# Physical constant (anomaly)

Specific combinations of the 6 dimensioned constants (G, h, c, e, me, kB) suggest a mathematical relationship linking the 5 SI units (kg⇔15, m⇔-13, s⇔-30, A⇔3, K⇔20). This permits to reduce the required number of SI units, and so required dimensioned constants, to 2. Natural Planck units as geometrical objects are implied.

A dimensioned physical constant, sometimes denoted a fundamental physical constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. Common examples being the speed of light c, the gravitational constant G, the Planck constant h and the elementary charge e. These constants are usually measured in terms of SI units mass (kilogram), length (meter), time (second), charge (ampere), temperature (Kelvin) ... (kg, m, s, A, K ...).

These constants form the scaffolding around which the theories of physics are erected, and they define the fabric of our universe, but science has no idea why they take the special numerical values that they do, for these constants follow no discernible pattern. The desire to explain the constants has been one of the driving forces behind efforts to develop a complete unified description of nature, or "theory of everything". Physicists have hoped that such a theory would show that each of the constants of nature could have only one logically possible value. It would reveal an underlying order to the seeming arbitrariness of nature [1].

Notably a physical universe, as opposed to a mathematical universe, has as a fundamental premise the concept that the universe scaffolding (of mass, space and time) exists, that somehow mass is, space is, time is ... and that these units are independent of each other, we cannot measure distance in kilograms and amperes or mass in length and temperature. The 2019 redefinition of SI base units resulted in 4 physical constants (h, c, e, kB) having independently assigned exact values (they cannot be derived in terms of each other), and this confirmed the independence of their associated SI units.

2019 redefinition of SI base units
constant SI units
Speed of light c ${\displaystyle {\frac {m}{s}}}$
Planck constant h ${\displaystyle {\frac {kg\;m^{2}}{s}}}$
Elementary charge e ${\displaystyle C=As}$
Boltzmann constant kB ${\displaystyle {\frac {kg\;m^{2}}{s^{2}\;K}}}$

However there are anomalies which occur in certain combinations of the physical constants (G, h, c, e, me, kB) which suggest a mathematical relationship between the units (kg, m, s, A, K) [2]. In order for the dimensioned physical constants to be fundamental, the units must be independent of each other, there cannot be a unit number relationship ... however these anomalies question this fundamental assumption. Every combination predicted by the model returns an answer consistent with CODATA precision. Statistically therefore, can these anomalies be dismissed as coincidence?

## Anomalies

We can define this relationship between the units by assigning a number θ to each unit. [3].

Table 1. unit relationship
attribute unit number SI equivalent
M (mass) θ = 15 kg
T (time) θ = -30 s
V (velocity) θ = 17 m/s
L (length) θ = -13 m
A (ampere) θ = 3 A
K (temperature) θ = 20 K

Table 2. Physical constants
SI constant MLTA analogue unit number θ
Speed of light c* = V 17
Planck constant ${\displaystyle h^{*}=2\pi MVL}$ 15+17-13=19
Gravitational constant ${\displaystyle G^{*}={\frac {V^{2}L}{M}}}$ 34-13-15=6
Elementary charge ${\displaystyle e^{*}=AT}$ 3-30=-27
Boltzmann constant ${\displaystyle k_{B}^{*}={\frac {2\pi VM}{A}}}$ 17+15-3=29
Vacuum permeability ${\displaystyle \mu _{0}^{*}={\frac {4\pi V^{2}M}{\alpha LA^{2}}}}$ 34+15+13-6=56

### Planck units

The Planck units are measures of the SI units; Planck mass in kg, Planck length in m, Planck time in s ... and so they are analogues to the MLTA attributes in the above table. The SI Planck units have numerical values, however for there to be a mathematical relation between these SI units we cannot use numerical values, this is because numerical values are simply dimensionless frequencies of the SI unit, 299792458 could refer to the speed of light 299792458m/s or equally the number of apples in a container, the number 299792458 itself carries no unit-specific information.

This can be resolved by assigning to each Planck unit a geometrical object, and for which the geometry embeds the attribute (for example, the geometry of the time object embeds the function time and so a descriptive s is not required). We may then combine these objects to form more complex objects; from electrons to galaxies, while still retaining the underlying attributes (of mass, wavelength, frequency …). As this particular geometrical approach requires that the objects be interrelated (they are not independent of each other), a unit number relationship hypothesis can be tested. This is because, if there are natural Planck objects, then these objects will be embedded within our dimensioned SI physical constants.

Table 3. is an example of object orientated units; it assigns MLTA objects as the geometry of 2 dimensionless physical constants; the Sommerfeld fine structure constant alpha and Omega. As alpha and Omega are dimensionless, so too are these objects.

Table 3. MLTVA Geometrical objects
attribute geometrical object
mass ${\displaystyle M=(1)}$
time ${\displaystyle T=(\pi )}$
velocity ${\displaystyle V=(2\pi \Omega ^{2})}$
length ${\displaystyle L=(2\pi ^{2}\Omega ^{2})}$
ampere ${\displaystyle A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})}$

Alpha and Omega can have numerical values (alpha = 137.035999139, Omega = 2.0071349496), and so the MLTVA objects can have numerical values (i.e.: V = 2πΩ2 = 25.3123819), and so we can use dimensioned numerical scalars to convert from the MLTVA objects to SI equivalents.

For example, we can use scalar v to convert from dimensionless geometrical object V to dimensioned c.

scalar vSI = 11843707.905 m/s gives c = V*vSI = 25.3123819 * 11843707.905 m/s = 299792458 m/s (SI units)
scalar vimp = 7359.3232155 miles/s gives c = V*vimp = 186282 miles/s (imperial units)

As scalar v carries the unit designation m/s (v = 11843707.905 m/s), scalar v is dimensioned, and so we can assign a unit number to v (θ = 17).

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

As the scalars include dimensioned units, they will also follow this unit number relationship (included in formulas as unit = uθ), consequently we need only 2 scalars to define the others, for example if we know the numerical values for a and l then we know the numerical value for t (a3l3/t = 1 and so a3l3 = t), and from l and t we know k … and so we can solve the SI Planck units (α and Ω have fixed values), and from these, we can solve (G, h, c, e, me, kB).

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

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; from c is v (θ = 17), and from μ0 is r (θ = 8).

Table 5. 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 6. Comparison θ; SI units and scalars
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

### Calculating from (α, Ω)

If we can reduce the 5 SI units to 2 scalars (example; r, v in tables 5, 6), 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 [4][5]. For example

${\displaystyle {\frac {(2\pi MVL)^{3}}{(AT)^{13}(V)^{24}}}={\frac {\alpha ^{13}}{2^{106}\pi ^{64}(\color {red}\Omega ^{15})^{5}\color {black}}}=}$ 0.228 473 759... 10-58
${\displaystyle {\frac {(h^{*})^{3}}{(e^{*})^{13}(c^{*})^{24}}}=(2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}})^{3}/({\frac {2^{7}\pi ^{4}\Omega ^{3}r^{3}}{\alpha v^{3}}})^{7}.(2\pi \Omega ^{2}v)^{24}={\frac {\alpha ^{13}}{2^{106}\pi ^{64}(\color {red}\Omega ^{15})^{5}\color {black}}}}$
${\displaystyle {\frac {h^{3}}{e^{13}c^{24}}}=}$ 0.228 473 639... 10-58

Here we solve physical constant combinations using only α, Ω (and the mathematical constants 2, 3, π). As the scalars (v, r) have cancelled, we do not need to know their values or the 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 7. 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 ${\displaystyle {\frac {(u^{19})(u^{17})^{2}(u^{-27})(u^{15})}{(u^{6})^{2}(u^{29})}}=1}$ ${\displaystyle ({\frac {r^{13}}{v^{5}}})v^{2}({\frac {r^{3}}{v^{3}}})({\frac {r^{4}}{v^{1}}})/({\frac {r^{5}}{v^{2}}})^{2}({\frac {r^{10}}{v^{3}}})=1}$

### Calculating from (α, Ω, v, r)

Using α, Ω and the CODATA 2014 SI values/units for scalars v (θ=17), r (θ=8).

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

Table 8. Dimensioned constants (α, Ω, v, r)
constant geometrical object calculated (α, Ω, r, v) CODATA 2014 [6]
Planck constant ${\displaystyle h^{*}=2\pi MVL=2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}}}$ 6.626 069 134 e-34, u19 6.626 070 040(81) e-34
Gravitational constant ${\displaystyle G^{*}={\frac {V^{2}L}{M}}=2^{3}\pi ^{4}\Omega ^{6}{\frac {r^{5}}{v^{2}}}}$ 6.672 497 192 29 e11, u6 6.674 08(31) e-11
Elementary charge ${\displaystyle e^{*}=AT={\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }}{\frac {r^{3}}{v^{3}}}}$ 1.602 176 511 30 e-19, u-27 1.602 176 620 8(98) e-19
Boltzmann constant ${\displaystyle k_{B}^{*}={\frac {2\pi VM}{A}}={\frac {\alpha }{2^{5}\pi \Omega }}{\frac {r^{10}}{v^{3}}}}$ 1.379 510 147 52 e-23, u29 1.380 648 52(79) e-23
Vacuum permeability ${\displaystyle \mu _{0}^{*}={\frac {4\pi V^{2}M}{\alpha LA^{2}}}={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7}}$ 4π/10^7, u56 4π/10^7 (exact)

### Calculating from (α)

Combinations which reduce to the dimensionless (no scalars or units) fine structure constant. For example;

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

Table 9. fine structure constant
CODATA 2014 geometrical (α)
${\displaystyle {\frac {2h}{\mu _{0}e^{2}c}}=\color {red}\alpha \color {black}=}$ 137.035 999 139(31) ${\displaystyle {\frac {2(h^{*})}{(\mu _{0}^{*})(e^{*})^{2}(c^{*})}}=\color {red}\alpha \color {black}}$
${\displaystyle {\sqrt {\frac {2^{11}\pi ^{3}G^{2}k_{B}}{hc^{2}em_{p}}}}=}$ 137.133 167 47 ${\displaystyle {\sqrt {\frac {2^{11}\pi ^{3}(G^{*})^{2}(k_{B}^{*})}{(h^{*})(c^{*})^{2}(e^{*})(m_{p}^{*})}}}=\color {red}\alpha \color {black}}$
${\displaystyle ({\frac {2^{3}\pi ^{5}k_{B}}{3^{3}e^{2}m_{e}c^{4}}})^{1/6}=}$ 137.054 833 44 ${\displaystyle ({\frac {2^{3}\pi ^{5}(k_{B}^{*})}{3^{3}(e^{*})^{2}(m_{e}^{*})(c^{*})^{4}}})^{1/6}=\color {red}\alpha \color {black}}$
${\displaystyle {\frac {2^{9}\pi ^{2}t_{p}}{m_{P}^{4}\epsilon _{0}}}=}$ 137.119 576 89 ${\displaystyle {\frac {2^{9}\pi ^{2}(t_{p})^{*}}{(m_{P}^{*})^{4}(\epsilon _{0}^{*})}}=\color {red}\alpha \color {black}}$

### Calculating the electron

The electron is an example of a combination that is dimensionless (units = scalars = 1). The electron function (the mathematical formula for the electron) fe can be defined, for example, in terms of ALT where AL as an ampere-meter (ampere-length = e*c) are the units for a magnetic monopole.

${\displaystyle T=\pi {\frac {r^{9}}{v^{6}}},\;u^{-30}}$
${\displaystyle \sigma _{e}={\frac {3\alpha ^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}}{\frac {r^{3}}{v^{2}}},\;u^{-10}}$
${\displaystyle f_{e}={\frac {\sigma _{e}^{3}}{2T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }},\;unit={\frac {(u^{-10})^{3}}{u^{-30}}}=1,scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1}$
${\displaystyle f_{e}=4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23},\;unit=1}$ (unit-less)

#### Electron parameters

Associated with the electron are dimensioned parameters, these parameters however are a function of the base MLTVA units, the formula fe dictating the frequency of these units. By setting MLTVA 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

### Calculating from (c, R, μ0, α)

By matching the unit numbers we can numerically solve the least precise dimensioned physical constants (G, h, e, me, kB ...) using the 3 most precise (CODATA 2014); speed of light c (exact value), vacuum permeability μ0 (exact value), Rydberg constant R (12-13 digits) and the dimensionless fine structure constant alpha.

R = 10973731.568508 (θ=13)

c = 299792458 (θ=17)

μ0 = 4π/107 (θ=56)

α = 137.035999139 (θ=0)

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. R, c, μ0, α ... (CODATA 2014 mean)
constant formula* calculated θ CODATA 2014 [7] Units
Planck constant ${\displaystyle {(h^{*})}^{3}={\frac {2\pi ^{10}{\mu _{0}}^{3}}{3^{6}{c}^{5}\alpha ^{13}{R}^{2}}}}$ h* = 6.626 069 134 e-34 ${\displaystyle {\frac {kg^{3}}{A^{6}s}}}$, 15*3-3*6+30 = 57 h = 6.626 070 040(81) e-34 ${\displaystyle {\frac {kg\;m^{2}}{s}}}$, θ = 15-13*2+30 = 19
Gravitational constant ${\displaystyle {(G^{*})}^{5}={\frac {\pi ^{3}{\mu _{0}}}{2^{20}3^{6}\alpha ^{11}{R}^{2}}}}$ G* = 6.672 497 192 29 e11 ${\displaystyle {\frac {kg\;m^{3}}{A^{2}s^{2}}}}$, 15-13*3-3*2+30*2 = 30 G = 6.674 08(31) e-11 ${\displaystyle {\frac {m^{3}}{kg\;s^{2}}}}$, θ = -13*3-15+30*2 = 6
Elementary charge ${\displaystyle {(e^{*})}^{3}={\frac {4\pi ^{5}}{3^{3}{c}^{4}\alpha ^{8}{R}}}}$ e* = 1.602 176 511 30 e-19 ${\displaystyle {\frac {s^{3}}{m^{3}}}}$, -30*4+13*3 = -81 e = 1.602 176 620 8(98) e-19 ${\displaystyle As}$, θ = 3-30 = -27
Boltzmann constant ${\displaystyle {(k_{B}^{*})}^{3}={\frac {\pi ^{5}{\mu _{0}}^{3}}{3^{3}2{c}^{4}\alpha ^{5}{R}}}}$ kB* = 1.379 510 147 52 e-23 ${\displaystyle {\frac {kg^{3}}{s^{2}A^{6}}}}$, 15*3+30*2-3*6 = 87 kB = 1.380 648 52(79) e-23 ${\displaystyle {\frac {kg\;m^{2}}{s^{2}\;K}}}$, θ = 15-26+60-20 = 29
Electron mass ${\displaystyle {(m_{e}^{*})}^{3}={\frac {16\pi ^{10}{R}{\mu _{0}}^{3}}{3^{6}{c}^{8}\alpha ^{7}}}}$ me* = 9.109 382 312 56 e-31, u = 15 ${\displaystyle {\frac {kg^{3}s^{2}}{m^{6}A^{6}}}}$, 15*3-30*2+13*6-3*6 = 45 me = 9.109 383 56(11) e-31 ${\displaystyle kg}$, θ = 15
${\displaystyle {\frac {2^{3}\pi ^{5}({k_{B}}^{*})}{3^{3}\alpha ^{6}({e^{*}})^{2}({m_{e}}^{*})c^{4}}}}$ 1.0 ${\displaystyle {\frac {1}{m^{2}A^{2}K}}}$, 13*2-3*2-20 = 0 5kB/(27α6e2mec4) = 1.000 825 ${\displaystyle {\frac {1}{m^{2}A^{2}K}}}$, θ = 0
Gyromagnetic ratio ${\displaystyle ({(\gamma _{e}^{*})/2\pi })^{3}={\frac {g_{e}^{3}3^{3}c^{4}}{2^{8}\pi ^{8}\alpha \mu _{0}^{3}R_{\infty }^{2}}}}$ e*/2π) = 28024.953 55 ${\displaystyle {\frac {m^{3}s^{2}A^{6}}{kg^{3}}}}$, -13*3-30*2+3*6-15*3 = -126 γe/2π = 28024.951 64(17) ${\displaystyle {\frac {A\;s}{kg}}}$, θ = -42
Planck length ${\displaystyle ({l_{p}^{*}})^{15}={\frac {\pi ^{22}{\mu _{0}}^{9}}{2^{35}3^{24}\alpha ^{49}c^{35}R^{8}}}}$ lp* = 0.161 603 660 096 e-34 ${\displaystyle {\frac {kg^{9}s^{17}}{m^{18}A^{18}}}}$, 15*9-30*17+13*18-3*18 = -195 lp = 0.161 622 9(38) e-34 ${\displaystyle m}$, θ = -13
Planck mass ${\displaystyle ({m_{P}^{*}})^{15}={\frac {2^{25}\pi ^{13}{\mu _{0}}^{6}}{3^{6}c^{5}\alpha ^{16}R^{2}}}}$ mP* = 0.217 672 817 580 e-7 u = ${\displaystyle {\frac {kg^{6}m^{3}}{s^{7}A^{12}}}}$, 15*6-13*3+30*7-3*12 = 225 mP = 0.217 647 0(51) e-7 ${\displaystyle kg}$, θ = 15
${\displaystyle {\frac {2^{11}\pi ^{3}(G^{*})^{2}(k_{B}^{*})}{\alpha ^{2}(h^{*})(c^{*})^{2}(e^{*})(m_{P}^{*})}}}$ 1.0 ${\displaystyle {\frac {m^{4}}{kg^{3}s^{4}A\;K}}}$, -13*4-15*3+30*4-3-20 = 0 211π3G2kB/(α2hc2emP) = 1.001 418 ${\displaystyle {\frac {m^{4}}{kg^{3}s^{4}A\;K}}}$, θ = 0

### Table of constants (i, xθ, y)

Constants in ascending order (θ) in terms of these 3 geometries

-x as the base unit (θ = 1, scalars > 1)

${\displaystyle x=\Omega {\frac {v}{r^{2}}}}$ , units = ${\displaystyle {\sqrt {\frac {L}{MT}}}}$ (θ = -13 -15 +30 = 2/2 = 1)

-y (θ = 0, scalars > 1) defines the numerical limit to the constants (depending on the system of units used)

${\displaystyle y={\frac {r^{17}}{v^{8}}}}$ , units = ${\displaystyle M^{2}T}$ (θ = 15*2 -30 = 0)

-i as a dimensionless constant (θ = 0, scalars = 0)

${\displaystyle i=\Omega ^{15}}$, units = ${\displaystyle {\sqrt {\frac {L^{15}}{M^{9}T^{11}}}}}$ (θ = -13*15 -15*9 +11*30 = 0, scalars have cancelled)

Table 11. 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