# Planck units (geometrical)

Natural Planck units as geometrical objects (mathematical electron model)

Planck unit theories use basic units for Planck mass, Planck length, Planck time and charge, and operate at the Planck scale. In a geometrical Planck unit model, these units are assigned geometrical objects rather than numerical values, for the geometries themselves can encode the attribute (mass, length, time, charge) of the unit, whereas for SI units (kg, m, s, A) the numerical values are simply dimensionless frequencies of the unit, 3kg refers to 3 of the unit kg, the number 3 carries no mass-specific information.

This however requires that the geometry of the (Planck) objects are selected whereby they may interact with each other, and so a physical universe can be constructed Lego-style by combining the Planck objects 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) [1]. This in turn requires a (mathematical) unit relationship linking the objects.

### Geometrical objects

In this (the mathematical electron) model [2], the base units for mass ${\displaystyle M}$, length ${\displaystyle L}$, time ${\displaystyle T}$, and ampere ${\displaystyle A}$ are assigned geometrical objects from the geometry of 2 dimensionless physical constants, the (inverse) fine structure constant α = 137.035 999 139 and Omega Ω = 2.007 134 949 636. Embedded into each object is the object function (attribute) with a unit number (θ) assigned whereby relationships between the objects can be defined.

Table 1. Geometrical units
Attribute Geometrical object Unit number θ
mass ${\displaystyle M=(1)}$ ${\displaystyle 15}$
time ${\displaystyle T=(\pi )}$ ${\displaystyle -30}$
sqrt(momentum) ${\displaystyle P=(\Omega )}$ ${\displaystyle 16}$
velocity ${\displaystyle V=(2\pi \Omega ^{2})}$ ${\displaystyle 17}$
length ${\displaystyle L=(2\pi ^{2}\Omega ^{2})}$ ${\displaystyle -13}$
ampere ${\displaystyle A={\frac {16V^{3}}{\alpha P^{3}}}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})}$ ${\displaystyle 3}$

Being dimensionless they are independent of any system of units, and as geometrical objects they are independent of any numerical system, and so could qualify as "natural 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 [3][4]

As alpha and Omega both have numerical solutions, we can assign to MLTA numerical values, i.e.: V = 2πΩ2 = 25.3123819.

Table 2. Comparison: Dimensioned physical constants and geometrical objects
CODATA 2014 [5] SI unit Geometrical constant Unit number θ
c = 299 792 458 (exact) ${\displaystyle {\frac {m}{s}}}$ c* = V = 25.312381933 17
h = 6.626 070 040(81) e-34 ${\displaystyle {\frac {kg\;m^{2}}{s}}}$ h* = ${\displaystyle 2\pi MVL}$ = 12647.2403 15+17-13 = 19
G = 6.674 08(31) e-11 ${\displaystyle {\frac {m^{3}}{kg\;s^{2}}}}$ G* = ${\displaystyle {\frac {V^{2}L}{M}}}$ = 50950.55478 34-13-15 = 6
e = 1.602 176 620 8(98) e-19 ${\displaystyle C=As}$ e* = ${\displaystyle AT}$ = 735.70635849 3-30 = -27
kB = 1.380 648 52(79) e-23 ${\displaystyle {\frac {kg\;m^{2}}{s^{2}\;K}}}$ kB* = ${\displaystyle {\frac {2\pi VM}{A}}}$ = 0.679138336 17+15-3 = 29

#### Scalars

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 = ${\displaystyle m_{P}}$), 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 3. Geometrical units
Attribute Geometrical object Scalar Unit uθ
mass ${\displaystyle M=(1)}$ k ${\displaystyle u^{15}}$
time ${\displaystyle T=(\pi )}$ t ${\displaystyle u^{-30}}$
sqrt(momentum) ${\displaystyle P=(\Omega )}$ r2 ${\displaystyle u^{16}}$
velocity ${\displaystyle V=(2\pi \Omega ^{2})}$ v ${\displaystyle u^{17}}$
length ${\displaystyle L=(2\pi ^{2}\Omega ^{2})}$ l ${\displaystyle u^{-13}}$
ampere ${\displaystyle A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})}$ a ${\displaystyle u^{3}}$

#### Scalar relationships

Because the scalars also include the unit, v = 11843707.905m/s ... they follow the unit number relationship. 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 a and l then we know t (t = a3l3), and from l and t we know k.

${\displaystyle {\frac {u^{3*3}u^{-13*3}}{u^{-30}}}\;({\frac {a^{3}l^{3}}{t}})={\frac {u^{-13*15}}{u^{15*9}u^{-30*11}}}\;({\frac {l^{15}}{k^{9}t^{11}}})=\;...\;=1}$

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. Here the attributes are defined in terms of 2 scalars (r, v) as v (θ = 17) can be derived from c and r (θ = 8) can be derived from μ0.

Table 4. Geometrical objects
attribute geometrical object unit number θ scalar r(8), v(17)
mass ${\displaystyle M=(1)}$ 15 = 8*4-17 ${\displaystyle k={\frac {r^{4}}{v}}}$
time ${\displaystyle T=(\pi )}$ -30 = 8*9-17*6 ${\displaystyle t={\frac {r^{9}}{v^{6}}}}$
velocity ${\displaystyle V=(2\pi \Omega ^{2})}$ 17 v
length ${\displaystyle L=(2\pi ^{2}\Omega ^{2})}$ -13 = 8*9-17*5 ${\displaystyle l={\frac {r^{9}}{v^{5}}}}$
ampere ${\displaystyle A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})}$ 3 = 17*3-8*6 ${\displaystyle a={\frac {v^{3}}{r^{6}}}}$

Table 5. Comparison; SI and θ
constant θ (SI unit) MLTVA scalar r(8), v(17)
c ${\displaystyle {\frac {m}{s}}}$ (-13+30 = 17) c* = ${\displaystyle V*v}$ 17
h ${\displaystyle {\frac {kg\;m^{2}}{s}}}$ (15-26+30=19) h* = ${\displaystyle 2\pi MVL*{\frac {r^{13}}{v^{5}}}}$ 8*13-17*5=19
G ${\displaystyle {\frac {m^{3}}{kg\;s^{2}}}}$ (-39-15+60=6) G* = ${\displaystyle {\frac {V^{2}L}{M}}*{\frac {r^{5}}{v^{2}}}}$ 8*5-17*2=6
e ${\displaystyle C=As}$ (3-30=-27) e* = ${\displaystyle AT*{\frac {r^{3}}{v^{3}}}}$ 8*3-17*3=-27
kB ${\displaystyle {\frac {kg\;m^{2}}{s^{2}\;K}}}$ (15-26+60-20=29) kB* = ${\displaystyle {\frac {2\pi VM}{A}}*{\frac {r^{10}}{v^{3}}}}$ 8*10-17*3=29
μ0 ${\displaystyle {\frac {kg\;m}{s^{2}\;A^{2}}}}$ (15-13+60-6=56) μ0* = ${\displaystyle {\frac {4\pi V^{2}M}{\alpha LA^{2}}}*r^{7}}$ 8*7=56

#### Dimensionless combinations

If we can reduce the 5 SI units to 2 scalars (example; r, v in tables 4, 5), then we can find combinations of the physical constants where the unit numbers θ and the scalars will cancel, these combinations, which are unit-less (units = 1), will then return the same numerical value as the (Planck) MLTVA object equivalents. This is because if the scalars have cancelled, and as the scalars embed the SI conversion values as well as the SI units, then these combinations are defaulting to the underlying MLTVA objects (the SI component has cancelled). This should therefore apply to any set of units, even extraterrestrial and non-human ones, suggesting that these MLTVA objects could be 'natural' units [6][4]. For example

${\displaystyle {\frac {(h^{*})^{3}}{(e^{*})^{13}(c^{*})^{24}}}=(2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}})^{3}/({\frac {2^{7}\pi ^{4}\Omega ^{3}r^{3}}{\alpha v^{3}}})^{7}.(2\pi \Omega ^{2}v)^{24}={\frac {\alpha ^{13}}{2^{106}\pi ^{64}(\color {red}\Omega ^{15})^{5}\color {black}}}=}$ 0.228 473 759... 10-58

Here we solve physical constant combinations using only α, Ω (and the mathematical constants 2, 3, π). As the scalars have cancelled, we do not need to know their values or units. The precision of the results depends on the precision of the SI constants; combinations with G and kB return the least precise values.

Note: the geometry ${\displaystyle \color {red}(\Omega ^{15})^{n}\color {black}}$ (integer n ≥ 0) is common to all ratios where units and scalars cancel


Table 6. Dimensionless combinations (α, Ω)
CODATA 2014 (mean) (α, Ω) units uΘ = 1 scalars = 1
${\displaystyle {\frac {k_{B}ec}{h}}=}$ 1.000 8254 ${\displaystyle {\frac {(k_{B}^{*})(e^{*})(c^{*})}{(h^{*})}}}$ = 1.0 ${\displaystyle {\frac {(u^{29})(u^{-27})(u^{17})}{(u^{19})}}=1}$ ${\displaystyle ({\frac {r^{10}}{v^{3}}})({\frac {r^{3}}{v^{3}}})(v)/({\frac {r^{13}}{v^{5}}})=1}$
${\displaystyle {\frac {h^{3}}{e^{13}c^{24}}}=}$ 0.228 473 639... 10-58 ${\displaystyle {\frac {(h^{*})^{3}}{(e^{*})^{13}(c^{*})^{24}}}={\frac {\alpha ^{13}}{2^{106}\pi ^{64}(\color {red}\Omega ^{15})^{5}\color {black}}}=}$ 0.228 473 759... 10-58 ${\displaystyle {\frac {(u^{19})^{3}}{(u^{-27})^{13}(u^{17})^{24}}}=1}$ ${\displaystyle ({\frac {r^{13}}{v^{5}}})^{3}/({\frac {r^{3}}{v^{3}}})^{13}(v^{24})=1}$
${\displaystyle {\frac {c^{35}}{\mu _{0}^{9}R^{7}}}=}$ 0.326 103 528 6170... 10301 ${\displaystyle {\frac {(c^{*})^{35}}{(\mu _{0}^{*})^{9}(R^{*})^{7}}}=2^{295}\pi ^{157}3^{21}\alpha ^{26}\color {red}(\Omega ^{15})^{15}\color {black}=}$ 0.326 103 528 6170... 10301 ${\displaystyle {\frac {(u^{17})^{35}}{(u^{56})^{9}(u^{13})^{7}}}=1}$ ${\displaystyle (v^{35})/(r^{7})^{9}({\frac {v^{5}}{r^{9}}})^{7}=1}$
${\displaystyle {\frac {c^{9}e^{4}}{m_{e}^{3}}}=}$ 0.170 514 342... 1092 ${\displaystyle {\frac {(c^{*})^{9}(e^{*})^{4}}{(m_{e}^{*})^{3}}}=2^{97}\pi ^{49}3^{9}\alpha ^{5}(\color {red}\Omega ^{15})^{5}\color {black}=}$ 0.170 514 368... 1092 ${\displaystyle {\frac {(u^{29})(u^{-27})(u^{17})}{(u^{19})}}=1}$ ${\displaystyle (v^{9})({\frac {r^{3}}{v^{3}}})^{4}/({\frac {r^{4}}{v}})^{3}=1}$
${\displaystyle {\frac {k_{B}}{e^{2}m_{e}c^{4}}}=}$ 73 095 507 858. ${\displaystyle {\frac {(k_{B}^{*})}{(e^{*})^{2}(m_{e}^{*})(c^{*})^{4}}}={\frac {3^{3}\alpha ^{6}}{2^{3}\pi ^{5}}}=}$ 73 035 235 897. ${\displaystyle {\frac {(u^{29})}{(u^{-27})^{2}(u^{15})(u^{17})^{4}}}=1}$ ${\displaystyle ({\frac {r^{10}}{v^{3}}})/({\frac {r^{3}}{v^{3}}})^{2}({\frac {r^{4}}{v}})(v)^{4}=1}$
${\displaystyle {\frac {hc^{2}em_{p}}{G^{2}k_{B}}}=}$ 3.376 716 ${\displaystyle {\frac {(h^{*})(c^{*})^{2}(e^{*})(m_{p}^{*})}{(G^{*})^{2}(k_{B}^{*})}}={\frac {2^{11}\pi ^{3}}{\alpha ^{2}}}=}$ 3.381 506 ${\displaystyle {\frac {(u^{19})(u^{17})^{2}(u^{-27})(u^{15})}{(u^{6})^{2}(u^{29})}}=1}$ ${\displaystyle ({\frac {r^{13}}{v^{5}}})v^{2}({\frac {r^{3}}{v^{3}}})({\frac {r^{4}}{v^{1}}})/({\frac {r^{5}}{v^{2}}})^{2}({\frac {r^{10}}{v^{3}}})=1}$

#### Electron formula

The electron object (formula fe) is a classical example of a dimensionless combination. It is a mathematical particle (it is dimensionless and the scalars cancel). As it is the geometry of the 2 dimensionless physical constants, it may also be defined as a dimensionless physical constant.

${\displaystyle f_{e}=4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23}}$

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.

${\displaystyle T=\pi {\frac {r^{9}}{v^{6}}},\;u^{-30}}$
${\displaystyle \sigma _{e}={\frac {3\alpha ^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}}{\frac {r^{3}}{v^{2}}},\;u^{-10}}$
${\displaystyle f_{e}={\frac {\sigma _{e}^{3}}{2T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }},\;units={\frac {(u^{-10})^{3}}{u^{-30}}}=1,scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1}$

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

electron mass ${\displaystyle m_{e}^{*}={\frac {M}{f_{e}}}}$ (M = Planck mass) = 0.910 938 232 11 e-30

electron wavelength ${\displaystyle \lambda _{e}^{*}=2\pi Lf_{e}}$ (L = Planck length) = 0.242 631 023 86 e-11

elementary charge ${\displaystyle e^{*}=A\;T}$ (T = Planck time) = 0.160 217 651 30 e-18

Rydberg constant ${\displaystyle R^{*}=({\frac {m_{e}}{4\pi L\alpha ^{2}M}})={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}}\;u^{13}}$ = 10 973 731.568 508

#### Fine structure constant

The Sommerfeld fine structure constant alpha can also be derived from combinations of dimensioned constants. The most commonly cited example;

${\displaystyle {\frac {2(h^{*})}{(\mu _{0}^{*})(e^{*})^{2}(c^{*})}}=2({2^{3}\pi ^{4}\Omega ^{4}})/({\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}})({\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }})^{2}(2\pi \Omega ^{2})=\color {blue}\alpha \color {black},\;{\frac {u^{19}}{u^{56}(u^{-27})^{2}u^{17}}}=1,\;({\frac {r^{13}}{v^{5}}})({\frac {1}{r^{7}}})({\frac {v^{6}}{r^{6}}})({\frac {1}{v}})=1}$

#### Omega

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;

${\displaystyle \mu _{0}^{*}={\frac {4\pi V^{2}M}{\alpha LA^{2}}}={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7},\;u^{56}}$
${\displaystyle R^{*}=({\frac {m_{e}}{4\pi L\alpha ^{2}M}})={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}},\;u^{13}}$

${\displaystyle {\frac {(c^{*})^{35}}{(\mu _{0}^{*})^{9}(R^{*})^{7}}}=(2\pi \Omega ^{2})^{35}/({\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}})^{9}.({\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}})^{7},\;units={\frac {(u^{17})^{35}}{(u^{56})^{9}(u^{13})^{7}}}=1}$
${\displaystyle {\frac {(c^{*})^{35}}{(\mu _{0}^{*})^{9}(R^{*})^{7}}}=2^{295}\pi ^{157}3^{21}\alpha ^{26}(\Omega ^{15})^{15}}$

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).

${\displaystyle \Omega ^{225}={\frac {(c^{*})^{35}}{2^{295}3^{21}\pi ^{157}(\mu _{0}^{*})^{9}(R^{*})^{7}\alpha ^{26}}},\;units=1}$
${\displaystyle \Omega =2.007\;134\;949\;636...,\;units=1}$ (CODATA 2014 mean values)
${\displaystyle \Omega =2.007\;134\;949\;687...,\;units=1}$ (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. This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants).

${\displaystyle \Omega ={\sqrt {\left(\pi ^{e}e^{(1-e)}\right)}}=2.007\;134\;9543...}$

#### Table of Constants

Scalars r and v were chosen as they can be derived directly from 2 precise constants c and μ0.

${\displaystyle v={\frac {c}{2\pi \Omega ^{2}}}=11843707.905...,\;units=m/s}$
${\displaystyle r^{7}={\frac {2^{11}\pi ^{5}\Omega ^{4}\mu _{0}}{\alpha }};\;r=0.712562514304...,\;units=({\frac {kg.m}{s}})^{1/4}}$

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

${\displaystyle i=\Omega ^{15}}$, units = ${\displaystyle {\sqrt {\frac {L^{15}}{M^{9}T^{11}}}}}$ = 1 (unit number = -13*15 -15*9 +11*30 = 0, no scalars)
${\displaystyle x=\Omega {\frac {v}{r^{2}}}}$ , units = ${\displaystyle {\sqrt {\frac {L}{MT}}}}$ = u1 = u (unit number = -13 -15 +30 = 2/2 = 1) ... ${\displaystyle ({\sqrt {\frac {L^{15}}{M^{9}T^{11}}}}={\sqrt {\frac {L}{MT}}}{\frac {L^{7}}{M^{4}T^{5}}}),\;{\frac {l^{7}}{k^{4}t^{5}}}={\frac {r^{2}}{v}}}$
${\displaystyle y={\frac {r^{17}}{v^{8}}}}$ , units = ${\displaystyle M^{2}T}$ = 1, (unit number = 15*2 -30 = 0) ... ${\displaystyle ({\sqrt {\frac {L^{15}}{M^{9}T^{11}}}}=M^{2}T{\sqrt {\frac {L^{15}}{M^{13}T^{13}}}}),\;{\sqrt {\frac {l^{15}}{k^{13}t^{13}}}}={\frac {v^{8}}{r^{17}}}}$

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

${\displaystyle j={\frac {r^{17}}{v^{8}}}=k^{2}t={\frac {k^{17/4}}{v^{15/4}}}=...}$ gives a range from 0.812997... x10-59 to 0.123... x1060
${\displaystyle a^{1/3}={\frac {v}{r^{2}}}={\frac {1}{t^{2/15}k^{1/5}}}={\frac {\sqrt {v}}{\sqrt {k}}}}$ ... = 23326079.1...; unit = u

Table 7. Table of Constants
Constant θ Geometrical object (α, Ω, v, r) Unit Calculated CODATA 2014
Time (Planck) ${\displaystyle \color {red}-30\color {black}}$ ${\displaystyle T=(\pi )\color {red}{\frac {x^{\theta }i^{2}}{y^{3}}}\color {black}=(\pi ){\frac {r^{9}}{v^{6}}}}$ ${\displaystyle T}$ T = 5.390 517 866 e-44 tp = 5.391 247(60) e-44
Elementary charge ${\displaystyle \color {red}-27\color {black}}$ ${\displaystyle e^{*}=({\frac {2^{7}\pi ^{4}}{\alpha }})\color {red}{\frac {x^{\theta }i^{2}}{y^{3}}}\color {black}=({\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }}){\frac {r^{3}}{v^{3}}}}$ ${\displaystyle {\frac {L^{3/2}}{T^{1/2}M^{3/2}}}=AT}$ e* = 1.602 176 511 30 e-19 e = 1.602 176 620 8(98) e-19
Planck constant ${\displaystyle \color {red}-19\color {black}}$ ${\displaystyle (h^{*})^{-1}=({\frac {1}{2^{3}\pi ^{4}}})\color {red}{\frac {x^{\theta }i}{y^{3}}}\color {black}=({\frac {1}{2^{3}\pi ^{4}\Omega ^{4}}}){\frac {v^{5}}{r^{13}}}}$ ${\displaystyle {\frac {T}{L^{2}M}}}$ h* = 6.626 069 134 e-34 h = 6.626 070 040(81) e-34
Length (Planck) ${\displaystyle \color {red}-13\color {black}}$ ${\displaystyle L=(2\pi ^{2})\color {red}{\frac {x^{\theta }i}{y}}\color {black}=(2\pi ^{2}\Omega ^{2}){\frac {r^{9}}{v^{5}}}}$ ${\displaystyle L}$ L = 0.161 603 660 096 e-34 lp = 0.161 622 9(38) e-34
Gravitational constant ${\displaystyle \color {red}-6\color {black}}$ ${\displaystyle (G^{*})^{-1}=({\frac {1}{2^{3}\pi ^{4}}})\color {red}\color {red}{\frac {x^{\theta }}{y}}\color {black}=({\frac {1}{2^{3}\pi ^{4}\Omega ^{6}}}){\frac {v^{2}}{r^{5}}}}$ ${\displaystyle {\frac {MT^{2}}{L^{3}}}}$ G* = 6.672 497 192 29 e1 G = 6.674 08(31) e-11
Ampere ${\displaystyle \color {red}3\color {black}}$ ${\displaystyle A=({\frac {2^{7}\pi ^{3}}{\alpha }})\color {red}x^{\theta }\color {black}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }}){\frac {v^{3}}{r^{6}}}}$ ${\displaystyle A={\frac {L^{3/2}}{M^{3/2}T^{3/2}}}}$ A = 0.297 221 e25 e/tp = 0.297 181 e25
Mass (Planck) ${\displaystyle \color {red}\color {red}15\color {black}}$ ${\displaystyle M=(1)\color {red}\color {red}{\frac {x^{\theta }y^{2}}{i}}\color {black}=(1){\frac {r^{4}}{v}}}$ ${\displaystyle M}$ M = .217 672 817 580 e-7 mP = .217 647 0(51) e-7
sqrt(momentum) ${\displaystyle \color {red}16\color {black}}$ ${\displaystyle P=\color {red}\color {red}{\frac {x^{\theta }y^{2}}{i}}\color {black}=(\Omega )r^{2}}$ ${\displaystyle {\frac {M^{1/2}L^{1/2}}{T^{1/2}}}}$
Velocity ${\displaystyle \color {red}\color {red}17\color {black}}$ ${\displaystyle V=(2\pi )\color {red}\color {red}{\frac {x^{\theta }y^{2}}{i}}\color {black}=(2\pi \Omega ^{2})v}$ ${\displaystyle V={\frac {L}{T}}}$ V = 299 792 458 c = 299 792 458
Planck temperature ${\displaystyle \color {red}\color {red}20\color {black}}$ ${\displaystyle {T_{p}}^{*}=({\frac {2^{7}\pi ^{3}}{\alpha }})\color {red}\color {red}{\frac {x^{\theta }y^{2}}{i}}\color {black}=({\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha }}){\frac {v^{4}}{r^{6}}}}$ ${\displaystyle {\frac {L^{5/2}}{M^{3/2}T^{5/2}}}=AV}$ Tp* = 1.418 145 219 e32 Tp = 1.416 784(16) e32
Boltzmann constant ${\displaystyle \color {red}\color {red}29\color {black}}$ ${\displaystyle {k_{B}}^{*}=({\frac {\alpha }{2^{5}\pi }})\color {red}{\frac {x^{\theta }y^{4}}{i^{2}}}\color {black}=({\frac {\alpha }{2^{5}\pi \Omega }}){\frac {r^{10}}{v^{3}}}}$ ${\displaystyle {\frac {M^{5/2}T^{1/2}}{L^{1/2}}}={\frac {ML}{TA}}}$ kB* = 1.379 510 147 52 e-23 kB = 1.380 648 52(79) e-23
Vacuum permeability ${\displaystyle \color {red}56\color {black}}$ ${\displaystyle {\mu _{0}}^{*}=({\frac {\alpha }{2^{11}\pi ^{5}}})\color {red}{\frac {x^{\theta }y^{7}}{i^{4}}}\color {black}=({\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}})r^{7}}$ ${\displaystyle {\frac {M^{4}T}{L^{2}}}}$ μ0* = 4π/10^7 μ0 = 4π/10^7

From the perspective of geometries

note: ${\displaystyle \color {red}(u^{15})^{n}\color {black}}$ constants have no Omega term.

Table 8. Dimensioned constants; geometrical vs CODATA 2014
Constant In Planck units Geometrical object SI calculated (r, v, Ω, α*) SI CODATA 2014 [7]
Speed of light V ${\displaystyle c^{*}=(2\pi \Omega ^{2})v,\;u^{17}}$ 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 ${\displaystyle R^{*}=({\frac {m_{e}}{4\pi L\alpha ^{2}M}})}$ ${\displaystyle R^{*}={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}},\;u^{13}}$ R* = 10 973 731.568 508, unit = u13 R = 10 973 731.568 508(65)
Vacuum permeability ${\displaystyle \mu _{0}^{*}={\frac {4\pi V^{2}M}{\alpha LA^{2}}}}$ ${\displaystyle \mu _{0}^{*}={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7},\;u^{56}}$ μ0* = 4π/10^7, unit = u56 μ0 = 4π/10^7 (exact)
Vacuum permittivity ${\displaystyle \epsilon _{0}^{*}={\frac {1}{\mu _{0}^{*}(c^{*})^{2}}}}$ ${\displaystyle \epsilon _{0}^{*}={\frac {2^{9}\pi ^{3}}{\alpha }}{\frac {1}{r^{7}v^{2}}},\;\color {red}1/(u^{15})^{6}\color {black}=u^{-90}}$
Planck constant ${\displaystyle h^{*}=2\pi MVL}$ ${\displaystyle h^{*}=2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}},\;u^{19}}$ h* = 6.626 069 134 e-34, unit = u19 h = 6.626 070 040(81) e-34
Gravitational constant ${\displaystyle G^{*}={\frac {V^{2}L}{M}}}$ ${\displaystyle G^{*}=2^{3}\pi ^{4}\Omega ^{6}{\frac {r^{5}}{v^{2}}},\;u^{6}}$ G* = 6.672 497 192 29 e11, unit = u6 G = 6.674 08(31) e-11
Elementary charge ${\displaystyle e^{*}=AT}$ ${\displaystyle e^{*}={\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }}{\frac {r^{3}}{v^{3}}},\;u^{-27}}$ e* = 1.602 176 511 30 e-19, unit = u-27 e = 1.602 176 620 8(98) e-19
Boltzmann constant ${\displaystyle k_{B}^{*}={\frac {2\pi VM}{A}}}$ ${\displaystyle k_{B}^{*}={\frac {\alpha }{2^{5}\pi \Omega }}{\frac {r^{10}}{v^{3}}},\;u^{29}}$ kB* = 1.379 510 147 52 e-23, unit = u29 kB = 1.380 648 52(79) e-23
Electron mass ${\displaystyle m_{e}^{*}={\frac {M}{f_{e}}},\;u^{15}}$ me* = 9.109 382 312 56 e-31, unit = u15 me = 9.109 383 56(11) e-31
Classical electron radius ${\displaystyle \lambda _{e}^{*}=2\pi Lf_{e},\;u^{-13}}$ λe* = 2.426 310 2366 e-12, unit = u-13 λe = 2.426 310 236 7(11) e-12
Planck temperature ${\displaystyle T_{p}^{*}={\frac {AV}{\pi }}}$ ${\displaystyle T_{p}^{*}={\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha }}{\frac {v^{4}}{r^{6}}},\;u^{20}}$ Tp* = 1.418 145 219 e32, unit = u20 Tp = 1.416 784(16) e32
Planck mass M ${\displaystyle m_{P}^{*}=(1){\frac {r^{4}}{v}},\;\color {red}\color {red}(u^{15})^{1}\color {black}}$ mP* = .217 672 817 580 e-7, unit = u15 mP = .217 647 0(51) e-7
Planck length L ${\displaystyle l_{p}^{*}=(2\pi ^{2}\Omega ^{2}){\frac {r^{9}}{v^{5}}},\;u^{-13}}$ lp* = .161 603 660 096 e-34, unit = u-13 lp = .161 622 9(38) e-34
Planck time T ${\displaystyle t_{p}^{*}=(\pi ){\frac {r^{9}}{v^{6}}},\;\color {red}\color {red}1/(u^{15})^{2}\color {black}}$ tp* = 5.390 517 866 e-44, unit = u-30 tp = 5.391 247(60) e-44
Ampere ${\displaystyle A={\frac {16V^{3}}{\alpha P^{3}}}}$ ${\displaystyle A^{*}={\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }}{\frac {v^{3}}{r^{6}}},\;u^{3}}$ A* = 0.297 221 e25, unit = u3 e/tp = 0.297 181 e25
Von Klitzing constant ${\displaystyle R_{K}^{*}=({\frac {h}{e^{2}}})^{*}}$ ${\displaystyle R_{K}^{*}={\frac {\alpha ^{2}}{2^{11}\pi ^{4}\Omega ^{2}}}r^{7}v,\;u^{73}}$ RK* = 25812.807 455 59, unit = u73 RK = 25812.807 455 5(59)
Gyromagnetic ratio ${\displaystyle \gamma _{e}/2\pi ={\frac {gl_{p}^{*}m_{P}^{*}}{2k_{B}^{*}m_{e}^{*}}},\;unit=u^{-42}}$ 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). If this mathematical relationship can therefore be identified within the SI units themselves, then we have an argument for a Planck scale mathematical universe

#### 2019 SI unit revision

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 9. Physical constants
Constant CODATA 2018 [8]
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;

${\displaystyle R^{*}={\frac {4\pi ^{5}}{3^{3}c^{4}\alpha ^{8}e^{3}}}=10973\;729.082\;465}$

${\displaystyle {(m_{e}^{*})}^{3}={\frac {2^{4}\pi ^{10}R\mu _{0}^{3}}{3^{6}c^{8}\alpha ^{7}}},\;m_{e}^{*}=9.109\;382\;3259\;10^{-31}}$

${\displaystyle {(\mu _{0}^{*})}^{3}={\frac {3^{6}h^{3}c^{5}\alpha ^{13}R^{2}}{2\pi ^{10}}},\;\mu _{0}^{*}=1.256\;637\;251\;88\;10^{-6}}$

${\displaystyle {(h^{*})}^{3}={\frac {2\pi ^{10}\mu _{0}^{3}}{3^{6}c^{5}\alpha ^{13}R^{2}}},\;h^{*}=6.626\;069\;149\;10^{-34}}$

${\displaystyle {(e^{*})}^{3}={\frac {4\pi ^{5}}{3^{3}c^{4}\alpha ^{8}R}},\;e^{*}=1.602\;176\;513\;10^{-19}}$

### Anomalies

Regardless of which system of units we use, alien or terrestrial, any combination of constants where scalars = 1 (i.e.: the scalars overlap and cancel) will give the same numerical result, they will default to the MLTPA objects. This implies that these objects are Planck's 'natural' units, i.e.: that all possible systems of units are based on these objects, and so, given that these are dimensionless geometrical objects, they can be construed as evidence of a mathematical universe. The following are examples of units = scalars = 1 ratios using SI units [9]. Note: the geometry ${\displaystyle \color {red}(\Omega ^{15})^{n}\color {black}}$ (integer n ≥ 0) is common to all (units = 1) ratios that include an Omega term.

###### mP, lp, tp

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
${\displaystyle {\frac {L^{15}}{M^{9}T^{11}}}={\frac {(2\pi ^{2}\Omega ^{2})^{15}}{(1)^{9}(\pi )^{11}}}({\frac {l^{15}}{k^{9}t^{11}}})={\frac {l_{p}^{15}}{m_{P}^{9}t_{p}^{11}}}}$ (CODATA 2018 mean)

The klt scalars cancel, leaving;

${\displaystyle {\frac {L^{15}}{M^{9}T^{11}}}={\frac {(2\pi ^{2}\Omega ^{2})^{15}}{(1)^{9}(\pi )^{11}}}({\frac {l^{15}}{k^{9}t^{11}}})=2^{15}\pi ^{19}\color {red}(\Omega ^{15})^{2}\color {black}=}$0.109 293... 1024 , ${\displaystyle ({\frac {l^{15}}{k^{9}t^{11}}})=1,\;{\frac {u^{-13*15}}{u^{15*9}u^{-30*11}}}=1}$

Solving for the SI units;

${\displaystyle {\frac {l_{p}^{15}}{m_{P}^{9}t_{p}^{11}}}={\frac {(1.616255e-35)^{15}}{(2.176434e-8)^{9}(5.391247e-44)^{11}}}=}$ 0.109 485... 1024

###### A, lp, tp
a = 0.126 918 588 592... x 1023A
${\displaystyle {\frac {A^{3}L^{3}}{T}}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})^{3}{\frac {(2\pi ^{2}\Omega ^{2})^{3}}{(\pi )}}({\frac {a^{3}l^{3}}{t}})={\frac {2^{24}\pi ^{14}\color {red}(\Omega ^{15})^{1}\color {black}}{\alpha ^{3}}}=}$ 0.205 571... 1013, ${\displaystyle ({\frac {a^{3}l^{3}}{t}})=1,\;{\frac {u^{3*3}u^{-13*3}}{u^{-30}}}=1}$
${\displaystyle {\frac {(e/t_{p})^{3}l_{p}^{3}}{t_{p}}}={\frac {(1.602176634e-19/5.391247e-44)^{3}(1.616255e-35)^{3}}{(5.391247e-44)}}=}$ 0.205 543... 1013, ${\displaystyle units={\frac {(C/s)^{3}m^{3}}{s}}}$

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, ${\displaystyle \mu _{0}}$; CODATA 2014 mean values), we can test to greater precision;

###### c, μ0, R
${\displaystyle {\frac {(c^{*})^{35}}{(\mu _{0}^{*})^{9}(R^{*})^{7}}}=(2\pi \Omega ^{2}v)^{35}/({\frac {\alpha r^{7}}{2^{11}\pi ^{5}\Omega ^{4}}})^{9}.({\frac {v^{5}}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}r^{9}}})^{7}=2^{295}\pi ^{157}3^{21}\alpha ^{26}\color {red}(\Omega ^{15})^{15}\color {black}=}$ 0.326 103 528 6170... 10301, ${\displaystyle {\frac {(u^{17})^{35}}{(u^{56})^{9}(u^{13})^{7}}}=1,\;(v^{35})/(r^{7})^{9}({\frac {v^{5}}{r^{9}}})^{7}=1}$
${\displaystyle {\frac {c^{35}}{\mu _{0}^{9}R^{7}}}={\frac {(299792458)^{35}}{(4\pi /10^{7})^{9}(10973731.568160)^{7}}}=}$ 0.326 103 528 6170... 10301, ${\displaystyle units={\frac {m^{33}A^{18}}{s^{17}kg^{9}}}=={\frac {(u^{-13})^{33}(u^{3})^{18}}{(u^{-30})^{17}(u^{15})^{9}}}=1}$

###### c, e, kB, h
${\displaystyle {\frac {(k_{B}^{*})(e^{*})(c^{*})}{(h^{*})}}=({\frac {\alpha }{2^{5}\pi \Omega }}{\frac {r^{10}}{v^{3}}})({\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }}{\frac {r^{3}}{v^{3}}})(2\pi \Omega ^{2}v)/(2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}})}$ = 1.0, ${\displaystyle {\frac {(u^{29})(u^{-27})(u^{17})}{(u^{19})}}=1,\;({\frac {r^{10}}{v^{3}}})({\frac {r^{3}}{v^{3}}})(v)/({\frac {r^{13}}{v^{5}}})=1}$
${\displaystyle {\frac {k_{B}ec}{h}}=}$ 1.000 8254, ${\displaystyle units={\frac {mC}{s^{2}K}}=={\frac {(u^{-13})(u^{-27})}{(u^{-30})^{2}(u^{20})}}=1}$

###### c, h, e
${\displaystyle {\frac {(h^{*})^{3}}{(e^{*})^{13}(c^{*})^{24}}}=(2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}})^{3}/({\frac {2^{7}\pi ^{4}\Omega ^{3}r^{3}}{\alpha v^{3}}})^{7}.(2\pi \Omega ^{2}v)^{24}={\frac {\alpha ^{13}}{2^{106}\pi ^{64}(\color {red}\Omega ^{15})^{5}\color {black}}}=}$ 0.228 473 759... 10-58, ${\displaystyle {\frac {(u^{19})^{3}}{(u^{-27})^{13}(u^{17})^{24}}}=1,\;({\frac {r^{13}}{v^{5}}})^{3}/({\frac {r^{3}}{v^{3}}})^{13}(v^{24})=1}$
${\displaystyle {\frac {h^{3}}{e^{13}c^{24}}}=}$ 0.228 473 639... 10-58, ${\displaystyle units={\frac {kg^{3}s^{21}}{m^{18}C^{13}}}=={\frac {(u^{15})^{3}(u^{-30})^{21}}{(u^{-13})^{18}(u^{-27})^{13}}}=1}$

###### me, λe
${\displaystyle \sigma _{e}={\frac {3\alpha ^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}}{\frac {r^{3}}{v^{2}}},\;u^{-10}}$
${\displaystyle f_{e}={\frac {\sigma _{e}^{3}}{2T}}=2^{20}3^{3}\pi ^{8}\alpha ^{3}(\color {red}\Omega ^{15})\color {black},\;{\frac {(u^{-10})^{3}}{u^{-30}}}=1,\;({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1}$
${\displaystyle (m_{e}^{*})={\frac {M}{f_{e}}}=\color {blue}9.109\;382\;3227\;10^{-31}\color {black}\;u^{15}}$
${\displaystyle (m_{e}^{*})={\frac {2^{3}\pi ^{5}(h^{*})}{3^{3}\alpha ^{6}(e^{*})^{3}(c^{*})^{5}}}={\frac {1}{2^{20}\pi ^{8}3^{3}\alpha ^{3}(\color {red}\Omega ^{15})\color {black}}}{\frac {r^{4}u^{15}}{v}}=\color {blue}9.109\;382\;3227\;10^{-31}\color {black}\;u^{15}}$
${\displaystyle m_{e}=\color {blue}9.109\;383\;7015...\;10^{-31}\color {black}\;kg}$
${\displaystyle (\lambda _{e}^{*})=2\pi Lf_{e}=\color {purple}2.426\;310\;238\;667\;10^{-12}\color {black}\;u^{-13}}$
${\displaystyle \lambda _{e}={\frac {h}{m_{e}c}}=\color {purple}2.426\;310\;238\;67\;10^{-12}\color {black}\;m}$

###### c, e, me
${\displaystyle (m_{e}^{*})={\frac {M}{f_{e}}},\;f_{e}=2^{20}3^{3}\pi ^{8}\alpha ^{3}(\color {red}\Omega ^{15})^{1}\color {black}}$, units = scalars = 1 (me formula)
${\displaystyle {\frac {(c^{*})^{9}(e^{*})^{4}}{(m_{e}^{*})^{3}}}=2^{97}\pi ^{49}3^{9}\alpha ^{5}(\color {red}\Omega ^{15})^{5}\color {black}=}$ 0.170 514 368... 1092, ${\displaystyle {\frac {(u^{17})^{9}(u^{-27})^{4}}{(u^{15})^{3}}}=1,\;(v^{9})({\frac {r^{3}}{v^{3}}})^{4}/({\frac {r^{4}}{v}})^{3}=1}$
${\displaystyle {\frac {c^{9}e^{4}}{m_{e}^{3}}}=}$ 0.170 514 342... 1092, ${\displaystyle units={\frac {m^{9}C^{4}}{s^{9}kg^{3}}}=={\frac {(u^{-13})^{9}(u^{-27})^{4}}{(u^{-30})^{9}(u^{15})^{3}}}=1}$

###### kB, c, e, me
${\displaystyle {\frac {(k_{B}^{*})}{(e^{*})^{2}(m_{e}^{*})(c^{*})^{4}}}={\frac {3^{3}\alpha ^{6}}{2^{3}\pi ^{5}}}=}$ 73 035 235 897., ${\displaystyle {\frac {(u^{29})}{(u^{-27})^{2}(u^{15})(u^{17})^{4}}}=1,\;({\frac {r^{10}}{v^{3}}})/({\frac {r^{3}}{v^{3}}})^{2}({\frac {r^{4}}{v}})(v)^{4}=1}$
${\displaystyle {\frac {k_{B}}{e^{2}m_{e}c^{4}}}=}$ 73 095 507 858., ${\displaystyle units={\frac {s^{2}}{m^{2}KC^{2}}}=={\frac {(u^{-30})^{2}}{(u^{-13})^{2}(u^{20})(u^{-27})^{2}}}=1}$

###### mP, tp, ε0

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

${\displaystyle {\frac {M^{4}(\epsilon _{0}^{*})}{T}}=(1)({\frac {2^{9}\pi ^{3}}{\alpha }})/(\pi )={\frac {2^{9}\pi ^{2}}{\alpha }}=}$ 36.875, ${\displaystyle {\frac {(u^{15})^{4}(u^{-90})}{(u^{-30})}}=1,\;({\frac {r^{4}}{v}})^{4}({\frac {1}{r^{7}v^{2}}})/({\frac {r^{9}}{v^{6}}})=1}$
${\displaystyle {\frac {m_{p}^{4}(\epsilon _{0})}{t_{p}}}=}$ 36.850, ${\displaystyle units={\frac {kg^{4}}{s}}{\frac {s^{4}A^{2}}{m^{3}kg}}={\frac {kg^{3}A^{2}s^{3}}{m^{3}}}=={\frac {(u^{15})^{3}(u^{3})^{2}(u^{-30})^{3}}{(u^{-13})^{3}}}=1}$

###### G, h, c, e, me, KB
${\displaystyle {\frac {(h^{*})(c^{*})^{2}(e^{*})(m_{e}^{*})}{(G^{*})^{2}(k_{B}^{*})}}=(m_{e}^{*})({\frac {2^{11}\pi ^{3}}{\alpha ^{2}}})=}$ 0.1415... 10-21, ${\displaystyle {\frac {(u^{19})(u^{17})^{2}(u^{-27})(u^{15})}{(u^{6})^{2}(u^{29})}}=1,\;({\frac {r^{13}}{v^{5}}})v^{2}({\frac {r^{3}}{v^{3}}})({\frac {r^{4}}{v^{1}}})/({\frac {r^{5}}{v^{2}}})^{2}({\frac {r^{10}}{v^{3}}})=1}$
${\displaystyle {\frac {hc^{2}em_{e}}{G^{2}k_{B}}}=}$ 0.1413... 10-21, ${\displaystyle units={\frac {kg^{3}s^{3}CK}{m^{4}}}=={\frac {(u^{15})^{3}(u^{-30})^{3}(u^{-27})(u^{20})}{(u^{-13})^{4}}}=1}$

###### α
${\displaystyle {\frac {2(h^{*})}{(\mu _{0}^{*})(e^{*})^{2}(c^{*})}}=2({2^{3}\pi ^{4}\Omega ^{4}})/({\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}})({\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }})^{2}(2\pi \Omega ^{2})=\color {blue}\alpha \color {black},\;{\frac {u^{19}}{u^{56}(u^{-27})^{2}u^{17}}}=1,\;({\frac {r^{13}}{v^{5}}})({\frac {1}{r^{7}}})({\frac {v^{6}}{r^{6}}})({\frac {1}{v}})=1}$

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

##### SI Planck unit scalars
${\displaystyle M=m_{P}=(1)k;\;k=m_{P}=.217\;672\;817\;58...\;10^{-7},\;u^{15}\;(kg)}$
${\displaystyle T=t_{p}={\pi }t;\;t={\frac {t_{p}}{\pi }}=.171\;585\;512\;84...10^{-43},\;u^{-30}\;(s)}$
${\displaystyle L=l_{p}={2\pi ^{2}\Omega ^{2}}l;\;l={\frac {l_{p}}{2\pi ^{2}\Omega ^{2}}}=.203\;220\;869\;48...10^{-36},\;u^{-13}\;(m)}$
${\displaystyle V=c={2\pi \Omega ^{2}}v;\;v={\frac {c}{2\pi \Omega ^{2}}}=11\;843\;707.905...,\;u^{17}\;(m/s)}$
${\displaystyle A=e/t_{p}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})a=.126\;918\;588\;59...10^{23},\;u^{3}\;(A)}$

###### MT to LPVA

In this example LPVA are derived from MT. The formulas for MT;

${\displaystyle M=(1)k,\;unit=u^{15}}$
${\displaystyle T=(\pi )t,\;unit=u^{-30}}$

Replacing scalars pvla with kt

${\displaystyle P=(\Omega )\;{\frac {k^{12/15}}{t^{2/15}}},\;unit=u^{12/15*15-2/15*(-30)=16}}$
${\displaystyle V={\frac {2\pi P^{2}}{M}}=(2\pi \Omega ^{2})\;{\frac {k^{9/15}}{t^{4/15}}},\;unit=u^{9/15*15-4/15*(-30)=17}}$
${\displaystyle L=TV=(2\pi ^{2}\Omega ^{2})\;k^{9/15}t^{11/15},\;unit=u^{9/15*15+11/15*(-30)=-13}}$
${\displaystyle A={\frac {2^{4}V^{3}}{\alpha P^{3}}}=\left({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }}\right)\;{\frac {1}{k^{3/5}t^{2/5}}},\;unit=u^{9/15*(-15)+6/15*30=3}}$

###### PV to MTLA

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

${\displaystyle P=(\Omega )p,\;unit=u^{16}}$
${\displaystyle V=(2\pi \Omega ^{2})v,\;unit=u^{17}}$

Replacing scalars klta with pv

${\displaystyle M={\frac {2\pi P^{2}}{V}}=(1){\frac {p^{2}}{v}},\;unit=u^{16*2-17=15}}$
${\displaystyle T=(\pi ){\frac {p^{9/2}}{v^{6}}},\;unit=u^{16*9/2-17*6=-30}}$
${\displaystyle L=TV=(2\pi ^{2}\Omega ^{2}){\frac {p^{9/2}}{v^{5}}},\;unit=u^{16*9/2-17*5=-13}}$
${\displaystyle A={\frac {2^{4}V^{3}}{\alpha P^{3}}}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }}){\frac {v^{3}}{p^{3}}},\;unit=u^{17*3-16*3=3}}$

##### G, h, e, me, kB

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;

${\displaystyle {(h^{*})}^{3}=(2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}u^{19}}{v^{5}}})^{3}={\frac {3^{19}\pi ^{12}\Omega ^{12}r^{39}u^{57}}{v^{15}}},\;\theta =57}$ ... and ... ${\displaystyle {\frac {2\pi ^{10}{(\mu _{0}^{*})}^{3}}{3^{6}{(c^{*})}^{5}\alpha ^{13}{(R^{*})}^{2}}}={\frac {3^{19}\pi ^{12}\Omega ^{12}r^{39}u^{57}}{v^{15}}},\;\theta =57}$

Table 10. 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 [10] Units
Planck constant ${\displaystyle {(h^{*})}^{3}={\frac {2\pi ^{10}{\mu _{0}}^{3}}{3^{6}{c}^{5}\alpha ^{13}{R}^{2}}}}$ ${\displaystyle {\frac {kg^{3}}{A^{6}s}}}$, θ = 57 h* = 6.626 069 134 e-34, θ = 19 h = 6.626 070 040(81) e-34 ${\displaystyle {\frac {kg\;m^{2}}{s}}}$, θ = 19
Gravitational constant ${\displaystyle {(G^{*})}^{5}={\frac {\pi ^{3}{\mu _{0}}}{2^{20}3^{6}\alpha ^{11}{R}^{2}}}}$ ${\displaystyle {\frac {kg\;m^{3}}{A^{2}s^{2}}}}$, θ = 30 G* = 6.672 497 192 29 e11, θ = 6 G = 6.674 08(31) e-11 ${\displaystyle {\frac {m^{3}}{kg\;s^{2}}}}$, θ = 6
Elementary charge ${\displaystyle {(e^{*})}^{3}={\frac {4\pi ^{5}}{3^{3}{c}^{4}\alpha ^{8}{R}}}}$ ${\displaystyle {\frac {s^{4}}{A^{3}}}}$, θ = -81 e* = 1.602 176 511 30 e-19, θ = -27 e = 1.602 176 620 8(98) e-19 ${\displaystyle As}$, θ = -27
Boltzmann constant ${\displaystyle {(k_{B}^{*})}^{3}={\frac {\pi ^{5}{\mu _{0}}^{3}}{3^{3}2{c}^{4}\alpha ^{5}{R}}}}$ ${\displaystyle {\frac {kg^{3}}{s^{2}A^{6}}}}$, θ = 87 kB* = 1.379 510 147 52 e-23, θ = 29 kB = 1.380 648 52(79) e-23 ${\displaystyle {\frac {kg\;m^{2}}{s^{2}\;K}}}$, θ = 29
Electron mass ${\displaystyle {(m_{e}^{*})}^{3}={\frac {16\pi ^{10}{R}{\mu _{0}}^{3}}{3^{6}{c}^{8}\alpha ^{7}}}}$ ${\displaystyle {\frac {kg^{3}s^{2}}{m^{6}A^{6}}}}$, θ = 45 me* = 9.109 382 312 56 e-31, θ = 15 me = 9.109 383 56(11) e-31 ${\displaystyle kg}$, θ = 15
Gyromagnetic ratio ${\displaystyle ({(\gamma _{e}^{*})/2\pi })^{3}={\frac {g_{e}^{3}3^{3}c^{4}}{2^{8}\pi ^{8}\alpha \mu _{0}^{3}R_{\infty }^{2}}}}$ ${\displaystyle {\frac {m^{3}s^{2}A^{6}}{kg^{3}}}}$, θ = -126 e*/2π) = 28024.953 55, θ = -42 γe/2π = 28024.951 64(17) ${\displaystyle {\frac {A\;s}{kg}}}$, θ = -42
Planck mass ${\displaystyle ({m_{P}^{*}})^{15}={\frac {2^{25}\pi ^{13}{\mu _{0}}^{6}}{3^{6}c^{5}\alpha ^{16}R^{2}}}}$ ${\displaystyle {\frac {kg^{6}m^{3}}{s^{7}A^{12}}}}$, θ = 225 mP* = 0.217 672 817 580 e-7, θ = 15 mP = 0.217 647 0(51) e-7 ${\displaystyle kg}$, θ = 15
Planck length ${\displaystyle ({l_{p}^{*}})^{15}={\frac {\pi ^{22}{\mu _{0}}^{9}}{2^{35}3^{24}\alpha ^{49}c^{35}R^{8}}}}$ ${\displaystyle {\frac {kg^{9}s^{17}}{m^{18}A^{18}}}}$, θ = -195 lp* = 0.161 603 660 096 e-34, θ = -13 lp = 0.161 622 9(38) e-34 ${\displaystyle m}$, θ = -13