The UnitSystem

Physical unit system constants (Metric, English, Natural, etc...) PDF 2020-2025

In 2019 the SI2019 standardization was completed, based on the 7 physics dimensions specific to the Metric system. That is actually an inadequate and insufficient unit system standard, as it is mathematically impossible to unify all historical units with that standard. In 2020, Michael Reed set out to work around that impossibility with a new project called UnitSystems.jl, which ended up completely solving the problem with a brand new 11 dimensional Unified System of Quantities (USQ) for physics.

By default, UnitSystems provides a modern unified re-interpretation of various historical unit systems which were previously incompatible. In order to make each UnitSystem consistently compatible with each other, a few convenience assumptions are made. Specifically, it is assumed that all default modern unit systems share the same common Universe of dimensionless constants, although this can be optionally changed. Therefore, the philosophy is to characterize differences among UnitSystem instances by means of dimensional constants. As a result, all the defaults are ideal modern variants of these historical unit systems based on a common underlying Universe, which are completely consistent and compatible with each other. These default UnitSystem values are to be taken as a newly defined mutually-compatible recommended standard, verified to be consistent and coherent.

Metric SI Unit Systems

In the Systeme International d'Unites (the SI units) the UnitSystem constants are derived from the most accurate possible physical measurements and a few exactly defined constants. Exact values are the avogadro number, boltzmann constant, planck constant, lightspeed definition, and elementary charge definition.

Construction of UnitSystem instances based on specifying the the constants molarmass, the vacuumpermeability, and the molargas along with some other options is facilitated by MetricSystem. This construction helps characterize the differences between

MeasureSystems.Metric — Constant 
Metric = MetricSystem(milli,𝟐*τ/𝟏𝟎^7)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Standard Metric system based on exact molarmass and vacuumpermeability.

julia> boltzmann(Metric) # J⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³ = 1.38064899953(43) × 10⁻²³ [J⋅K⁻¹] Metric

julia> planckreduced(Metric) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹ = 1.0545718176461565×10⁻³⁴ [J⋅s] Metric

julia> lightspeed(Metric) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] Metric

julia> vacuumpermeability(Metric) # H⋅m⁻¹
τ⋅2⁻⁶5⁻⁷ = 1.2566370614359173×10⁻⁶ [H⋅m⁻¹] Metric

julia> electronmass(Metric) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] Metric

julia> molarmass(Metric) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] Metric

julia> luminousefficacy(Metric) # lm⋅W⁻¹
Kcd = 683.01969009009 [lm⋅W⁻¹] Metric
MeasureSystems.SI2019 — Constant 
SI2019 = MetricSystem(Mᵤ,μ₀)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Systeme International d'Unites based on approximate molarmass and vacuumpermeability.

julia> boltzmann(SI2019) # J⋅K⁻¹
kB = 1.380649×10⁻²³ [J⋅K⁻¹] SI2019

julia> planckreduced(SI2019) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹ = 1.0545718176461565×10⁻³⁴ [J⋅s] SI2019

julia> lightspeed(SI2019) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] SI2019

julia> vacuumpermeability(SI2019) # H⋅m⁻¹
𝘩⋅𝘤⁻¹𝘦⁻²α⋅2 = 1.25663706212(19) × 10⁻⁶ [H⋅m⁻¹] SI2019

julia> electronmass(SI2019) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] SI2019

julia> molarmass(SI2019) # kg⋅mol⁻¹
NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2 = 0.00099999999966(31) [kg⋅mol⁻¹] SI2019

julia> luminousefficacy(SI2019) # lm⋅W⁻¹
Kcd = 683.01969009009 [lm⋅W⁻¹] SI2019
MeasureSystems.SI1976 — Constant 
SI1976 = MetricSystem(milli,𝟐*τ/𝟏𝟎^7,8.31432)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Reference UnitSystem with universal gas constant of 8.31432 from 1976 standard atmosphere.

julia> boltzmann(SI1976) # J⋅K⁻¹
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³⋅8.31432 = 1.38062531722(43) × 10⁻²³ [kg⋅m²s⁻²K⁻¹] SI1976

julia> planckreduced(SI1976) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹ = 1.0545718176461565×10⁻³⁴ [kg⋅m²s⁻¹] SI1976

julia> lightspeed(SI1976) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] SI1976

julia> vacuumpermeability(SI1976) # H⋅m⁻¹
τ⋅2⁻⁶5⁻⁷ = 1.2566370614359173×10⁻⁶ [kg⋅m⋅C⁻²] SI1976

julia> electronmass(SI1976) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] SI1976

julia> molarmass(SI1976) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] SI1976

julia> luminousefficacy(SI1976) # lm⋅W⁻¹
Kcd = 683.01969009009 [kg⁻¹m⁻²s³lm] SI1976
MeasureSystems.Engineering — Constant 
Engineering = MetricSystem(milli,𝟐*τ/𝟏𝟎^7,Rᵤ,g₀)
F=F, M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=A, R=𝟙, C=𝟙

Standard Metric Engineering system based on kilogram and kilopond (kilogram-force) units.

julia> boltzmann(Engineering) # kgf⋅m⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹2⁴5³ = 1.40787016925(43) × 10⁻²⁴ [kgf⋅m⋅K⁻¹] Engineering

julia> planckreduced(Engineering) # kgf⋅m⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹τ⁻¹ = 1.0753639802033891×10⁻³⁵ [kgf⋅m⋅s⋅rad⁻¹] Engineering

julia> lightspeed(Engineering) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] Engineering

julia> vacuumpermeability(Engineering) # kgf⋅s²⋅C⁻²
g₀⁻¹τ⋅2⁻⁶5⁻⁷ = 1.2814131853751459×10⁻⁷ [kgf⋅s²C⁻²] Engineering

julia> electronmass(Engineering) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] Engineering

julia> molarmass(Engineering) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] Engineering

julia> luminousefficacy(Engineering) # lm⋅s⋅m⁻¹⋅kgf⁻¹
Kcd⋅g₀ = 6698.135043821981 [kgf⁻¹m⁻¹s⋅lm] Engineering

julia> gravity(Engineering) # kg⋅m⋅kgf⁻¹⋅s⁻²
g₀ = 9.80665 [kgf⁻¹kg⋅m⋅s⁻²] Engineering

Additional Metric variants with angle scaling include MetricTurn, MetricSpatian, MetricGradian, MetricDegree, MetricArcminute, MetricArcsecond.

Historically, the josephson and klitzing constants have been used to define Conventional and CODATA variants.

MeasureSystems.Conventional — Constant 
Conventional = ConventionalSystem(RK1990,KJ1990)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Conventional electronic UnitSystem with 1990 tuned josephson and klitzing constants.

julia> josephson(Conventional) # Hz⋅V⁻¹
KJ90 = 4.835979×10¹⁴ [Hz⋅V⁻¹] Conventional

julia> klitzing(Conventional) # Ω
RK90 = 25812.807 [Ω] Conventional

julia> boltzmann(Conventional) # J⋅K⁻¹
kB⋅NA⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹RK90⁻¹KJ90⁻²2⁶5³ = 1.38064872956(43) × 10⁻²³ [J⋅K⁻¹] Conventional

julia> planckreduced(Conventional) # J⋅s⋅rad⁻¹
RK90⁻¹KJ90⁻²τ⁻¹2² = 1.0545716114388567×10⁻³⁴ [J⋅s] Conventional

julia> lightspeed(Conventional) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] Conventional

julia> vacuumpermeability(Conventional) # H⋅m⁻¹
𝘤⁻¹α⋅RK90⋅2 = 1.25663703976(19) × 10⁻⁶ [H⋅m⁻¹] Conventional

julia> electronmass(Conventional) # kg
𝘤⁻¹R∞⋅α⁻²RK90⁻¹KJ90⁻²2³ = 9.1093819203(28) × 10⁻³¹ [kg] Conventional

julia> molarmass(Conventional) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] Conventional

julia> luminousefficacy(Conventional) # lm⋅W⁻¹
𝘩⋅Kcd⋅RK90⋅KJ90²2⁻² = 683.0198236454071 [lm⋅W⁻¹] Conventional
MeasureSystems.CODATA — Constant 
CODATA = ConventionalSystem(RK2014,KJ2014,Rᵤ2014)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Reference UnitSystem based on Committee on Data of the International Science Council.

julia> josephson(CODATA) # Hz⋅V⁻¹
KJ = 4.835978525(30) × 10¹⁴ [Hz⋅V⁻¹] CODATA

julia> klitzing(CODATA) # Ω
RK = 25812.8074555(59) [Ω] CODATA

julia> boltzmann(CODATA) # J⋅K⁻¹
𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹RK⁻¹KJ⁻²Rᵤ2014⋅2⁶5³ = 1.38064851(80) × 10⁻²³ [J⋅K⁻¹] CODATA

julia> planckreduced(CODATA) # J⋅s⋅rad⁻¹
RK⁻¹KJ⁻²τ⁻¹2² = 1.054571800(13) × 10⁻³⁴ [J⋅s] CODATA

julia> lightspeed(CODATA) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] CODATA

julia> vacuumpermeability(CODATA) # H⋅m⁻¹
𝘤⁻¹α⋅RK⋅2 = 1.25663706194(35) × 10⁻⁶ [H⋅m⁻¹] CODATA

julia> electronmass(CODATA) # kg
𝘤⁻¹R∞⋅α⁻²RK⁻¹KJ⁻²2³ = 9.10938355(11) × 10⁻³¹ [kg] CODATA

julia> molarmass(CODATA) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] CODATA

julia> luminousefficacy(CODATA) # lm⋅W⁻¹
𝘩⋅Kcd⋅RK⋅KJ²2⁻² = 683.0197015(85) [lm⋅W⁻¹] CODATA

Originally, the practical units where specified by resistance and electricpotential.

MeasureSystems.International — Constant 
International = ElectricSystem(Metric,Ωᵢₜ = 1.000495,Vᵢₜ = 1.00033)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

International UnitSystem with United States measurements of Ωᵢₜ and Vᵢₜ.

julia> resistance(International,Metric) # Ω⋅Ω⁻¹
Ωᵢₜ = 1.000495 [kg⋅m⋅s⁻²C⁻²]/[kg⋅m⋅s⁻²C⁻²] International -> Metric

julia> electricpotential(International,Metric) # V⋅V⁻¹
Vᵢₜ = 1.00033 [V⋅m⁻¹]/[V⋅m⁻¹] International -> Metric

julia> boltzmann(International) # J⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹Ωᵢₜ⋅Vᵢₜ⁻²2⁴5³ = 1.38042119247(42) × 10⁻²³ [J⋅K⁻¹] International

julia> planckreduced(International) # J⋅s⋅rad⁻¹
𝘩⋅Ωᵢₜ⋅Vᵢₜ⁻²τ⁻¹ = 1.0543978133151816×10⁻³⁴ [J⋅s] International

julia> lightspeed(International) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] International

julia> vacuumpermeability(International) # H⋅m⁻¹
Ωᵢₜ⁻¹τ⋅2⁻⁶5⁻⁷ = 1.2560153338456637×10⁻⁶ [H⋅m⁻¹] International

julia> electronmass(International) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²Ωᵢₜ⋅Vᵢₜ⁻²2 = 9.1078806534(28) × 10⁻³¹ [kg] International

julia> molarmass(International) # kg⋅mol⁻¹
Ωᵢₜ⋅Vᵢₜ⁻²2⁻³5⁻³ = 0.0009998350000179567 [kg⋅mol⁻¹] International

julia> luminousefficacy(International) # lm⋅W⁻¹
Kcd⋅Ωᵢₜ⁻¹Vᵢₜ² = 683.1324069249656 [lm⋅W⁻¹] International
MeasureSystems.InternationalMean — Constant 
InternationalMean = ElectricSystem(Metric,1.00049,1.00034)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

International UnitSystem with mean measurements of Ωᵢₜ and Vᵢₜ.

julia> resistance(InternationalMean,Metric) # Ω⋅Ω⁻¹
1.00049 = 1.00049 [kg⋅m⋅s⁻²C⁻²]/[kg⋅m⋅s⁻²C⁻²] InternationalMean -> Metric

julia> electricpotential(InternationalMean,Metric) # V⋅V⁻¹
1.00034 = 1.00034 [V⋅m⁻¹]/[V⋅m⁻¹] InternationalMean -> Metric

julia> boltzmann(InternationalMean) # J⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³/1.0001900224889804 = 1.38038669501(42) × 10⁻²³ [J⋅K⁻¹] InternationalMean

julia> planckreduced(InternationalMean) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹/1.0001900224889804 = 1.0543714633563797×10⁻³⁴ [J⋅s] InternationalMean

julia> lightspeed(InternationalMean) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] InternationalMean

julia> vacuumpermeability(InternationalMean) # H⋅m⁻¹
τ⋅2⁻⁶5⁻⁷/1.00049 = 1.2560216108466022×10⁻⁶ [H⋅m⁻¹] InternationalMean

julia> electronmass(InternationalMean) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2/1.0001900224889804 = 9.1076530427(28) × 10⁻³¹ [kg] InternationalMean

julia> molarmass(InternationalMean) # kg⋅mol⁻¹
2⁻³5⁻³/1.0001900224889804 = 0.0009998100136127059 [kg⋅mol⁻¹] InternationalMean

julia> luminousefficacy(International) # lm⋅W⁻¹
Kcd⋅Ωᵢₜ⁻¹Vᵢₜ² = 683.1324069249656 [lm⋅W⁻¹] International

Electromagnetic CGS Systems

Alternatives to the SI unit system are the centimetre-gram-second variants, where the constants are rescaled with centi*meter and milli kilogram units along with introduction of additional rationalization and lorentz constants or electromagnetic units.

MeasureSystems.EMU — Constant 
EMU = GaussSystem(Metric,𝟏,𝟐*τ)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L¹ᐟ², Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Centimetre-gram-second UnitSystem variant based on EMU (non-rationalized).

julia> boltzmann(EMU) # erg⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2¹¹5¹⁰ = 1.38064899953(43) × 10⁻¹⁶ [erg⋅K⁻¹] EMU

julia> planckreduced(EMU) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] EMU

julia> lightspeed(EMU) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] EMU

julia> vacuumpermeability(EMU) # abH⋅cm⁻¹
𝟏 = 1.0 [𝟙] EMU

julia> electronmass(EMU) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] EMU

julia> molarmass(EMU) # g⋅mol⁻¹
𝟏 = 1.0 [g⋅mol⁻¹] EMU

julia> luminousefficacy(EMU) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.830196900900899×10⁻⁵ [lm⋅s⋅erg⁻¹] EMU

julia> rationalization(EMU)
τ⋅2 = 12.566370614359172 [𝟙] EMU
MeasureSystems.ESU — Constant 
ESU = GaussSystem(Metric,(𝟏𝟎*𝘤)^-2,𝟐*τ)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L³ᐟ²T⁻¹, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Centimetre-gram-second UnitSystem variant based on ESU (non-rationalized).

julia> boltzmann(ESU) # erg⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2¹¹5¹⁰ = 1.38064899953(43) × 10⁻¹⁶ [erg⋅K⁻¹] ESU

julia> planckreduced(ESU) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] ESU

julia> lightspeed(ESU) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] ESU

julia> vacuumpermeability(ESU) # statH⋅cm⁻¹
𝘤⁻²2⁻⁴5⁻⁴ = 1.1126500560536184×10⁻²¹ [cm⁻²s²] ESU

julia> electronmass(ESU) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] ESU

julia> molarmass(ESU) # g⋅mol⁻¹
𝟏 = 1.0 [g⋅mol⁻¹] ESU

julia> luminousefficacy(ESU) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.830196900900899×10⁻⁵ [lm⋅s⋅erg⁻¹] ESU

julia> rationalization(ESU)
τ⋅2 = 12.566370614359172 [𝟙] ESU
MeasureSystems.Gauss — Constant 
Gauss = GaussSystem(Metric,𝟏,𝟐*τ,𝟏𝟎^-2/𝘤)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L³ᐟ²T⁻¹, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=LT⁻¹

Centimetre-gram-second UnitSystem variant CGS (Gauss-Lorentz, non-rationalized).

julia> boltzmann(Gauss) # erg⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2¹¹5¹⁰ = 1.38064899953(43) × 10⁻¹⁶ [erg⋅K⁻¹] Gauss

julia> planckreduced(Gauss) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] Gauss

julia> lightspeed(Gauss) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] Gauss

julia> vacuumpermeability(Gauss) # statH⋅cm⁻¹
𝟏 = 1.0 [𝟙] Gauss

julia> electronmass(Gauss) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] Gauss

julia> molarmass(Gauss) # g⋅mol⁻¹
𝟏 = 1.0 [g⋅mol⁻¹] Gauss

julia> luminousefficacy(Gauss) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.830196900900899×10⁻⁵ [lm⋅s⋅erg⁻¹] Gauss

julia> rationalization(Gauss)
τ⋅2 = 12.566370614359172 [𝟙] Gauss

julia> lorentz(Gauss)
𝘤⁻¹2⁻²5⁻² = 3.335640951981521×10⁻¹¹ [cm⁻¹s] Gauss
MeasureSystems.LorentzHeaviside — Constant 
LorentzHeaviside = GaussSystem(Metric,𝟏,𝟏,centi/𝘤)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L³ᐟ²T⁻¹, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=LT⁻¹

Centimetre-gram-second UnitSystem variant HLU (Heaviside-Lorentz, rationalized).

julia> boltzmann(LorentzHeaviside) # erg⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2¹¹5¹⁰ = 1.38064899953(43) × 10⁻¹⁶ [erg⋅K⁻¹] LorentzHeaviside

julia> planckreduced(LorentzHeaviside) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] LorentzHeaviside

julia> lightspeed(LorentzHeaviside) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] LorentzHeaviside

julia> vacuumpermeability(HLU) # hlH⋅cm⁻¹
𝟏 = 1.0 [𝟙] LorentzHeaviside

julia> electronmass(LorentzHeaviside) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] LorentzHeaviside

julia> molarmass(LorentzHeaviside) # g⋅mol⁻¹
𝟏 = 1.0 [g⋅mol⁻¹] LorentzHeaviside

julia> luminousefficacy(LorentzHeaviside) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.830196900900899×10⁻⁵ [lm⋅s⋅erg⁻¹] LorentzHeaviside

julia> rationalization(LorentzHeaviside)
𝟏 = 1.0 [𝟙] LorentzHeaviside

julia> lorentz(LorentzHeaviside)
𝘤⁻¹2⁻²5⁻² = 3.335640951981521×10⁻¹¹ [cm⁻¹s] LorentzHeaviside

There are multiple choices of elctromagnetic units for these variants based on electromagnetic units, electrostatic units, Gaussian non-rationalized units, and Lorentz-Heaviside rationalized units. Note that CGS is an alias for the Gauss system.

Modified (Entropy) Unit Systems

Most other un-natural unit systems are derived from the construction above by rescaling time, length, mass, temperature, and gravity; which results in modified entropy constants:

MeasureSystems.Gravitational — Constant 
Gravitational = EntropySystem(Metric,𝟏,𝟏,g₀)
F=F, M=FL⁻¹T², L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Standard Gravitational system based on hyl and kilopond units.

julia> boltzmann(Gravitational) # kgf⋅m⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹2⁴5³ = 1.40787016925(43) × 10⁻²⁴ [kgf⋅m⋅K⁻¹] Gravitational

julia> planckreduced(Gravitational) # kgf⋅m⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹τ⁻¹ = 1.0753639802033891×10⁻³⁵ [kgf⋅m⋅s] Gravitational

julia> lightspeed(Gravitational) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] Gravitational

julia> vacuumpermeability(Gravitational) # H⋅m⁻¹
g₀⁻¹τ⋅2⁻⁶5⁻⁷ = 1.2814131853751459×10⁻⁷ [kgf⋅s²C⁻²] Gravitational

julia> electronmass(Gravitational) # hyl
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀⁻¹2 = 9.2889862507(28) × 10⁻³² [hyl] Gravitational

julia> molarmass(Gravitational) # hyl⋅mol⁻¹
g₀⁻¹2⁻³5⁻³ = 0.00010197162129779284 [hyl⋅mol⁻¹] Gravitational

julia> luminousefficacy(Gravitational) # lm⋅s⋅m⁻¹⋅kgf⁻¹
Kcd⋅g₀ = 6698.135043821981 [kgf⁻¹m⁻¹s⋅lm] Gravitational
MeasureSystems.MTS — Constant 
MTS = EntropySystem(SI2019,𝟏,𝟏,kilo)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Metre-tonne-second UnitSystem variant of Metric system.

julia> boltzmann(MTS) # kJ⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2 = 1.38064899953(43) × 10⁻²⁶ [t⋅m²s⁻²K⁻¹] MTS

julia> planckreduced(MTS) # kJ⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁻³5⁻³ = 1.0545718176461566×10⁻³⁷ [t⋅m²s⁻¹] MTS

julia> lightspeed(MTS) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] MTS

julia> vacuumpermeability(MTS) # kH⋅m⁻¹
τ⋅2⁻⁹5⁻¹⁰ = 1.2566370614359174×10⁻⁹ [t⋅m⋅C⁻²] MTS

julia> electronmass(MTS) # t
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁻²5⁻³ = 9.1093837016(28) × 10⁻³⁴ [t] MTS

julia> molarmass(MTS) # t⋅mol⁻¹
2⁻⁶5⁻⁶ = 1.0×10⁻⁶ [t⋅mol⁻¹] MTS

julia> luminousefficacy(MTS) # lm⋅kW⁻¹
Kcd⋅2³5³ = 683019.6900900899 [t⁻¹m⁻²s³lm] MTS
MeasureSystems.KKH — Constant 
KKH = EntropySystem(Metric,HOUR,kilo,𝟏)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Kilometer-kilogram-hour UnitSystem variant of Metric system.

julia> boltzmann(KKH) # kg⋅km²⋅h⁻²⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁶3⁴5 = 1.78932110338(55) × 10⁻²² [kg⋅km²h⁻²K⁻¹] KKH

julia> planckreduced(KKH) # kg⋅km²⋅h⁻¹
𝘩⋅τ⁻¹2⁻²3²5⁻⁴ = 3.7964585435261634×10⁻³⁷ [kg⋅km²h⁻¹] KKH

julia> lightspeed(KKH) # km⋅hr⁻¹
𝘤⋅2⋅3²5⁻¹ = 1.0792528488×10⁹ [km⋅h⁻¹] KKH

julia> vacuumpermeability(KKH) # kg⋅km⋅C⁻²
τ⋅2⁻⁹5⁻¹⁰ = 1.2566370614359174×10⁻⁹ [kg⋅km⋅C⁻²] KKH

julia> electronmass(KKH) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] KKH

julia> molarmass(KKH) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] KKH

julia> luminousefficacy(KKH) # lm⋅h³⋅kg⁻¹⋅km⁻²
Kcd⋅2⁻⁶3⁻⁶ = 0.014639482383618183 [kg⁻¹km⁻²h³lm] KKH
MeasureSystems.MPH — Constant 
MPH = EntropySystem(FPS,HOUR,mi,𝟏)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Mile-pound-hour specification based on FPS absolute UnitSystem.

julia> boltzmann(MPH) # lbf⋅mi²⋅hr⁻²⋅F⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹ft⁻²lb⁻¹2²5⁶11⁻² = 8.4615956484(26) × 10⁻²³ [lb⋅mi²h⁻²°R⁻¹] MPH

julia> planckreduced(MPH) # lbf⋅mi²⋅hr⁻¹⋅rad⁻¹
𝘩⋅ft⁻²lb⁻¹τ⁻¹2⁻⁶11⁻² = 3.2315817800735083×10⁻³⁷ [lb⋅mi²h⁻¹] MPH

julia> lightspeed(MPH) # mi⋅hr⁻¹
𝘤⋅ft⁻¹2⁻¹3⋅5⋅11⁻¹ = 6.706166293843951×10⁸ [mi⋅h⁻¹] MPH

julia> vacuumpermeability(MPH) # lbm⋅mi⋅C⁻²
ft⁻¹lb⁻¹τ⋅2⁻¹¹3⁻¹5⁻⁸11⁻¹ = 1.7214532710813804×10⁻⁹ [lb⋅mi⋅C⁻²] MPH

julia> electronmass(MPH) # lbm
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹2 = 2.00827533796(62) × 10⁻³⁰ [lb] MPH

julia> molarmass(MPH) # lbm⋅lb-mol⁻¹
𝟏 = 1.0 [lb⋅lb-mol⁻¹] MPH

julia> luminousefficacy(MPH) # lm⋅h³⋅lb⁻¹⋅mi⁻²
Kcd⋅ft²lb⋅2⁻²3⁻⁴5⁻⁴11² = 0.017198446999173198 [lb⁻¹mi⁻²h³lm] MPH
MeasureSystems.Nautical — Constant 
Nautical = EntropySystem(Metric,HOUR,nm,em^3,𝟏,τ*𝟑^3/𝟐^10/𝟓^12,milli)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Nautical miles, kilo-earthgram, hour specification based on Meridian definition.

julia> greatcircle(Nautical) # nm
2⁵3³5² = 21600.0 [nm] Nautical

julia> boltzmann(Nautical) # keg⋅nm²⋅hr⁻²⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁵2⁴⁹3¹⁰5³² = 5.180046618(26) × 10⁻²³ [keg⋅nm²h⁻²K⁻¹] Nautical

julia> planckreduced(Nautical) # keg⋅nm²⋅hr⁻¹⋅rad⁻¹
𝘩⋅g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁶2⁴¹3⁸5²⁷ = 1.0990666907(55) × 10⁻³⁷ [keg⋅nm²h⁻¹] Nautical

julia> lightspeed(Nautical) # nm⋅hr⁻¹
𝘤⋅g₀¹ᐟ²GME⁻¹ᐟ²τ⁻¹2⁹3⁵5⁴ = 5.8195383759(58) × 10⁸ [nm⋅h⁻¹] Nautical

julia> vacuumpermeability(Nautical) # keg⋅nm⋅eC⁻²
τ⋅2⁻¹⁰3³5⁻¹² = 6.785840131753953×10⁻¹⁰ [keg⋅nm⋅eC⁻²] Nautical

julia> electronmass(Nautical) # keg
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀³ᐟ²GME⁻³ᐟ²τ⁻³2²⁸5²¹ = 9.069925385(27) × 10⁻³¹ [keg] Nautical

julia> molarmass(Nautical) # keg⋅eg-mol⁻¹
2⁻³5⁻³ = 0.001 [keg⋅eg-mol⁻¹] Nautical

julia> luminousefficacy(Nautical) # lm⋅h³⋅keg⁻¹⋅nm⁻²
Kcd⋅g₀⁻⁵ᐟ²GME⁵ᐟ²τ⁵2⁻⁴⁹3⁻¹²5⁻³¹ = 0.05056853095(25) [keg⁻¹nm⁻²h³lm] Nautical
MeasureSystems.Meridian — Constant 
Meridian = EntropySystem(Metric,𝟏,em,em^3,𝟏,τ/𝟐^6/𝟓^7,milli)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Modern ideal Meridian system defined by France's original earthmeter definition.

julia> greatcircle(Meridian) # em
2⁹5⁷ = 4.0×10⁷ [em] Meridian

julia> boltzmann(Meridian) # eJ⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁵2⁴⁹5³⁸ = 1.3706960050(69) × 10⁻²³ [eJ⋅K⁻¹] Meridian

julia> planckreduced(Meridian) # eJ⋅s⋅rad⁻¹
𝘩⋅g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁶2⁴⁵5³⁵ = 1.0469694890(53) × 10⁻³⁴ [eJ⋅s] Meridian

julia> lightspeed(Meridian) # em⋅s⁻¹
𝘤⋅g₀¹ᐟ²GME⁻¹ᐟ²τ⁻¹2⁹5⁷ = 2.9935896996(3) × 10⁸ [em⋅s⁻¹] Meridian

julia> vacuumpermeability(Meridian) # kegf⋅s²⋅eC⁻²
τ⋅2⁻⁶5⁻⁷ = 1.2566370614359173×10⁻⁶ [eH⋅em⁻¹] Meridian

julia> electronmass(Meridian) # keg
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀³ᐟ²GME⁻³ᐟ²τ⁻³2²⁸5²¹ = 9.069925385(27) × 10⁻³¹ [keg] Meridian

julia> molarmass(Meridian) # keg⋅eg-mol⁻¹
2⁻³5⁻³ = 0.001 [keg⋅eg-mol⁻¹] Meridian

julia> luminousefficacy(Meridian) # lm⋅W⁻¹
Kcd⋅g₀⁻⁵ᐟ²GME⁵ᐟ²τ⁵2⁻⁴⁵5⁻³⁵ = 687.9792808(35) [lm⋅eW⁻¹] Meridian

Foot-Pound-Second-Rankine

In Britain and the United States an English system of engineering units was commonly used.

MeasureSystems.FPS — Constant 
FPS = RankineSystem(Metric,ft,lb)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Absolute English UnitSystem based on the foot, pound, second, and poundal.

julia> boltzmann(FPS) # ft⋅pdl⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹ft⁻²lb⁻¹2⁴3⁻²5⁴ = 1.82018324169(56) × 10⁻²² [lb⋅ft²s⁻²°R⁻¹] FPS

julia> planckreduced(FPS) # ft⋅pdl⋅s⋅rad⁻¹
𝘩⋅ft⁻²lb⁻¹τ⁻¹ = 2.5025369304889247×10⁻³³ [lb⋅ft²s⁻¹] FPS

julia> lightspeed(FPS) # ft⋅s⁻¹
𝘤⋅ft⁻¹ = 9.835710564304461×10⁸ [ft⋅s⁻¹] FPS

julia> vacuumpermeability(FPS) # lb⋅ft⋅C⁻²
ft⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 9.089273271309687×10⁻⁶ [lb⋅ft⋅C⁻²] FPS

julia> electronmass(FPS) # lb
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹2 = 2.00827533796(62) × 10⁻³⁰ [lb] FPS

julia> molarmass(FPS) # lb⋅lb-mol⁻¹
𝟏 = 1.0 [lb⋅lb-mol⁻¹] FPS

julia> luminousefficacy(FPS) # lm⋅s³⋅lb⁻¹⋅ft⁻²
Kcd⋅ft²lb = 28.78252493663283 [lb⁻¹ft⁻²s³lm] FPS
MeasureSystems.IPS — Constant 
IPS = RankineSystem(Metric,ft/𝟐^2/𝟑,lb*g₀*𝟐^2*𝟑/ft)
F=F, M=FL⁻¹T², L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

British Gravitational UnitSystem historically used in the United States of America.

julia> boltzmann(IPS) # in⋅lb⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ft⁻¹lb⁻¹2⁶3⁻¹5⁴ = 6.7887629566(21) × 10⁻²³ [lb⋅in⋅°R⁻¹] IPS

julia> planckreduced(IPS) # in⋅lb⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ft⁻¹lb⁻¹τ⁻¹2²3 = 9.333747076683978×10⁻³⁴ [lb⋅in⋅s] IPS

julia> lightspeed(IPS) # in⋅s⁻¹
𝘤⋅ft⁻¹2²3 = 1.1802852677165354×10¹⁰ [in⋅s⁻¹] IPS

julia> vacuumpermeability(IPS) # slinch⋅in⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.8250324964133447×10⁻⁷ [lb⋅s²C⁻²] IPS

julia> electronmass(IPS) # slinch
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀⁻¹ft⋅lb⁻¹2⁻¹3⁻¹ = 5.2015921425(16) × 10⁻³³ [slinch] IPS

julia> molarmass(IPS) # slinch⋅slinch-mol⁻¹
𝟏 = 1.0 [slinch-slinch-mol⁻¹] IPS

julia> luminousefficacy(IPS) # lm⋅s⋅in⁻¹⋅lb⁻¹
Kcd⋅g₀⋅ft⋅lb⋅2⁻²3⁻¹ = 77.17086290732456 [lb⁻¹in⁻¹s⋅lm] IPS
MeasureSystems.British — Constant 
British = RankineSystem(Metric,ft,slug)
F=F, M=FL⁻¹T², L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

British Gravitational UnitSystem historically used by Britain and United States.

julia> boltzmann(British) # ft⋅lb⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ft⁻¹lb⁻¹2⁴3⁻²5⁴ = 5.6573024638(17) × 10⁻²⁴ [lb⋅ft⋅°R⁻¹] British

julia> planckreduced(British) # ft⋅lb⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ft⁻¹lb⁻¹τ⁻¹ = 7.778122563903315×10⁻³⁵ [lb⋅ft⋅s] British

julia> lightspeed(British) # ft⋅s⁻¹
𝘤⋅ft⁻¹ = 9.835710564304461×10⁸ [ft⋅s⁻¹] British

julia> vacuumpermeability(British) # slug⋅ft⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.8250324964133447×10⁻⁷ [lb⋅s²C⁻²] British

julia> electronmass(British) # slugs
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀⁻¹ft⋅lb⁻¹2 = 6.2419105710(19) × 10⁻³² [slug] British

julia> molarmass(British) # slug⋅slug-mol⁻¹
𝟏 = 1.0 [slug⋅slug-mol⁻¹] British

julia> luminousefficacy(British) # lm⋅s⋅ft⁻¹⋅lb⁻¹
Kcd⋅g₀⋅ft⋅lb = 926.0503548878947 [lb⁻¹ft⁻¹s⋅lm] British
MeasureSystems.English — Constant 
English = RankineSystem(Metric,ft,lb,g₀/ft)
F=F, M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=A, R=𝟙, C=𝟙

English Engineering UnitSystem historically used in the United States of America.

julia> boltzmann(English) # ft⋅lbf⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ft⁻¹lb⁻¹2⁴3⁻²5⁴ = 5.6573024638(17) × 10⁻²⁴ [lbf⋅ft⋅°R⁻¹] English

julia> planckreduced(English) # ft⋅lbf⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ft⁻¹lb⁻¹τ⁻¹ = 7.778122563903315×10⁻³⁵ [lbf⋅ft⋅s⋅rad⁻¹] English

julia> lightspeed(English) # ft⋅s⁻¹
𝘤⋅ft⁻¹ = 9.835710564304461×10⁸ [ft⋅s⁻¹] English

julia> vacuumpermeability(English) # lbm⋅ft⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.8250324964133447×10⁻⁷ [lbf⋅s²C⁻²] English

julia> electronmass(English) # lbm
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹2 = 2.00827533796(62) × 10⁻³⁰ [lbm] English

julia> molarmass(English) # lbm⋅lb-mol⁻¹
𝟏 = 1.0 [lbm⋅lb-mol⁻¹] English

julia> luminousefficacy(English) # lm⋅s⋅ft⁻¹⋅lbf⁻¹
Kcd⋅g₀⋅ft⋅lb = 926.0503548878947 [lbf⁻¹ft⁻¹s⋅lm] English

julia> gravity(English) # lbm⋅ft⋅lbf⁻¹⋅s⁻²
g₀⋅ft⁻¹ = 32.17404855643044 [lbf⁻¹lbm⋅ft⋅s⁻²] English
MeasureSystems.Survey — Constant 
Survey = RankineSystem(Metric,ftUS,lb,g₀/ftUS)
F=F, M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=A, R=𝟙, C=𝟙

English Engineering UnitSystem based on the geophysical US survey foot (1200/3937).

julia> boltzmann(Survey) # ftUS⋅lbf⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ftUS⁻¹lb⁻¹2⁴3⁻²5⁴ = 5.6572911492(17) × 10⁻²⁴ [lbf⋅ft⋅°R⁻¹] Survey

julia> planckreduced(Survey) # ftUS⋅lbf⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ftUS⁻¹lb⁻¹τ⁻¹ = 7.77810700765819×10⁻³⁵ [lbf⋅ft⋅s⋅rad⁻¹] Survey

julia> lightspeed(Survey) # ftUS⋅s⁻¹
𝘤⋅ftUS⁻¹ = 9.835690892883334×10⁸ [ft⋅s⁻¹] Survey

julia> vacuumpermeability(Survey) # lbm⋅ftUS⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.8250324964133447×10⁻⁷ [lbf⋅s²C⁻²] Survey

julia> electronmass(Survey) # lbm
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹2 = 2.00827533796(62) × 10⁻³⁰ [lbm] Survey

julia> molarmass(Survey) # lbm⋅lb-mol⁻¹
𝟏 = 1.0 [lbm⋅lb-mol⁻¹] Survey

julia> luminousefficacy(Survey) # lm⋅s⋅ft⁻¹⋅lbf⁻¹
Kcd⋅g₀⋅ftUS⋅lb = 926.0522069923087 [lbf⁻¹ft⁻¹s⋅lm] Survey

julia> gravity(Survey) # lbm⋅ftUS⋅lbf⁻¹⋅s⁻²
g₀⋅ftUS⁻¹ = 32.17398420833334 [lbf⁻¹lbm⋅ft⋅s⁻²] Survey
MeasureSystems.FFF — Constant 
FFF = EntropySystem(Metric,𝟐*𝟕*DAY,fur,𝟐*𝟑^2*𝟓*lb,°R,0,𝟏)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Furlong–firkin–fortnight FFF is a humorous UnitSystem based on unusal impractical units.

julia> boltzmann(FFF) # fir⋅fur²⋅ftn⁻²⋅F⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹ft⁻²lb⁻¹2¹⁵5⁵7²11⁻² = 6.7931043720(21) × 10⁻¹⁸ [fir⋅fur²ftn⁻²°R⁻¹] FFF

julia> planckreduced(FFF) # fir⋅fur²⋅ftn⁻¹⋅rad⁻¹
𝘩⋅ft⁻²lb⁻¹τ⁻¹2³3⁻¹5⁻¹7⋅11⁻² = 7.721326066522302×10⁻³⁵ [fir⋅fur²ftn⁻¹] FFF

julia> lightspeed(FFF) # fur⋅ftn⁻¹
𝘤⋅ft⁻¹2⁶3²5⋅7⋅11⁻¹ = 1.8026174997852542×10¹² [fur⋅ftn⁻¹] FFF

julia> vacuumpermeability(FFF) # fir⋅fur⋅Inf⁻²
𝟏/Inf = 0.0 [fir⋅fur⋅Inf⁻²] FFF

julia> electronmass(FFF) # fir
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹3⁻²5⁻¹ = 2.23141704217(68) × 10⁻³² [fir] FFF

julia> molarmass(FFF) # fir⋅fir-mol⁻¹
𝟏 = 1.0 [fir⋅fir-mol⁻¹] FFF

julia> luminousefficacy(FFF) # lm⋅ftn³⋅fir⁻¹⋅fur⁻²
Kcd⋅ft²lb⋅2⁻¹⁹3⁻⁵5⁻³7⁻³11² = 6.375788993269436×10⁻¹⁰ [fir⁻¹fur⁻²ftn³lm] FFF

Astronomical Unit Systems

The International Astronomical Union (IAU) units are based on the solar mass, distance from the sun to the earth, and the length of a terrestrial day.

MeasureSystems.IAU — Constant 
IAU☉ = EntropySystem(Metric,DAY,au,GM☉/G)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Solar UnitSystem defined by International Astronomical Union and solarmass.

julia> boltzmann(IAU) # M⊙⋅au²⋅D⁻²⋅K⁻¹
kB⋅NA⋅𝘩²R∞⋅α⁻²μₑᵤ⁻¹au⁻⁵kG⁻²mP⁻²τ⁻³2⁴⁶3²⁰5¹⁷ = 2.316083(51) × 10⁻⁶⁶ [M☉⋅au²D⁻²K⁻¹] IAU☉

julia> planckreduced(IAU) # M⊙⋅au²⋅D⁻¹⋅rad⁻¹
𝘩²𝘤⋅au⁻⁵kG⁻²mP⁻²τ⁻⁴2³⁵3¹⁷5¹² = 2.047544(45) × 10⁻⁸² [M☉⋅au²D⁻¹] IAU☉

julia> lightspeed(IAU) # au⋅D⁻¹
𝘤⋅au⁻¹2⁷3³5² = 173.1446326742(35) [au⋅D⁻¹] IAU☉

julia> vacuumpermeability(IAU) # M⊙⋅au²⋅C⁻²
𝘩⋅𝘤⋅au⁻⁴kG⁻²mP⁻²τ⁻²2²²3¹⁴5³ = 4.224533(93) × 10⁻⁴⁸ [M☉⋅au⋅C⁻²] IAU☉

julia> electronmass(IAU) # M⊙
𝘩²R∞⋅α⁻²au⁻³kG⁻²mP⁻²τ⁻³2²⁹3¹⁴5¹⁰ = 4.58124(10) × 10⁻⁶¹ [M☉] IAU☉

julia> molarmass(IAU) # M☉⋅mol⁻¹
𝘩⋅𝘤⋅au⁻³kG⁻²mP⁻²τ⁻³2²⁵3¹⁴5⁷ = 5.02915(11) × 10⁻³⁴ [M☉⋅mol⁻¹] IAU☉

julia> luminousefficacy(IAU) # lm⋅D³⋅M☉⁻¹⋅au⁻²
𝘩⁻¹𝘤⁻¹Kcd⋅au⁵kG²mP²τ³2⁻⁴⁹3⁻²³5⁻¹⁶ = 4.71247(10) × 10⁴⁰ [M☉⁻¹au⁻²D³lm] IAU☉

julia> gaussgravitation(IAU) # D⁻¹
kG⋅τ⋅2⁻⁷3⁻⁴5⁻³ = 0.017202098964713464 [D⁻¹] IAU☉
MeasureSystems.IAUE — Constant 
IAUE = EntropySystem(Metric,DAY,LD,GME/G)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Astronomical (Earth) UnitSystem defined by lunardistance around the earthmass.

julia> boltzmann(IAUE) # ME⋅LD²⋅D⁻²⋅K⁻¹
kB⋅NA⋅𝘩²R∞⋅α⁻²μₑᵤ⁻¹mP⁻²GME⁻¹τ⁻¹2¹²5/202692169 = 1.167923(26) × 10⁻⁵⁵ [ME⋅LD²D⁻²K⁻¹] IAUE

julia> planckreduced(IAUE) # ME⋅LD²⋅D⁻¹⋅rad⁻¹
𝘩²𝘤⋅mP⁻²GME⁻¹τ⁻²2⋅3⁻³5⁻⁴/202692169 = 1.032508(23) × 10⁻⁷¹ [ME⋅LD²D⁻¹] IAUE

julia> lightspeed(IAUE) # LD⋅D⁻¹
𝘤⋅2⁴5⁻¹/14237 = 67383.2876027253 [LD⋅D⁻¹] IAUE

julia> vacuumpermeability(IAUE) # ME⋅LD²⋅C⁻²
𝘩⋅𝘤⋅mP⁻²GME⁻¹2⁻⁹3⁻³5⁻¹⁰/14237 = 5.47389(12) × 10⁻⁴⁰ [ME⋅LD⋅C⁻²] IAUE

julia> electronmass(IAUE) # ME
𝘩²R∞⋅α⁻²mP⁻²GME⁻¹τ⁻¹2 = 1.525306(34) × 10⁻⁵⁵ [ME] IAUE

julia> molarmass(IAUE) # ME⋅mol⁻¹
𝘩⋅𝘤⋅mP⁻²GME⁻¹τ⁻¹2⁻³5⁻³ = 1.674434(37) × 10⁻²⁸ [ME⋅mol⁻¹] IAUE

julia> luminousefficacy(IAUE) # lm⋅D³⋅ME⁻¹⋅LD⁻²
𝘩⁻¹𝘤⁻¹Kcd⋅mP²GME⋅τ⋅2⁻¹⁵3⁻³⋅202692169 = 9.34520(21) × 10²⁹ [ME⁻¹LD⁻²D³lm] IAUE

julia> turn(IAU)/gaussianmonth(IAU) # D⁻¹
GME¹ᐟ²2⁵ᐟ²3⁻³ᐟ²5⁻⁵ᐟ²/1.6987431854323947×10⁶ = 0.22888074402(23) [D⁻¹] IAU☉
MeasureSystems.IAUJ — Constant 
IAUJ = EntropySystem(Metric,DAY,JD,GMJ/G)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Astronomical (Jupiter) UnitSystem defined by jupiterdistance around the solarmass.

julia> boltzmann(IAUJ) # MJ⋅JD²⋅D⁻²⋅K⁻¹
kB⋅NA⋅𝘩²R∞⋅α⁻²μₑᵤ⁻¹mP⁻²GMJ⁻¹τ⁻¹2⁶3⁴5⁻⁵/67336617049 = 8.95968(20) × 10⁻⁶⁵ [MJ⋅JD²D⁻²K⁻¹] IAUJ

julia> planckreduced(IAUJ) # MJ⋅JD²⋅D⁻¹⋅rad⁻¹
𝘩²𝘤⋅mP⁻²GMJ⁻¹τ⁻²2⁻⁵3⋅5⁻¹⁰/67336617049 = 7.92084(17) × 10⁻⁸¹ [MJ⋅JD²D⁻¹] IAUJ

julia> lightspeed(IAUJ) # JD⋅D⁻¹
𝘤⋅2⋅3²5⁻⁴/259493 = 33.272661653300865 [JD⋅D⁻¹] IAUJ

julia> vacuumpermeability(IAUJ) # MJ⋅JD²⋅C⁻²
𝘩⋅𝘤⋅mP⁻²GMJ⁻¹2⁻¹²3⁻¹5⁻¹³/259493 = 8.50430(19) × 10⁻⁴⁶ [MJ⋅JD⋅C⁻²] IAUJ

julia> electronmass(IAUJ) # MJ
𝘩²R∞⋅α⁻²mP⁻²GMJ⁻¹τ⁻¹2 = 4.79915(11) × 10⁻⁵⁸ [MJ] IAUJ

julia> molarmass(IAUJ) # MJ⋅mol⁻¹
𝘩⋅𝘤⋅mP⁻²GMJ⁻¹τ⁻¹2⁻³5⁻³ = 5.26836(12) × 10⁻³¹ [MJ⋅mol⁻¹] IAUJ

julia> luminousefficacy(IAUJ) # lm⋅D³⋅MJ⁻¹⋅JD⁻²
𝘩⁻¹𝘤⁻¹Kcd⋅mP²GMJ⋅τ⋅2⁻⁹3⁻⁷5⁶⋅67336617049 = 1.218177(27) × 10³⁹ [MJ⁻¹JD⁻²D³lm] IAUJ

julia> sqrt(gravitation(IAUJ)*solarmass(IAUJ)/jupiterdistance(IAUJ)^3) # D⁻¹
au³ᐟ²kG⋅τ⋅2⁻¹⁶3⁻¹¹ᐟ²5⁻¹²/1.3218691602384917×10⁸ = 0.001449102839405(44) [D⁻¹] IAUJ
MeasureSystems.Hubble — Constant 
Hubble = AstronomicalSystem(Metric,th,𝘤*th,mₑ)
F=T⁻¹, M=𝟙, L=T, T, Q, Θ=𝟙, N=𝟙, J=T⁻¹, A=𝟙, R=𝟙, C=𝟙

Hubble UnitSystem defined by hubble parameter.

julia> boltzmann(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> planckreduced(Hubble)
𝘤⁻¹R∞⁻¹α²H0⋅au⁻¹2⁻¹¹3⁻⁴5⁻⁶ = 2.824(18) × 10⁻³⁹ [T] Hubble

julia> lightspeed(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> vacuumpermeability(Hubble)
τ⋅2 = 12.566370614359172 [TQ⁻²] Hubble

julia> electronmass(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> molarmass(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> luminousefficacy(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> hubble(Hubble)
𝟏 = 1.0 [T⁻¹] Hubble

julia> cosmological(Hubble)
ΩΛ⋅3 = 2.067(17) [T⁻²] Hubble
MeasureSystems.Cosmological — Constant 
Cosmological = AstronomicalSystem(Metric,lc/𝘤,lc,mc)
F=MT⁻¹, M, L=T, T, Q, Θ=M, N=M, J, A=𝟙, R=𝟙, C=𝟙

Cosmological scale UnitSystem defined by darkenergydensity.

julia> boltzmann(Cosmological)
𝟏 = 1.0 [𝟙] Cosmological

julia> planckreduced(Cosmological)
𝘩²𝘤⁻⁴ΩΛ⋅H0²au⁻²mP⁻²2⁻²⁰3⁻⁷5⁻¹² = 2.888(43) × 10⁻¹²² [MT] Cosmological

julia> lightspeed(Cosmological)
𝟏 = 1.0 [𝟙] Cosmological

julia> vacuumpermeability(Cosmological)
τ⋅2 = 12.566370614359172 [MTQ⁻²] Cosmological

julia> electronmass(Cosmological)
𝘩²𝘤⁻³R∞⋅α⁻²ΩΛ¹ᐟ²H0⋅au⁻¹mP⁻²τ¹ᐟ²2⁻⁸3⁻⁷ᐟ²5⁻⁶ = 3.566(26) × 10⁻⁸³ [M] Cosmological

julia> molarmass(Cosmological)
𝟏 = 1.0 [𝟙] Cosmological

julia> luminousefficacy(Cosmological)
𝟏 = 1.0 [M⁻¹TJ] Cosmological

julia> hubble(Cosmological)
ΩΛ⁻¹ᐟ²τ¹ᐟ²2⋅3⁻¹ᐟ² = 3.487(14) [T⁻¹] Cosmological

julia> cosmological(Cosmological)
τ⋅2² = 25.132741228718345 [T⁻²] Cosmological
MeasureSystems.CosmologicalQuantum — Constant 
CosmologicalQuantum = AstronomicalSystem(Metric,tcq,lcq,mcq)
F=M², M, L=M⁻¹, T=M⁻¹, Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Cosmological quantum scale UnitSystem defined by darkenergydensity.

julia> boltzmann(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> planckreduced(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> lightspeed(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> vacuumpermeability(CosmologicalQuantum)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁼²] CosmologicalQuantum

julia> electronmass(CosmologicalQuantum)
𝘩¹ᐟ²R∞⋅α⁻²ΩΛ⁻¹ᐟ⁴H0⁻¹ᐟ²au¹ᐟ²mP⁻¹ᐟ²τ¹ᐟ⁴2¹³ᐟ²3⁷ᐟ⁴5³ = 2.2733(84) × 10⁸ [M] CosmologicalQuantum

julia> molarmass(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> luminousefficacy(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

Natural Unit Systems

With the introduction of the planckmass a set of natural atomic unit systems can be derived in terms of the gravitational coupling constant.

\[\alpha_G = \left(\frac{m_e}{m_P}\right)^2, \qquad \tilde k_B = 1, \qquad (\tilde M_u = 1, \quad \tilde \lambda = 1, \quad \tilde\alpha_L = 1)\]

julia> αG # (mₑ/mP)^2
𝘩²𝘤⁻²mP⁻²R∞²α⁻⁴2² = 1.75181e-45 ± 3.9e-50

Some of the notable variants include

Planck       ::UnitSystem{1,1,1,1,√(4π*αG)}
PlanckGauss  ::UnitSystem{1,1,1,4π,√αG}
Stoney       ::UnitSystem{1,1/α,1,4π,√(αG/α)}
Hartree      ::UnitSystem{1,1,1/α,4π*α^2,1}
Rydberg      ::UnitSystem{1,1,2/,π*α^2,1/2}
Schrodinger  ::UnitSystem{1,1,1/α,4π*α^2,√(αG/α)}
Electronic   ::UnitSystem{1,1/α,1,4π,1}
Natural      ::UnitSystem{1,1,1,1,1}
NaturalGauss ::UnitSystem{1,1,1,4π,1}
QCD          ::UnitSystem{1,1,1,1,1/μₚₑ}
QCDGauss     ::UnitSystem{1,1,1,4π,1/μₚₑ}
QCDoriginal  ::UnitSystem{1,1,1,4π*α,1/μₚₑ}

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 1, \qquad \widetilde{m_e} = \sqrt{4\pi\alpha_G}\]

MeasureSystems.Planck — Constant 
Planck = UnitSystem(𝟏,𝟏,𝟏,𝟏,√(𝟐*τ*αG))
F=M², M, L=M⁻¹, T=M⁻¹, Q=𝟙, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Planck UnitSystem with the electronmass value √(4π*αG) using gravitational coupling.

julia> boltzmann(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> planckreduced(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> lightspeed(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> vacuumpermeability(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> electronmass(Planck)
𝘩⋅𝘤⁻¹R∞⋅α⁻²mP⁻¹τ¹ᐟ²2³ᐟ² = 1.483708(16) × 10⁻²² [M] Planck

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 4\pi, \qquad \widetilde{m_e} = \sqrt{\alpha_G}\]

MeasureSystems.PlanckGauss — Constant 
PlanckGauss = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ,√αG)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Planck (Gauss) UnitSystem with permeability of 4π and electronmass coupling √αG.

julia> boltzmann(PlanckGauss)
𝟏 = 1.0 [𝟙] PlanckGauss

julia> planckreduced(PlanckGauss)
𝟏 = 1.0 [𝟙] PlanckGauss

julia> lightspeed(PlanckGauss)
𝟏 = 1.0 [𝟙] PlanckGauss

julia> vacuumpermeability(PlanckGauss)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁻²] PlanckGauss

julia> electronmass(PlanckGauss)
𝘩⋅𝘤⁻¹R∞⋅α⁻²mP⁻¹2 = 4.185463(46) × 10⁻²³ [mP] PlanckGauss

The well known PlanckGauss values for length, time, mass, and temperature are:

julia> length(PlanckGauss,SI2019) # ℓP
𝘩⋅𝘤⁻¹mP⁻¹τ⁻¹ = 1.616255(18) × 10⁻³⁵ [m]/[mP⁻¹] PlanckGauss -> SI2019

julia> time(PlanckGauss,SI2019) # tP
𝘩⋅𝘤⁻²mP⁻¹τ⁻¹ = 5.391247(59) × 10⁻⁴⁴ [s]/[mP⁻¹] PlanckGauss -> SI2019

julia> mass(PlanckGauss,SI2019) # mP
mP = 2.176434(24) × 10⁻⁸ [kg]/[mP] PlanckGauss -> SI2019

julia> temperature(PlanckGauss,SI2019) # TP
kB⁻¹𝘤²mP = 1.416784(16) × 10³² [K]/[mP] PlanckGauss -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = \frac{1}{\alpha}, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 4\pi, \qquad \widetilde{m_e} = \sqrt{\frac{\alpha_G}{\alpha}}\]

MeasureSystems.Stoney — Constant 
Stoney = UnitSystem(𝟏,𝟏/α,𝟏,𝟐*τ,√(αG/α))
F=MT⁻¹, M, L=T, T, Q, Θ=M, N=M, J, A=𝟙, R=𝟙, C=𝟙

Stoney UnitSystem with permeability of 4π and electronmass coupling √(αG/α).

julia> boltzmann(Stoney)
𝟏 = 1.0 [𝟙] Stoney

julia> planckreduced(Stoney)
α⁻¹ = 137.035999084(21) [MT] Stoney

julia> lightspeed(Stoney)
𝟏 = 1.0 [𝟙] Stoney

julia> vacuumpermeability(Stoney)
τ⋅2 = 12.566370614359172 [MTQ⁻²] Stoney

julia> electronmass(Stoney)
𝘩⋅𝘤⁻¹R∞⋅α⁻⁵ᐟ²mP⁻¹2 = 4.899602(54) × 10⁻²² [M] Stoney

The well known Stoney values for length, time, mass, and charge are:

julia> length(Stoney,SI2019) # lS
𝘩⋅𝘤⁻¹α¹ᐟ²mP⁻¹τ⁻¹ = 1.380679(15) × 10⁻³⁶ [m]/[T] Stoney -> SI2019

julia> time(Stoney,SI2019) # tS
𝘩⋅𝘤⁻²α¹ᐟ²mP⁻¹τ⁻¹ = 4.605448(51) × 10⁻⁴⁵ [s]/[T] Stoney -> SI2019

julia> mass(Stoney,SI2019) # mS
α¹ᐟ²mP = 1.859209(21) × 10⁻⁹ [kg]/[M] Stoney -> SI2019

julia> charge(Stoney,SI2019) # qS
𝘦 = 1.602176634×10⁻¹⁹ [C]/[𝘦] Stoney -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = \frac{1}{\alpha}, \qquad \widetilde{\mu_0} = 4\pi\alpha^2, \qquad \widetilde{m_e} = 1\]

MeasureSystems.Hartree — Constant 
Hartree = UnitSystem(𝟏,𝟏,𝟏/α,𝟐*τ*α^2,𝟏)
F=L⁻³, M=𝟙, L=L, T=L², Q=Q, Θ=L⁻², N=𝟙, J=L⁻⁴, A=𝟙, R=𝟙, C=𝟙

Hartree atomic UnitSystem based on bohr radius and elementarycharge scale.

julia> boltzmann(Hartree)
𝟏 = 1.0 [𝟙] Hartree

julia> planckreduced(Hartree)
𝟏 = 1.0 [𝟙] Hartree

julia> lightspeed(Hartree)
α⁻¹ = 137.035999084(21) [a₀⁻¹] Hartree

julia> vacuumpermeability(Hartree)
α²τ⋅2 = 0.00066917625662(21) [a₀⋅𝘦⁻²] Hartree

julia> electronmass(Hartree)
𝟏 = 1.0 [𝟙] Hartree

The well known Hartree atomic unit values for length, time, mass, and charge are:

julia> length(Hartree,SI2019) # lA
R∞⁻¹α⋅τ⁻¹2⁻¹ = 5.29177210902(81) × 10⁻¹¹ [m]/[a₀] Hartree -> SI2019

julia> time(Hartree,SI2019) # tA
𝘤⁻¹R∞⁻¹τ⁻¹2⁻¹ = 2.4188843265857(46) × 10⁻¹⁷ [s]/[a₀²] Hartree -> SI2019

julia> mass(Hartree,SI2019) # mA
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg]/[𝟙] Hartree -> SI2019

julia> charge(Hartree,SI2019) # qA
𝘦 = 1.602176634×10⁻¹⁹ [C]/[𝘦] Hartree -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = \frac{2}{\alpha}, \qquad \widetilde{\mu_0} = \pi\alpha^2, \qquad \widetilde{m_e} = \frac{1}{2}\]

MeasureSystems.Rydberg — Constant 
Rydberg = UnitSystem(𝟏,𝟏,𝟐/α,τ/𝟐*α^2,𝟏/𝟐)
F=MLT⁻², M, L, T, Q, Θ=T⁻¹, N=M, J=T², A=𝟙, R=𝟙, C=𝟙

Rydberg UnitSystem with lightspeed of 𝟐/α and permeability of π*α^2.

julia> boltzmann(Rydberg)
𝟏 = 1.0 [ML²T⁻¹] Rydberg

julia> planckreduced(Rydberg)
𝟏 = 1.0 [ML²T⁻¹] Rydberg

julia> lightspeed(Rydberg)
α⁻¹2 = 274.071998168(42) [LT⁻¹] Rydberg

julia> vacuumpermeability(Rydberg)
α²τ⋅2⁻¹ = 0.000167294064155(51) [MLQ⁻²] Rydberg

julia> electronmass(Rydberg)
2⁻¹ = 0.5 [M] Rydberg

The well known Rydberg atomic unit values for length, time, mass, and charge are:

julia> length(Rydberg,SI2019) # lR
R∞⁻¹α⋅τ⁻¹2⁻¹ = 5.29177210902(81) × 10⁻¹¹ [m]/[a₀] Rydberg -> SI2019

julia> time(Rydberg,SI2019) # tR
𝘤⁻¹R∞⁻¹τ⁻¹ = 4.8377686531713(93) × 10⁻¹⁷ [s]/[T] Rydberg -> SI2019

julia> mass(Rydberg,SI2019) # mR
𝘩⋅𝘤⁻¹R∞⋅α⁻²2² = 1.82187674031(56) × 10⁻³⁰ [kg]/[M] Rydberg -> SI2019

julia> charge(Rydberg,SI2019) # qR
𝘦⋅2⁻¹ᐟ² = 1.1329099625600371×10⁻¹⁹ [C]/[Q] Rydberg -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = \frac{1}{\alpha}, \qquad \widetilde{\mu_0} = 4\pi\alpha^2, \qquad \widetilde{m_e} = \sqrt{\frac{\alpha_G}{\alpha}}\]

MeasureSystems.Schrodinger — Constant 
Schrodinger = UnitSystem(𝟏,𝟏,𝟏/α,𝟐*τ*α^2,√(αG/α))
F=MLT⁻², M, L, T, Q, Θ=T⁻¹, N=M, J=T², A=𝟙, R=𝟙, C=𝟙

Schrodinger UnitSystem with permeability of 4π/αinv^2 and electronmass of √(αG*αinv).

julia> boltzmann(Schrodinger)
𝟏 = 1.0 [ML²T⁻¹] Schrodinger

julia> planckreduced(Schrodinger)
𝟏 = 1.0 [ML²T⁻¹] Schrodinger

julia> lightspeed(Schrodinger)
α⁻¹ = 137.035999084(21) [LT⁻¹] Schrodinger

julia> vacuumpermeability(Schrodinger)
α²τ⋅2 = 0.00066917625662(21) [MLQ⁻²] Schrodinger

julia> electronmass(Schrodinger)
𝘩⋅𝘤⁻¹R∞⋅α⁻⁵ᐟ²mP⁻¹2 = 4.899602(54) × 10⁻²² [M] Schrodinger

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = \frac{1}{\alpha}, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 4\pi, \qquad \widetilde{m_e} = 1\]

MeasureSystems.Electronic — Constant 
Electronic = UnitSystem(𝟏,𝟏/α,𝟏,𝟐*τ,𝟏)
F=T⁻¹, M=𝟙, L=T, T, Q, Θ=𝟙, N=𝟙, J=T⁻¹, A=𝟙, R=𝟙, C=𝟙

Electronic UnitSystem with planckreduced of 1/α and permeability of 4π.

julia> boltzmann(Electronic)
𝟏 = 1.0 [𝟙] Electronic

julia> planckreduced(Electronic)
α⁻¹ = 137.035999084(21) [T] Electronic

julia> lightspeed(Electronic)
𝟏 = 1.0 [𝟙] Electronic

julia> vacuumpermeability(Electronic)
τ⋅2 = 12.566370614359172 [TQ⁻²] Electronic

julia> electronmass(Electronic)
𝟏 = 1.0 [𝟙] Electronic

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 1, \qquad \widetilde{m_e} = 1\]

MeasureSystems.Natural — Constant 
Natural = UnitSystem(𝟏,𝟏,𝟏,𝟏,𝟏)
F=𝟙, M=𝟙, L=𝟙, T=𝟙, Q=𝟙, Θ=𝟙, N=𝟙, J=𝟙, A=𝟙, R=𝟙, C=𝟙

Natural UnitSystem with all primary constants having unit value.

julia> boltzmann(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> planckreduced(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> lightspeed(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> vacuumpermeability(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> electronmass(Natural)
𝟏 = 1.0 [𝟙] Natural

The well known Natural values for length, time, mass, and charge are:

julia> length(Natural,SI2019)
R∞⁻¹α²τ⁻¹2⁻¹ = 3.8615926796(12) × 10⁻¹³ [m]/[𝟙] Natural -> SI2019

julia> time(Natural,SI2019)
𝘤⁻¹R∞⁻¹α²τ⁻¹2⁻¹ = 1.28808866819(39) × 10⁻²¹ [s]/[𝟙] Natural -> SI2019

julia> mass(Natural,SI2019)
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg]/[𝟙] Natural -> SI2019

julia> charge(Natural,SI2019)
𝘦⋅α⁻¹ᐟ²τ⁻¹ᐟ²2⁻¹ᐟ² = 5.29081768990(41) × 10⁻¹⁹ [C]/[𝟙] Natural -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 4\pi, \qquad \widetilde{m_e} = 1\]

MeasureSystems.NaturalGauss — Constant 
NaturalGauss = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ,𝟏)
F=𝟙, M=𝟙, L=𝟙, T=𝟙, Q=Q, Θ=𝟙, N=𝟙, J=𝟙, A=𝟙, R=𝟙, C=𝟙

Natural (Gauss) UnitSystem with the Gaussian permeability value of 4π.

julia> boltzmann(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

julia> planckreduced(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

julia> lightspeed(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

julia> vacuumpermeability(NaturalGauss)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁻²] NaturalGauss

julia> electronmass(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 1, \qquad \widetilde{m_e} = \frac{1}{\mu_{pe}} = \frac{m_e}{m_p}\]

MeasureSystems.QCD — Constant 
QCD = UnitSystem(𝟏,𝟏,𝟏,𝟏,𝟏/μₚₑ)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=𝟙, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Qunatum chromodynamics UnitSystem based on the protonmass scale.

julia> boltzmann(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> planckreduced(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> lightspeed(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> vacuumpermeability(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> electronmass(QCD)
μₑᵤ⋅μₚᵤ⁻¹ = 0.000544617021487(33) [mₚ] QCD

The well known QCD values for length, time, mass, and charge are:

julia> length(QCD,SI2019) # lQCD
R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 2.10308910335(66) × 10⁻¹⁶ [m]/[mₚ⁻¹] QCD -> SI2019

julia> time(QCD,SI2019) # tQCD
𝘤⁻¹R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 7.0151501388(22) × 10⁻²⁵ [s]/[mₚ⁻¹] QCD -> SI2019

julia> mass(QCD,SI2019) # mQCD
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg]/[mₚ] QCD -> SI2019

julia> charge(QCD,SI2019) # qQCD
𝘦⋅α⁻¹ᐟ²τ⁻¹ᐟ²2⁻¹ᐟ² = 5.29081768990(41) × 10⁻¹⁹ [C]/[𝟙] QCD -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 4\pi, \qquad \widetilde{m_e} = \frac{1}{\mu_{pe}} = \frac{m_e}{m_p}\]

MeasureSystems.QCDGauss — Constant 
QCDGauss = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ,𝟏/μₚₑ)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Qunatum chromodynamics (Gauss) UnitSystem based on the protonmass scale.

julia> boltzmann(QCDGauss)
𝟏 = 1.0 [𝟙] QCDGauss

julia> planckreduced(QCDGauss)
𝟏 = 1.0 [𝟙] QCDGauss

julia> lightspeed(QCDGauss)
𝟏 = 1.0 [𝟙] QCDGauss

julia> vacuumpermeability(QCDGauss)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁻²] QCDGauss

julia> electronmass(QCDGauss)
μₑᵤ⋅μₚᵤ⁻¹ = 0.000544617021487(33) [mₚ] QCDGauss

The well known QCDGauss values for length, time, mass, and charge are:

julia> length(QCDGauss,SI2019) # lQCD
R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 2.10308910335(66) × 10⁻¹⁶ [m]/[mₚ⁻¹] QCDGauss -> SI2019

julia> time(QCDGauss,SI2019) # tQCD
𝘤⁻¹R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 7.0151501388(22) × 10⁻²⁵ [s]/[mₚ⁻¹] QCDGauss -> SI2019

julia> mass(QCDGauss,SI2019) # mQCD
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg]/[mₚ] QCDGauss -> SI2019

julia> charge(QCDGauss,SI2019) # qQCD
𝘦⋅α⁻¹ᐟ² = 1.87554603778(14) × 10⁻¹⁸ [C]/[𝘦ₙ] QCDGauss -> SI2019

\[\widetilde{k_B} = 1, \qquad \widetilde\hbar = 1, \qquad \widetilde c = 1, \qquad \widetilde{\mu_0} = 4\pi\alpha, \qquad \widetilde{m_e} = \frac{1}{\mu_{pe}} = \frac{m_e}{m_p}\]

MeasureSystems.QCDoriginal — Constant 
QCDoriginal = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ*α,𝟏/μₚₑ)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Qunatum chromodynamics (original) UnitSystem scaled by protonmass and elementarycharge.

julia> boltzmann(QCDoriginal)
𝟏 = 1.0 [𝟙] QCDoriginal

julia> planckreduced(QCDoriginal)
𝟏 = 1.0 [𝟙] QCDoriginal

julia> lightspeed(QCDoriginal)
𝟏 = 1.0 [𝟙] QCDoriginal

julia> vacuumpermeability(QCDoriginal)
α⋅τ⋅2 = 0.091701236889(14) [𝘦⁻²] QCDoriginal

julia> electronmass(QCDoriginal)
μₑᵤ⋅μₚᵤ⁻¹ = 0.000544617021487(33) [mₚ] QCDoriginal

The well known QCDoriginal values for length, time, mass, and charge are:

julia> length(QCDoriginal,SI2019) # lQCD
R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 2.10308910335(66) × 10⁻¹⁶ [m]/[mₚ⁻¹] QCDoriginal -> SI2019

julia> time(QCDoriginal,SI2019) # tQCD
𝘤⁻¹R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 7.0151501388(22) × 10⁻²⁵ [s]/[mₚ⁻¹] QCDoriginal -> SI2019

julia> mass(QCDoriginal,SI2019) # mQCD
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg]/[mₚ] QCDoriginal -> SI2019

julia> charge(QCDoriginal,SI2019) # qQCD
𝘦 = 1.602176634×10⁻¹⁹ [C]/[𝘦] QCDoriginal -> SI2019

UnitSystem Index

Wolfram plagiarism timeline

Timeline of UnitSystems.jl registration and Wolfram Research plagiarism:

  • 2019: The SI2019 standard is formalized with a primitive SI only unit-system based on 7 physics dimensions (massive collaboration).
  • 2020: Registered DOI 10.5281/zenodo.7145479, UnitSystems.jl
  • 2021: Discused with Ted Corcovilos about what the unsolved and nuanced issues are with defining physics units, which I then solved by independently creating the never before seen 11 dimensional Unified System of Quantities (USQ) for physics, which was standardized in detail and completely handcrafted by myself alone.
  • 2021: Wolfram Research invited me to their Summer School, where everyone was hinting at the fact I would be hired there.
  • 2022: Wolfram Research interviewed and then hired me, with an explicit interest in my UnitSystems.jl work from lead developers. They requested that I present them my independently discovered UnitSystems.jl results in the Wolfram Language to make a comparison with their existing system. While I was shortly an employee at Wolfram, I indeed directly handed them my newly discovered Unified System of Quantities. My work was already independently complete and prepared ready to incorporate into their stack. They acknowledged that their system was old and outdated compared to mine, as they only implemented a Metric and Imperial unit system, and neither of these was up to the standard of my UnitSystems.jl standard. However, they told me that I would not be allowed to work on this project further because they didn't want to upgrade their systems. Instead, they did the software equivalent of placing me in a backroom shed to mop the floors. After 6 months they ended the contract and it turned out they lied to me on the job application about what my role would be (they said I would be part of the core team with Jonathan Gorard, but this was a blatant lie).
  • 2023-2025: Wolfram keeps inviting me to their Winter/Summer schools to help mentor people, but I declined because I am too busy making progress in my research (why directly help mentor my competitors, who made it clear that they don't want to actually support my work); their use of social environments feels predatory.
  • 2025: Wolfram bribes Memes of Destruction at Wolfram Summer School to take my fully prepared work and use it to boost the Wolfram brand on social media, presenting my completed project with AI generated text as if it was Wolfram's idea, without crediting that I was the person who directly handed them the completed project years earlier (but without AI generated text they added).

Did Wolfam think that they can pluck low hanging fruits from my garden to build their brand on social media? My only goal here is to show that these low hanging fruit Wolfram plucked, these fruits came from my public garden and were not grown or developed by them from scratch, it's my solo-project.

Academic institutions should be direclty investing in my research instead of funding and enabling Wolfram Research to systematically gangstalk me with an army of employees. I can feel the presence of Wolfram looking over my shoulder and monitoring my every step. There seems to be an entire economy of people being paid to monitor and surveil me, while I struggle to survive with my resources. Stephen Wolfram never seemed to care about earning my respect. Every time I interacted with him, he was only focused on talking about himself and that was the only topic.

It's fascinating to me how unaware Stephen Wolfram is of the fact that people perceive him as textbook specimen of ultra-narcissism. This is because he is entirely surrounded by people with a salary depending on how much they inflate Stephen Wolfram's ego, which completely divorces these people from the reality of doing actual scientific research. Wolfram's premise seems to be that they can use gangstalking to target open source developers like me to data-mine our work, enabled by funding granted from academic institutions who don't check for Wolfram's plagiarism violations.

Combining the ultra-narcissism of Wolfram with the economic incentive to target open source developers with gangstalking by an army of employees, it becomes highly uncomfortable knowing that these people are incentivized to gangstalk me for the rest of my life with smear campaigns and so on.

I urge academic institutions to quit enabling and sponsoring Stephen Wolfram's systematic gangstalking of individuals like me. He shouldn't be rewarded for plucking fruit from my public gardens, which I handcrafted. % by myself. Wolfram's goal seems to be taking the fruit of my work in a cowardly and uncollaborative way. Wolfram does not acknowledge that my science research is what's boosting their brand in the social media campaign funding Memes of Destruction.

Julia Computing are no better stewards, they are also unehtical people, but at least their product is open source and therefore a solid foundation. My work on UnitSystems.jl and the entire process of creating the new 11 dimensional Unified System of Quantities (USQ) was all done entirely in public on GitHub and each release registered with several scientific websites. This is only one of my side projects, the mainstream of my research is my differential geometric algebra software development, Grassmann.jl and Cartan.jl, and various related work at the cutting edge of science, making me a bigger target for Wolfram's gangstalking. Wolfram is now constantly being observed in attempting to keep up with my research by systematically gangstalking me in a hush-hush way, not acknowledging me. With shady business practices, I have to wonder what other fraud is being commited.

It appears that Wolfram tends to resort to plagiarism of other people's works by data mining other people's creativity through employment, ghostwriters, summer schools, shady business practices, identity theft, bribes.

The incentive behind this systematic gangstalking appears to be this: instead of working with me directly, they all wish to ostracize, isolate, and erase me. Their eventual goal is replacing me and then retroactively claiming credit for my past achievements to boost their brands. Ironically, the temptation (to incorrectly eat the fruit of my labor like this) will be their downfall, as this choice is accompanied by firm evidence of plagiarism. Plagiarism is considered a violation of academic standards by the academic institutions funding Wolfram Research. My projects are effectively ego-traps, which will trigger the downfall of an ultra-narcissit ego if incorrectly consumed. I know the academic institutions don't acknowledge me either, so all I can do is to permanently add the Wolfram plagiarism disclaimer to the original sources.

Having a quick 0-60 speed in pathological lying is not necessarily a sign of high intelligence in long term thinking. Rather, it's an indicator of a complete lack of long term thinking, demonstrating optimization toward the short term illusions of success, which falls apart upon any scrutiny.

If Wolfram does not want to be perceived as confirmed plagiarist, then Wolfram must acknowledge Michael Reed as the original creator of the new Unified System of Quantities (USQ), which is the underlying foundation for the completed research project I handed them (and they padded with AI generated text). Wolfram is well known for the claims that LLMs will replace writing code and text, so we have to assume the foundations of their work rests in AI generated text, on top of my presented complete project foundation. The LLMs and AI models all know about my UnitSystems.jl work and mine was the only reference work in existence which completed this type of work. Therefore, if using AI or LLM generated text to manipulate my unique project, this is effectively transforming the original source data which was ingested from my work using my own knowledge embedded in the LLMs. Wolfram is regurgitating the fruits of my labor without acknowledging that I directly handed this to them as a completed project.

Memes of Destruction self proclaims to not be an expert on the topic and publicly discloses the paid sponsorship from Wolfram for the social media campaign, at least this is some transparency.

– Michael Reed's audience reaction to Wolfram's plagiarism

This preface was written in 2025, the UnitSystems.jl Appendix has been documented on my website and registered as Julia package since 2020.

Core UnitSystems.jl} standard was last updated in 2022, while Similitude.jl and MeasureSystems.jl have received minor software design updates since.