Planck units (geometrical)

Natural Planck units as geometrical objects (the mathematical electron model)

In a geometrical Planck unit theory, the dimensioned universe at the Planck scale is defined by discrete geometrical objects for the Planck units; Planck mass, Planck length, Planck time and Planck charge. The object embeds the attribute (mass, length, time, charge) of the unit, whereas for numerical based constants, the numerical values are dimensionless frequencies of the SI unit (kg, m, s, A), 3kg refers to 3 of the unit kg, the number 3 carries no mass-specific information.


Geometrical objects

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The mathematical electron [1] is a Planck unit model where mass  , length  , time  , and ampere   are each assigned discrete geometrical objects from the geometry of 2 dimensionless physical constants, the (inverse) fine structure constant α and Omega Ω. Embedded into each object is the object function (attribute).

Table 1. Geometrical units
Attribute Geometrical object
mass  
time  
sqrt(momentum)  
velocity  
length  
ampere  


As the geometries of dimensionless constants, these objects are also dimensionless and so are independent of any system of units, and of any numerical system, and so could qualify as "natural units" (naturally occuring units);

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck [2][3]



As geometrical objects, they may combine Lego-style to form more complex objects such as electrons (i.e.: by embedding mass and ampere objects into the geometry of the electron (the electron object), the electron can have wavelength and charge) [4]. This requires a mathematical (unit number) relationship that defines how the objects interact with each other.

Table 2. Unit number
Attribute Object Unit number θ
mass    
time    
sqrt(momentum)    
velocity    
length    
ampere    



As alpha (α = 137.035 999 084) and Omega (Ω = 2.007 134 949 636) both have numerical solutions, we can assign to MLTA numerical values, i.e.: V = 2πΩ2 = 25.3123819 and use to solve geometrical physical constant equivalents.

Table 3. Physical constant equivalents
CODATA 2014 [5] SI unit Geometrical constant unit uθ
c = 299 792 458 (exact)   c* = V = 25.312381933  
h = 6.626 070 040(81) e-34   h* =   = 12647.2403   =  
G = 6.674 08(31) e-11   G* =   = 50950.55478   =  
e = 1.602 176 620 8(98) e-19   e* =   = 735.70635849   =  
kB = 1.380 648 52(79) e-23   kB* =   = 0.679138336   =  



We then find that where the unit numbers cancel, the numerical solutions agree (see Table 8).

Table 4. Dimensionless combinations
CODATA 2014 (mean) (α, Ω) units uΘ = 1
  1.000 8254   = 1.0  
  0.228 473 639... 10-58   0.228 473 759... 10-58  
  3.376 716   3.381 506  



Scalars

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To translate from geometrical objects to a numerical system of units requires system dependent scalars (kltpva). For example;

If we use k to convert M to the SI Planck mass (M*kSI =  ), then kSI = 0.2176728e-7kg (SI units)
Using vSI = 11843707.905m/s gives c = V*vSI = 299792458m/s (SI units)
Using vimp = 7359.3232155miles/s gives c = V*vimp = 186282miles/s (imperial units)


Table 5. Geometrical units
Attribute Geometrical object Scalar Unit uθ
mass   k  
time   t  
sqrt(momentum)   r2  
velocity   v  
length   l  
ampere   a  



Scalar relationships

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Because the scalars also include the SI unit, v = 11843707.905m/s ... they follow the unit number relationship uθ. This means that we can find ratios where the scalars cancel. Here are examples (units = 1), as such only 2 scalars are required, for example, if we know the numerical value for a and for l then we know the numerical value for t (t = a3l3), and from l and t we know the value for k.

 


This means that once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, μ0) are assigned exact values, following the 2019 redefinition of SI base units 4 constants have been independently assigned exact values which is problematic in terms of this model.

Scalars r (θ = 8) and v (θ = 17) are chosen as they can be derived directly from the 2 constants with exact values; c and μ0.

 
 



Table 6. Geometrical objects
attribute geometrical object unit number θ scalar r(8), v(17)
mass   15 = 8*4-17  
time   -30 = 8*9-17*6  
velocity   17 v
length   -13 = 8*9-17*5  
ampere   3 = 17*3-8*6  


Table 7. Comparison; SI and θ
constant θ (SI unit) MLTVA scalar r(8), v(17)
c   (-13+30 = 17) c* =   17
h   (15-26+30=19) h* =   8*13-17*5=19
G   (-39-15+60=6) G* =   8*5-17*2=6
e   (3-30=-27) e* =   8*3-17*3=-27
kB   (15-26+60-20=29) kB* =   8*10-17*3=29
μ0   (15-13+60-6=56) μ0* =   8*7=56



Fine structure constant

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The fine structure constant can be derived from this formula (units and scalars cancel).

 
 
 




Electron formula

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The electron object (formula fe) is a mathematical particle (units and scalars cancel).

  units = 1


In this example, embedded within the electron are the objects for charge, length and time ALT. AL as an ampere-meter (ampere-length) are the units for a magnetic monopole.

 
 
 


Associated with the electron are dimensioned parameters, these parameters however are a function of the MLTA units, the formula fe dictating the frequency of these units. By setting MLTA to their SI Planck unit equivalents (Table 6.);

electron mass   (M = Planck mass =   = 0.910 938 232 11 e-30

electron wavelength   (L = Planck length =   = 0.242 631 023 86 e-11

elementary charge   (T = Planck time) =   = 0.160 217 651 30 e-18

Rydberg constant   = 10 973 731.568 508



Omega

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The most precise of the experimentally measured constants is the Rydberg constant R = 10973731.568508(65) 1/m. Here c (exact), Vacuum permeability μ0 = 4π/10^7 (exact) and R (12-13 digits) are combined into a unit-less ratio;

 
 


 
 , units = 1


We can now define Ω using the geometries for (c*, μ0*, R*) and then solve by replacing (c*, μ0*, R*) with the numerical (c, μ0, R).

 
  (CODATA 2014 mean values)
  (CODATA 2018 mean values)

There is a close natural number for Ω that is a square root implying that Ω can have a plus or a minus solution, and this agrees with theory (in the mass domain Ω occurs as Ω2 = plus only, in the charge domain Ω occurs as Ω3 = can be plus or minus; see sqrt(momentum)). This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants).

 




Dimensionless combinations

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Reference List of dimensionless combinations. These can be solved using only α, Ω (and the mathematical constants 2, 3, π) as the units and scalars have cancelled. The precision of the results depends on the precision of the SI constants; combinations with G and kB return the least precise values. These combinations can be used to test the veracity of the MLTA geometries as natural Planck units. See also Anomalies (below).


Example

  0.228 473 759... 10-58


Note: the geometry   (integer n ≥ 0) is common to all ratios where units and scalars cancel, suggesting a geometrical base-15.


Table 8. Dimensionless combinations
CODATA 2014 mean (α, Ω) mean units = 1 scalars = 1
  1.000 8254   = 1.0    
  0.228 473 639... 10-58   0.228 473 759... 10-58    
  0.326 103 528 6170... 10301   0.326 103 528 6170... 10301    
  0.170 514 342... 1092   0.170 514 368... 1092    
  73 095 507 858.   73 035 235 897.    
  3.376 716   3.381 506    






Table of Constants

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We can construct a table of constants using these 3 geometries. Setting

 

i.e.: unit number θ = (-13*15) - (15*9) - (-30*11) = 0


 , units =   = 1 (unit number = 0, no scalars)
  , units =   = u1 = u (unit number = -13 -15 +30 = 2/2 = 1, with scalars v, r)
  , units =   = 1, (unit number = 15*2 -30 = 0, with scalars v, r)


Note: The following suggests a numerical boundary to the values the SI constants can have.

  ... = 23326079.1...; unit = u
  gives a range from 0.812997... x10-59 to 0.123... x1060


Note: Influence of  , units = 1

 
 
 


Table 9. Table of Constants
Constant θ Geometrical object (α, Ω, v, r) Unit Calculated CODATA 2014
Time (Planck)       T = 5.390 517 866 e-44 tp = 5.391 247(60) e-44
Elementary charge       e* = 1.602 176 511 30 e-19 e = 1.602 176 620 8(98) e-19
Length (Planck)       L = 0.161 603 660 096 e-34 lp = 0.161 622 9(38) e-34
Ampere       A = 0.297 221 e25 e/tp = 0.297 181 e25
Gravitational constant       G* = 6.672 497 192 29 e11 G = 6.674 08(31) e-11
      X = 918 977.554 22
Mass (Planck)       M = .217 672 817 580 e-7 mP = .217 647 0(51) e-7
sqrt(momentum)      
Velocity       V = 299 792 458 c = 299 792 458
Planck constant       h* = 6.626 069 134 e-34 h = 6.626 070 040(81) e-34
Planck temperature       Tp* = 1.418 145 219 e32 Tp = 1.416 784(16) e32
Boltzmann constant       kB* = 1.379 510 147 52 e-23 kB = 1.380 648 52(79) e-23
Vacuum permeability       μ0* = 4π/10^7 μ0 = 4π/10^7





From the perspective of geometries

note:   constants have no Omega term.

Table 10. Dimensioned constants; geometrical vs CODATA 2014
Constant In Planck units Geometrical object SI calculated (r, v, Ω, α*) SI CODATA 2014 [6]
Speed of light V   c* = 299 792 458, unit = u17 c = 299 792 458 (exact)
Fine structure constant α* = 137.035 999 139 (mean) α = 137.035 999 139(31)
Rydberg constant     R* = 10 973 731.568 508, unit = u13 R = 10 973 731.568 508(65)
Vacuum permeability     μ0* = 4π/10^7, unit = u56 μ0 = 4π/10^7 (exact)
Vacuum permittivity    
Planck constant     h* = 6.626 069 134 e-34, unit = u19 h = 6.626 070 040(81) e-34
Gravitational constant     G* = 6.672 497 192 29 e11, unit = u6 G = 6.674 08(31) e-11
Elementary charge     e* = 1.602 176 511 30 e-19, unit = u-27 e = 1.602 176 620 8(98) e-19
Boltzmann constant     kB* = 1.379 510 147 52 e-23, unit = u29 kB = 1.380 648 52(79) e-23
Electron mass   me* = 9.109 382 312 56 e-31, unit = u15 me = 9.109 383 56(11) e-31
Classical electron radius   λe* = 2.426 310 2366 e-12, unit = u-13 λe = 2.426 310 236 7(11) e-12
Planck temperature     Tp* = 1.418 145 219 e32, unit = u20 Tp = 1.416 784(16) e32
Planck mass M   mP* = .217 672 817 580 e-7, unit = u15 mP = .217 647 0(51) e-7
Planck length L   lp* = .161 603 660 096 e-34, unit = u-13 lp = .161 622 9(38) e-34
Planck time T   tp* = 5.390 517 866 e-44, unit = u-30 tp = 5.391 247(60) e-44
Ampere     A* = 0.297 221 e25, unit = u3 e/tp = 0.297 181 e25
Von Klitzing constant     RK* = 25812.807 455 59, unit = u73 RK = 25812.807 455 5(59)
Gyromagnetic ratio   γe/2π* = 28024.953 55, unit = u-42 γe/2π = 28024.951 64(17)

Note that r, v, Ω, α are dimensionless numbers, however when we replace uθ with the SI unit equivalents (u15 → kg, u-13 → m, u-30 → s, ...), the geometrical objects (i.e.: c* = 2πΩ2v = 299792458, units = u17) become indistinguishable from their respective physical constants (i.e.: c = 299792458, units = m/s).



2019 SI unit revision

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Following the 26th General Conference on Weights and Measures (2019 redefinition of SI base units) are fixed the numerical values of the 4 physical constants (h, c, e, kB). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so the revision may lead to unintended consequences.

Table 11. Physical constants
Constant CODATA 2018 [7]
Speed of light c = 299 792 458 (exact)
Planck constant h = 6.626 070 15 e-34 (exact)
Elementary charge e = 1.602 176 634 e-19 (exact)
Boltzmann constant kB = 1.380 649 e-23 (exact)
Fine structure constant α = 137.035 999 084(21)
Rydberg constant R = 10973 731.568 160(21)
Electron mass me = 9.109 383 7015(28) e-31
Vacuum permeability μ0 = 1.256 637 062 12(19) e-6
Von Klitzing constant RK = 25812.807 45 (exact)

For example, if we solve using the above formulas;

 

 

 

 

 




Anomalies

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The following are notes on the anomalies as evidence of a simulation universe source code [8].


mP, lp, tp
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In this ratio, the MLT units and klt scalars both cancel; units = scalars = 1, reverting to the base MLT objects. Setting the scalars klt for SI Planck units;

k = 0.217 672 817 580... x 10-7kg
l = 0.203 220 869 487... x 10-36m
t = 0.171 585 512 841... x 10-43s
  (CODATA 2018 mean)

The klt scalars cancel, leaving;

 0.109 293... 1024 ,  

Solving for the SI units;

  0.109 485... 1024


A, lp, tp
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a = 0.126 918 588 592... x 1023A
  0.205 571... 1013,  
  0.205 543... 1013,  


The Planck units are known with low precision, and so by defining the 3 most accurately known dimensioned constants in terms of these objects (c, R = Rydberg constant,  ; CODATA 2014 mean values), we can test to greater precision;

c, μ0, R
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  0.326 103 528 6170... 10301,  
  0.326 103 528 6170... 10301,  


c, e, kB, h
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  = 1.0,  
  1.000 8254,  


c, h, e
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  0.228 473 759... 10-58,  
  0.228 473 639... 10-58,  


me, λe
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c, e, me
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 , units = scalars = 1 (me formula)
  0.170 514 368... 1092,  
  0.170 514 342... 1092,  


kB, c, e, me
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  73 035 235 897.,  
  73 095 507 858.,  


mP, tp, ε0
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These 3 constants, Planck mass, Planck time and the vacuum permittivity have no Omega term.

  36.875,  
  36.850,  


G, h, c, e, me, KB
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  0.1415... 10-21,  
  0.1413... 10-21,  


 


Note: The above will apply to any combinations of constants (alien or terrestrial) where scalars = 1.


SI Planck unit scalars
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MT to LPVA
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In this example LPVA are derived from MT. The formulas for MT;

 
 

Replacing scalars pvla with kt

 
 
 
 


PV to MTLA
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In this example MLTA are derived from PV. The formulas for PV;

 
 

Replacing scalars klta with pv

 
 
 
 




G, h, e, me, kB
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As geometrical objects, the physical constants (G, h, e, me, kB) can also be defined using the geometrical formulas for (c*, μ0*, R*) and solved using the numerical (mean) values for (c, μ0, R, α). For example;

  ... and ...
 


Table 12. Calculated from (R, c, μ0, α) columns 2, 3, 4 vs CODATA 2014 columns 5, 6
Constant Formula Units Calculated from (R, c, μ0, α) CODATA 2014 [9] Units
Planck constant    , θ = 57 h* = 6.626 069 134 e-34, θ = 19 h = 6.626 070 040(81) e-34  , θ = 19
Gravitational constant    , θ = 30 G* = 6.672 497 192 29 e11, θ = 6 G = 6.674 08(31) e-11  , θ = 6
Elementary charge    , θ = -81 e* = 1.602 176 511 30 e-19, θ = -27 e = 1.602 176 620 8(98) e-19  , θ = -27
Boltzmann constant    , θ = 87 kB* = 1.379 510 147 52 e-23, θ = 29 kB = 1.380 648 52(79) e-23  , θ = 29
Electron mass    , θ = 45 me* = 9.109 382 312 56 e-31, θ = 15 me = 9.109 383 56(11) e-31  , θ = 15
Gyromagnetic ratio    , θ = -126 e*/2π) = 28024.953 55, θ = -42 γe/2π = 28024.951 64(17)  , θ = -42
Planck mass    , θ = 225 mP* = 0.217 672 817 580 e-7, θ = 15 mP = 0.217 647 0(51) e-7  , θ = 15
Planck length    , θ = -195 lp* = 0.161 603 660 096 e-34, θ = -13 lp = 0.161 622 9(38) e-34  , θ = -13



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References

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  1. Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x. 
  2. Planck (1899), p. 479.
  3. *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System", 287–296.
  4. A Planck scale mathematical universe model
  5. [1] | CODATA, The Committee on Data for Science and Technology | (2014)
  6. [2] | CODATA, The Committee on Data for Science and Technology | (2014)
  7. [3] | CODATA, The Committee on Data for Science and Technology | (2018)
  8. Macleod, Malcolm J. "Physical constant anomalies suggest a mathematical relationship linking SI units". RG. doi:10.13140/RG.2.2.15874.15041/6. 
  9. [4] | CODATA, The Committee on Data for Science and Technology | (2014)