# 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
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, a quantity = a number × a unit.
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
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 here A asks for the unit [A].
When A belongs 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;
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.

## 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 brightness is 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

## 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.
When a derived dimension is expressed in basic dimensions
broken exponents may 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
Physical constantconnected dimension
velocity of light in vacuumspeed
the spin of the protonangular momentum
the electric charge of the protonelectric charge
the dielektric constant of the vacuumcapacity/length
the rest mass of some stable elementary particlesmass
the rest energy of some stable elementary particlesenergy
All kind of combinations as
Physical constantbijbehorende dimension
the magnetic permeability of the vacuumself induction/length
Wien's constantenergy × length *)
the rydbergconstante for Hunit-dimension/lengte
the bohrmagnetoncurrent × oppervlakte
the atomic massa-unitmass
the chemical massa-unit *)mass
......
*) only if thermo has been accepted. See later that theme.

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 constant R, the Avogadro number NA and the Boltzmann constant kB
have been omitted deliberately after the introduction of thermo.br>
Doubt exists about the gravity constant G.
Is it really constant or not??
That is why G is missing in the list above.
For the same reason Planck's constants have been left.
They are connecting to G.

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.

hcεo = 68,5181 × e2,
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 × c2.
Of course c is 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
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 √(a2 + b2 +c2 + d2).
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 sum is 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 symbool u.

Preliminary summary It 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 spinelectron / (masselectron × c)
mass masselectron
current chargeproton × massaelectron × c2 / spinelectron
time spinelectron / (massaelectron × c2)

Armed with this knowledge the next table can be composed.
It has been sorted on the exponents.

dimensionlengtemassastroomsterktetijd
permittivity-3-124
particle density-3000
electric charge density-3011
density-3100
dimensionlengtemassastroomsterktetijd
electric conductance-2-123
capacitance-2-124
density of the electrical current-2010
electric flux density-2011
electric flux density-2011
density of the mass current-210-1
mass per area-2100
dimensionlengtemassastroomsterktetijd
magnetic susceptibility-1-122
power of a lens-1000
repetence-1000
wave number-1000
magnetic field strength H-1010
power per volume-110-3
pressure-110-2
dynamic viscosity-110-1
mass per lenght-1100
dimensionlengtemassastroomsterktetijd
exposure0-111
frequency000-1
time0 00 1
electrical current0 01 0
electric charge0011
magnetic flux density = B01-1-2
power per area010-3
Hooke's constant010-2
mass current010-1
mass0 100
dimensionlengtemassastroomsterktetijd
acceleration100-2
velocity100-1
length1 0 00
electric dipole moment1011
permeability11-2-2
elektric field strength = E11-1-3
force110-2
momentum110-1
dimensionlengtemassastroomsterktetijd
absorbed dose rate200-3
absorbed dose200-2
specific energy200-2
coefficient of diffusion200-1
area2000
magnetic moment2010
impedance21-2-3
coefficient of (self)induction21-2-2
magnetic flux21-1-2
power210-3
energy210-2
impuls momentum210-1
moment of inertia2100
electric potential22-1-3
dimensionlengtemassastroomsterktetijd
gravitation constant3-10-2
specific volume3-100
volume current300-1
volume3000
specific resistance31-2-3

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 The beste base ever seem to be {mass, length, current, time}

I wonder: coincidence or not? The solution seems to be 'our' actual base.
I could not find the answer in the physical literature.

## From one base to another

Suppose you want to make the transition or transformation from
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
dimensiontetro
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:
0012
0101
1000
10-1-1
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 × speed1 × spin3.

Results of the research
I examined only a limited number of different bases in a spreadsheet.
The results are
basegreat
tetro
sum
size
of the
tetro sum
sum of the
size of
all tetroes
ssumm *)
{length mass current time}[19 16 -39 5]47142221
{charge mass speed spin}[72 -28 5 47]91190318
{energy charge speed spin}[-22 -28 5 47]59176292
{energy charge permittivity speed}[6 -47 25 47]71215369
{charge mass permittivity spin}[-100 149 -72 47]199354646
{charge mass permittivity speed}[100 -51 28 47]125283514
{energy charge permittivity spin}[-6 -39 22 47]65256430
*) ssumm = the sum of the absolute values of the four coordinates of all tetroes

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?