 1 density -3 1 0 0 dimension lengte massa stroomsterkte tijd electric conductance -2 -1 2 3 capacitance -2 -1 2 4 density of the electrical current -2 0 1 0 electric flux density -2 0 1 1 electric flux density -2 0 1 1 density of the mass current -2 1 0 -1 mass per area -2 1 0 0 dimension lengte massa stroomsterkte tijd magnetic susceptibility -1 -1 2 2 power of a lens -1 0 0 0 repetence -1 0 0 0 wave number -1 0 0 0 magnetic field strength H -1 0 1 0 power per volume -1 1 0 -3 pressure -1 1 0 -2 dynamic viscosity -1 1 0 -1 mass per lenght -1 1 0 0 dimension lengte massa stroomsterkte tijd exposure 0 -1 1 1 frequency 0 0 0 -1 time 0 0 0 1 electrical current 0 0 1 0 electric charge 0 0 1 1 magnetic flux density = B 0 1 -1 -2 power per area 0 1 0 -3 Hooke's constant 0 1 0 -2 mass current 0 1 0 -1 mass 0 1 0 0 dimension lengte massa stroomsterkte tijd acceleration 1 0 0 -2 velocity 1 0 0 -1 length 1 0 0 0 electric dipole moment 1 0 1 1 permeability 1 1 -2 -2 elektric field strength = E 1 1 -1 -3 force 1 1 0 -2 momentum 1 1 0 -1 dimension lengte massa stroomsterkte tijd absorbed dose rate 2 0 0 -3 absorbed dose 2 0 0 -2 specific energy 2 0 0 -2 coefficient of diffusion 2 0 0 -1 area 2 0 0 0 magnetic moment 2 0 1 0 impedance 2 1 -2 -3 coefficient of (self)induction 2 1 -2 -2 magnetic flux 2 1 -1 -2 power 2 1 0 -3 energy 2 1 0 -2 impuls momentum 2 1 0 -1 moment of inertia 2 1 0 0 electric potential 2 2 -1 -3 dimension lengte massa stroomsterkte tijd gravitation constant 3 -1 0 -2 specific volume 3 -1 0 0 volume current 3 0 0 -1 volume 3 0 0 0 specific resistance 3 1 -2 -3

Some tetroes can be seen twice. However they have

been counted only once during the calculation below:

The great tetro sum equals [16 19 5 -39 ].

Its modulus is .47

The sum of all the moduli of the tetroes is 142

The grand sum of all the absolute 'coordinates' is 221

I investigated seven possible bases for dimensions.

This base was the best having the lowest sums.

## (Almost) final conclusion

I wonder: coincidence or not? The solution seems to be 'our' actual base.

The beste base ever seem to be

{mass, length, current, time}

How that choice came about?

I could not find the answer in the physical literature.

## From one base to another

Suppose you want to make thetransitionortransformationfrom

base {length, mass, current, time} to

base {charge, mass, speed, spin}.

(For the memory: spin belongs to angular momentum.)

From the given table you can find

dimension tetro charge [0,0,1,1] mass [0,1,0,0] speed [1,0,0,-1] spin [2,1,0,-1] Matrix

Write the four tetroes as column vectors and set them in line

Bedoelde u: Dan ontstaat deze 4 times;4-matrix Then arises this 4×4-matrix:

0 0 1 2 0 1 0 1 1 0 0 0 1 0 -1 -1 To transform a derived quantity from the old base to the new one

In the old basis the force tetro is [1,1,0, -2).

Then the tetro in the new basis is

the above matrix times [1,1,0, -2] written as column vector.

The product is the searched tetro displayed as column vector.

This calculation should happen in the linear algebra.

The result is [-4, -1, 1, 3]

It means force = charge^{-4}× mass^{-1}× speed^{1}× spin^{3}.

Results of the research

I examined only a limited number of different bases in a spreadsheet.

The results are

*) ssumm = the sum of the absolute values of the four coordinates of all tetroes

base great

tetro

sumsize

of the

tetro sumsum of the

size of

all tetroesssumm *) {length mass current time} [19 16 -39 5] 47 142 221 {charge mass speed spin} [72 -28 5 47] 91 190 318 {energy charge speed spin} [-22 -28 5 47] 59 176 292 {energy charge permittivity speed} [6 -47 25 47] 71 215 369 {charge mass permittivity spin} [-100 149 -72 47] 199 354 646 {charge mass permittivity speed} [100 -51 28 47] 125 283 514 {energy charge permittivity spin} [-6 -39 22 47] 65 256 430

Only 7 bases have been investigated.

It is still possible that the best base has yet to be found.

Who is going to find the basic base by trying?

Or is there a strategy to find the best base?

Who will take the challenge?

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