RECREATIONAL GEOMETRY OF PHYSICAL CONSTANTS

through the "magnifying glass" ()()()()()()()()()()()()()()()()()()()()()()()()()()()() Past year I bought a nice device .

without trade mark we call it monday ,

because the day we bought it .

Like a special lens it's able

to watch scientific numbers

under a geometrical point of view,

applying one set of few parameters

(see images in this page)

To test monday's skills

we'll gather all type of data .

Searching in the internet's published issues

At the bottom of this page

there is a collection of

'' geometric - entertainment '' in wich

we're using , most of times , the significant figures

of some physical constants and other scientific values .

It's worth to say that the source of the values used

are The NIST  (That is  CODATA recomended values ).

Monday's tool-kit .

1- Starting values : Numbers described in the following figure :

Two physical concepts inspire us :

2- Fractal : Structure that has the same appearance at different scale . In reference to that will use the series (32+16+2+1) or any other numerical series akin to that .

3- Action : An atribute of a physical system product of the energy and duration of a process .Combining some of the ítems descibed before . For example take a look at an approximation to Pi that we obtained combining the series (51) and starting values ( g + j ) :

                            Right triangle in wich the sum of the legs is close to Pi .

According to the formula described above will draw other ways to depict Pi besides the typical circle :

                                                            Ways to draw Pi .


On the other hand will use two geometric algorithms clearly analogous to derivative and integral calculus  :

One formula we've used several times . It includes the
parameter  d described in ''starting values'' and Pi :

                                       
                                            MONDAY'S SHELL ALGORITHM  .

For instance will illustrate with the following equation in wich we use four constants .
Significant figures writed in brackets :

compton wavelength : [242631] proton wavelength :[132141]

Planck constant : [662607] electron volt : [16021765]

Planck constant , electron volt , compton wavelength , proton wavelength .

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PLANCK CONSTANT .

The significant figures of the Planck constant, h :[662607]

As for the elementary charge , e :[16021765]

                                               Planck constant in relation to electron volt .

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Planck constant - electron volt .

eV:[16021765]
h :[662607]

                                           Elementary charge - Planck constant .

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Boltzmann  constant approximation .

Significant digits of the values :

Planck constant , h :[662607]
Boltzmann constant , K : [138065]

                                               Boltzmann  constant , Planck constant .

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Proportionality between Planck mass and nucleon mass .

Using the significant figures of values , ( in eV units ).

Neutron mass : [939565]
Proton mass : [938272]
Planck mass : [ 1221 ]

                                                  Planck mass - Nucleon mass   .

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MONDAY'S  SIGH  ALGORITHM .

Will to explore relations among one couple of values (V1 , V2) , the series ''255'' and Avogadro's constant . The following figuration shows the basic rule .

                                          MONDAY'S  SIGH  ALGORITHM .


For instance we've performed one recreational view of Planck's length based upon ''sigh's algorithm''  :


Items used in the formula (significant figures) are :

Planck length : [161616]

Avogadro's number : [60221413]

                                           Planck length - Avogadro's number .


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Application of sigh algorithm to the mass of some primordial elements .

Values  involved in the formula :
Hydrogen :[1007825]
Helium :[40026]
Carbon :[ 12 ]
Nitrogen :[ 14 ]
Oxygen : [ 16 ]

                              Hydrogen , Helium , Carbon , Nitrogen , Oxygen mass .

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Relationship between Planck mass and proton mass .

Significant  values given in electronvolts : Planck mass :[122096]

                                                                     Proton mass : [938272]

                                                          Planck mass , proton mass  .


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The case in wich Black hole entropy is equal to Boltzmann constant :

Inspirated by Bekenstein-Hawking formula of black hole entropy , will describe
a numerical curiosity based upon series 51 and a surface equivalent to square root of two .
Lp , Planck length : [161616]

                                                     Black hole entropy , Boltzmann constant .

If we write two new items in the above formula , will obtain the inverse of fine structure constant :

                                            Fine structure constant-Planck constant (eV)

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Neutron - proton mass ratio .

Neutron mass , in eV : [939565] . Proton mass : [938272]

                                                        Neutron-proton mass ratio  .

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Relationship among Planck mass , proton mass , Planck constant and Boltzmann constant :

Planck mass : [21765]
Proton mass : [167262]

Boltzmann constant : [138065]
Planck constant : [662607]

Avogadro's number : [602214]

               Planck mass - proton mass . Boltzmann constant - Planck constant .

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Geometric curiosity about gravitation constant , speed of light , electron mass and Compton wavelength :

Significant figures of constants , G :[66734]  C :[299792458]  em :[9109382]
                                                  
                                                   /\c : [242631]

                                    Newtonian constant of gravitation , electron mass .

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Nucleon energy in relation to Planck constant gives frequency mode of vibration .


Neutron mass + Proton mass :[334755]
Planck constant : [662607]
Avogadro number : [602214]
Speed of light : [299792458]

                                               Nucleon frecuency mode of vibration .


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The critical density of the universe in relation to dark energy parameter .

Here we apply the parameter  ''omega-lambda''  :[0.696] as the Planck mission shows .
Also will write critical density formula as mass-energy equivalent .

G : [66734]   HO : [ 698 ]

                                          Universe critical density - dark energy .


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Relationship between Newtonian constant of gravitation over h bar c and nucleon mass :

G* :[6708]  , N , neutron mass : [939565] , P , proton mass : [938272]

                                         Newtonian constant of gravitation over h bar c .

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Volume called (301)  :

                                                              Volume (301)

V(301) combined with Pi  and placed in  a surface (artistical depiction) :

                                                              Surface Pi(301)



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Recreational  look at the quantum vacuum  force at an individual case of frequency mode of vibration .

Significant digits of values used in the following approximation :  h Planck constant :[662607]  
 C , speed of light in vacuum : [29979246]     LP , Planck length : [161616]

                                                  Quantum vacuum force .


Now let us compare both equations posted before , that is the Planck force and the above vacuum  force :

 

 

                                          Vacuum  force in relation to Planck force  .

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Obtaining the mass of the electron and the Compton wavelength from vacuum  force (and viceverse) :

em : [9109382]   /\c : [242631]   tp , Planck time : [539095]

                                                    Electron  force (electron acceleration) .

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Planck momentum :

mp , Planck mass : [21766]  C , speed of light in vacuum [299792458]

Alpha , fine structure constant : [7297353]

                                                           Planck  momentum  .

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Quantum energy at Planck temperature :

Here we'll apply Wien's law (peak wavelength)

h , Planck constant :[662607]  C , speed of light in vacuum :[299792458]

/\m = b/Tp  (Wien's law)  where  b = [289777]  and  Tp :[141676]

                                         Quantum energy at Planck temperature .

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Numerical curiosity among higgs boson , Planck mass , Proton mass and W boson mass .

After Particle data group published issue , the mass of the particles described below are :

Hº , Higgs boson :[126]  P , proton mass :[938272] W boson : [804]
 mp , Planck mass :[1221]

                                                Higgs boson , Planck mass .

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An individual case in wich Casimir force ''is equal'' to gravity force .

Howsoever our intention is merely recreational , we will refrain from using powers of ten .

h(bar) , reduced Planck constant : [1054572]
c , speed of light in vacuum : [299792458]

                                                          Casimir force - gravity force  .

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Proton-proton collision in relation to (g + j ) values and the series 51 .

Below , values of the particles involved :

 

     h/e Planck constant in eV : [41356675] 

 

 

                                          Proton colission geometry .

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Nuclear fusion akin to above expression .

(A rare case in this blog where it is not neccesary to use significant figures )

All units are MeV  :

n , neutron mass = 939.565     He , Helium-4 binding energy = 28.3   ER , energy released = 17.59

H2 , deuterium binding energy = 2.22452    H3 , tritium  b.e. = 8.482

 

                                             Nuclear  fusion  reaction  geometry .


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Numerical curiosity in Hawking formula for black hole entropy .

( It is striking that the right side of the equation is equal to the above formula of nuclear fusión ?)

Significant figures of constants :

Lp , Planck length : [161616]    KB , Boltzmann constant : [138065]  C , speed of light in vacuum : [299792458]   h , Planck constant : [662607]   G , Newtonian constant of gravitation : [66736]

                                                  Hawking's black hole entropy  .


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Gravitational potential energy in relation to the physical Action of one photon .

G : [66735]  mp : [21765]  h : [662607] C : [299792458]

                                          Gravitational potential energy - photon action .

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Gravitational potential energy between Earth - Moon

(Just for fun) what about gravitational potential energy between the Earth and the Moon , both separated by the distance light travels in one second .

Significant figures:

Mass of the Earth :[ 5972 ]   Mass of the Moon : [ 7347 ]

G : [6674]  C : [299792458]   h : [662607]

                                            Earth - Moon gravitational potential

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A geometric view at some coupling constants  .

The following formula relates  fine structure constant (alpha- EM ), Strong coupling constant (alpha-s)  and gravitational coupling constant (alpha''- G) that refers the ratio of the mass of the proton and the Planck mass , squared .
As for the values (according to Particle data group ) :

alpha-em = 0.007297353     alpha-s = 0.1189  alpha''-G = 5.905 x 10^-39 .

                                           Physical coupling constants geometry .

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A geometric view at some cosmological parameters .

According to Planck misión data the values of the cosmological parameters described below are :

omega-c = 0.268   omega-/\ = 0.683   omega-b = 0.049

                                          cosmological parameters geometry .

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A recreational look at the Planck scale.

When write Physical coupling constants and cosmological parameters in the way in wich is described below , we obtain a simple geometric value that agree with the Planck scale :

                                                Planck scale geometry  .

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                                               Powers of ten geometry .

                                                Starting length and Surface .


The formula described below is inspired by the two preceding images . The left side of the equation belongs to the physical world . The right side belongs to the geometric realm .

As for the Planck length , the recommended value in S.I. units : 1.6162 x 10^-35  m .

                                                   Planck's  area  geometry  .

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Geometric relationship among the mass of the electron , the neutron and the proton .

Symbol alpha refers to fine structure constant = 0.007297353
The mass expressed in electronvolts :

e = 0.51009989 MeV    N = 939.56536  MeV   P = 938.272 MeV

            Electron mass,neutron mass,proton mass ,fine structure constant .


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Geometric ratio between electron mass and Planck mass .

 electron mass = 0.511 x 10^6 electronvolts

 Planck mass = 1.221 x 10^28 electronvolts

Avogadro's number = 6.02214 x 10^23

 

Electron mass - Planck mass ratio

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Planck temperature,Planck frequency,today's temperature,today's frequency .


Wp = 1.855 x 10^43 Hz ,   Wt = 1.602 x 10^11 Hz

 Tp = 1.417 x 10^32 K  ,  Tt = 2.725 K

Avogadro's number = 6.022 x 10^23

      Planck frequency , today's frequency , Planck temperature , today's temperature.


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Today's energy density of the universe - Planck's energy density :

The CMB (cosmic microwave background) temperature  of  2.725 K coresponds to an energy

density of  0.26 eV .  On the other hand Planck's energy density is the value obtained by dividing

 Planck's energy over Planck's volume  =  2.892 x 10^132  eV . 
 
As for Avogadro's number = 6.02214 x 10^23

                                          Today's energy density - Planck energy density

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A geometric value resulting from relationship among density-temperature-frequency in Planck's era and in the present Universe .


Densities values were previously described . Today's temperature = 2.725 K

Planck temperature = 1.417 x 10^32 K  . Planck frequency = 1.855 x 10^43 Hz

Today's frequency , according to CMB (cosmic microwave background) = 1.602 x 10^11 Hz

Note the Planck scale squared .

                                          Density , Temperature , Frequency , Geometry .

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As for the age of Universe in relation to several physical constants .

According to the data of Planck mission the age of Universe  to  is close to  4.353 x 10^17 seconds

The other values described below are : speed of light , Planck's length and Avogadro's number .

Note that the Planck scale appears explicitly in the formula .

                                       Age of Universe , Planck length , speed of light .

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Particular view of ideal gas law :

Starting from ideal gas law will consider a special gas inside  a particular cylinder  under Planck

temperature . The following formula suggest that resulting pressure is the inverse of the gravitational

constant  G .

                                                            Ideal gas law geometry .

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Ideal gas law in a sphere .

Let's see what happens if the gas is in a sphere instead of a cylinder :

                                                   Gas law in a sphere .

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Ideal gas law in a sphere whose radius is Planck length .

Note that in the equation appears Avogadro number (in addition to the Avogadro number implicitly associated to the constant R) .

                                          Gas law in a Planck sphere  .

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Ideal gas law in the case in wich Planck force is involved .

After assigning different values to the variables of the equation , after making the relevant calculations appeared the value of the fine structure constant ( Alpha ) . By the way , Planck force  multiplied by Pi is equal to vacuum force if we consider Planck frequency .

                                           Ideal gas law , Planck force  .

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Planck temperature - today temperature .


According to series 51 and the rectangle   ( g + j ) depicted before , will describe the ratio between Planck temperature , 1.4168 x 10^32 K  and today's temperature (according the Cosmic microwave background) , 2.7255 K .

                                                  Planck temperature-today temperature .

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Production of Higgs boson from electron-positron collision .

Consider the case in wich one electron collides with a positron . The two can merge to form a virtual W boson wich can then emit a Higgs boson .

As for the masses involved (GeV) : electron = 0.000511 , W = 80.4
Note that the right side of the equation belongs to the same that those related to nuclear fusión , black hole entropy and proton - proton collision , all described before .

                                           Higgs boson from fermions .

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Production of Higgs boson from proton-proton collision :

Follows another one example of the use of the series (51) and the symmetry (g + j )although in this case appears only one value (square root of two). The case in wich two protons collide and a Higgs boson results after the event . The mass of proton = 0.938272 GeV
Higgs boson mass (published) = 124.99 GeV .

                                            Higgs boson from  Proton collision .

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RECREATIONAL SCIENCE .
EASY EQUATIONS FOR A COMPLEX UNIVERSE.

We recently made an article concerning the geometry of the Universe . The paper named : The Universe as a Torus .

                                                 UNIVERSE  GEOMETRY EQUIVALENCE .

In that article as in other will apply the most simple geometry , looking for  elementary symmetries.
Write down in this blog successive contributions on these issues .
In addition to numerical equivalence between two volumes ( torus-Avogadro) and (polyhedron-speed of light) there is a third equivalence : ( proton Compton wavelength cubed -Torus volume) .
 
Definition of Torus-3 :

Definition of C3 :

Numerical equivalence Torus-3 and C3 :

(Inverse) Numerical relationship Avogadro's number - Proton Compton wavelength :

 

 

(Inverse)Numerical equivalence . Torus 3 and Proton Compton wavelength cubed :                                           

 

 

 

 

An other universe geometry equivalence depiction :

 

 

 

Geometric relationship between cosmological constant and Avogadro number : 

 

 

Surface area equivalences . An alternative view of the above formula :

An 'elementary' Surface used as numerical distiller :

Geometric distilled from proton energy :

 

The Torus , the Hubble parameter and a numerical frame .Where  h is a dimensionless number  Assumed h = 0.7:

Alternative view when two volume are retated :

Schematic depiction of the above equation  :

 

An other  vectorial look in wich values involved include : Euler number and inverse of fine structure constant . Besides the Torus-3  :

 Development of the Torus-3 from symmetry (g + j ) ,  the Neutron-Proton mass ratio N/P and the  series akin to the series (32+16+2+1) used time ago for an numerical approximation to Pi . Symbol e refers to Euler number :

 

                                                   The evolution of a (particular) Universe .


Numerical equivalence between (A B^2 ) and an other type of Torus :

According to above equations will depict the development from (smooth) universe  , Torus-03 to a granulated one , ie Torus - 3 in wich Avogadro's number appears .

                                             geometric beginning and evolution of a Universe

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Torus-3 , Planck mass-energy and the mass-energy of the electron from the geometrical point of view .

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Hubble parameter , Proton Compton wavelength , Parsec , Speed of light .

Relationship between Planck energy density and (today) Dark energy density :

Biology and geometry por A . Coe se encuentra bajo una Licencia Creative Commons Atribución-NoComercial 3.0 Unported.