Similitude

UnitSystems.similitude() = haskey(ENV,"SIMILITUDE")

An optional environment variable ENV["SIMILITUDE"] induces UnitSystems.similitude() to return true, giving flexibility for building dependencies whenever it is desirable to toggle usage between UnitSystems (default) and Similitude (requires environment variable specification). For example, in MeasureSystems and Geophysics this option is used to increase flexibility with variety in local compilation workflow.

UnitSystem(kB, ħ, 𝘤, μ₀, mₑ, Mᵤ, Kcd, ϕ, λ, αL, g₀, Universe)

A UnitSystem is a consistent set of dimensional values selected to accomodate a particular use case or standardization. It is possible to convert derived physical quantities from any UnitSystem specification into any other using accurate values. Eleven fundamental constants kB, ħ, 𝘤, μ₀, mₑ, Mᵤ, Kcd, ϕ, λ, αL, g₀ are used to govern a specific unit system consistent scaling. Different choices of natural units or physical measurements result in a variety of unit systems for many purposes.

Fundamental constants of physics are: kB Boltzmann's constant, ħ reduced Planck's constant, 𝘤 speed of light, μ₀ vacuum permeability, mₑ electron rest mass, Mᵤ molar mass, Kcd luminous efficacy, ϕ radian angle, λ Gauss rationalization, αL Lorentz's constant, and g₀ gravitational force reference. Primarily the Metric SI unit system is used in addition to the historic English engineering unit system. These constants induce derived values for avogadro, boltzmann, molargas, planck, planckreduced, lightspeed, planckmass, dalton, protonmass, electronmass, newton, einstein, vacuumpermeability, vacuumpermittivity, electrostatic, and additional constants molarmass, luminousefficacy, gravity, radian, turn, spat, stefan, radiationdensity, magnetostatic, lorentz, biotsavart, rationalization, vacuumimpedance, elementarycharge, magneton, conductancequantum, faraday, magneticfluxquantum, josephson, klitzing, hartree, rydberg, bohr.

Standardized unit/derived quantities are hyperfine, loschmidt, wienwavelength, wienfrequency, mechanicalheat, eddington, solarmass, jupitermass, earthmass, lunarmass, earthradius, greatcircle, radarmile, hubble, cosmological, steradian, spatian, degree, squaredegree, gradian, bradian, arcminute, arcsecond, second, minute, hour, day, gaussianmonth, siderealmonth, synodicmonth, year, gaussianyear, siderealyear, jovianyear, angstrom, inch, foot, surveyfoot, yard, meter, earthmeter, mile, statutemile, meridianmile, admiraltymile, nauticalmile, lunardistance, astronomicalunit, jupiterdistance, lightyear, parsec, barn, hectare, acre, surveyacre, liter, gallon, quart, pint, cup, fluidounce, teaspoon, tablespoon, bubnoff, ips, fps, fpm, ms, kmh, mph, knot, mps, grain, gram, earthgram, kilogram, tonne, ton, pound, ounce, slug, slinch, hyl, dyne, newton, poundal, poundforce, kilopond, psi, pascal, bar, barye, technicalatmosphere, atmosphere, inchmercury, torr, electronvolt, erg, joule, footpound, calorie, kilocalorie, meancalorie, earthcalorie, thermalunit, gasgallon, tontnt, watt, horsepower, horsepowerwatt, horsepowermetric, electricalhorsepower, tonsrefrigeration, boilerhorsepower, coulomb, earthcoulomb, ampere, volt, henry, ohm, siemens, farad, weber, tesla, abcoulomb, abampere, abvolt, abhenry, abohm, abmho, abfarad, maxwell, gauss, oersted, gilbert, statcoulomb, statampere, statvolt, stathenry, statohm, statmho, statfarad, statweber, stattesla, kelvin, rankine, celsius, fahrenheit, sealevel, boiling, mole, earthmole, poundmole, slugmole, slinchmole, katal, amagat, lumen, candela, lux, phot, footcandle, nit, apostilb, stilb, lambert, footlambert, bril, talbot, lumerg, neper, bel, decibel, hertz, apm, rpm, kayser, diopter, rayleigh, flick, gforce, galileo, eotvos, darcy, poise, reyn, stokes, rayl, mpge, langley, jansky, solarflux, curie, gray, roentgen, rem.

Additional reference UnitSystem variants: EMU, ESU, Gauss, LorentzHeaviside, SI2019, SI1976, CODATA, Conventional, International, InternationalMean, Engineering, Gravitational, IAU, IAUE, IAUJ, FPS, IPS, British, Survey, Hubble, Cosmological, CosmologicalQuantum, Meridian, Nautical, MPH, KKH, MTS, FFF; and natural atomic units based on gravitational coupling and finestructure constant (Planck, PlanckGauss, Stoney, Hartree, Rydberg, Schrodinger, Electronic, Natural, NaturalGauss, QCD, QCDGauss, and QCDoriginal).

Derived dimensions can be obtained from multiplicative base of 11 fundamental dimension symbols F, M, L, T, Q, Θ, N, J, A, R, C corresponding to force, mass, length, time, charge, temperature, molaramount, luminousflux, angle, demagnetizingfactor, and a nonstandard dimension. Specification of a UnitSystem is in dimensions of entropy, angularmomentum, speed, permeability, mass, molarmass, luminousefficacy, angle, rationalization, lorentz, gravityforce; whose Constant values are interpreted by units.

Mechanics: angle, angle, solidangle, time, angulartime, length, angularlength, area, angulararea, volume, wavenumber, angularwavenumber, fuelefficiency, numberdensity, frequency, angularfrequency, frequencydrift, stagnance, speed, acceleration, jerk, snap, crackle, pop, volumeflow, etendue, photonintensity, photonirradiance, photonradiance, inertia, mass, massflow, lineardensity, areadensity, density, specificweight, specificvolume, force, specificforce, gravityforce, pressure, compressibility, viscosity, diffusivity, rotationalinertia, impulse, momentum, angularmomentum, yank, energy, specificenergy, action, fluence, power, powerdensity, irradiance, radiance, radiantintensity, spectralflux, spectralexposure, soundexposure, impedance, specificimpedance, admittance, compliance, inertance; Electromagnetics: charge, chargedensity, linearchargedensity, exposure, mobility, current, currentdensity, resistance, conductance, resistivity, conductivity, capacitance, inductance, reluctance, permeance, permittivity, permeability, susceptibility, specificsusceptibility, demagnetizingfactor, vectorpotential, electricpotential, magneticpotential, electricfield, magneticfield, electricflux, magneticflux, electricdisplacement, magneticfluxdensity, electricdipolemoment, magneticdipolemoment, electricpolarizability, magneticpolarizability, magneticmoment, specificmagnetization, polestrength; Thermodynamics: temperature, entropy, specificentropy, volumeheatcapacity, thermalconductivity, thermalconductance, thermalresistivity, thermalresistance, thermalexpansion, lapserate, molarmass, molality, molaramount, molarity, molarvolume, molarentropy, molarenergy, molarconductivity, molarsusceptibility, catalysis, specificity, diffusionflux, luminousflux, luminousintensity, luminance, illuminance, luminousenergy, luminousexposure, luminousefficacy.

In UnitSystems and Similitude, the spectrum of Constant values is generated by a group of 11 mathematical constants (7 Integer primes and 4 Irrational numbers) with 33 physical measurement definitions. These are 𝟐, 𝟑, 𝟓, 𝟕, 𝟏𝟏, 𝟏𝟗, 𝟒𝟑, φ, γ, ℯ, τ, kB, NA, 𝘩, 𝘤, 𝘦, Kcd, ΔνCs, R∞, α, μₑᵤ, μₚᵤ, ΩΛ, H0, g₀, aⱼ, au, ft, ftUS, lb, T₀, atm, inHg, RK90, KJ90, RK, KJ, Rᵤ2014, Ωᵢₜ, Vᵢₜ, kG, mP, GME, GMJ.

julia> one # 𝟏
𝟏 = 1.0

julia> two # 𝟐
2 = 2.0

julia> three # 𝟑
3 = 3.0

julia> five # 𝟓
5 = 5.0

julia> seven # 𝟕
7 = 7.0

julia> eleven # 𝟏𝟏
11 = 11.0

julia> nineteen # 𝟏𝟗
19 = 19.0

julia> fourtythree # 𝟒𝟑
43 = 43.0

julia> sixty # 𝟔𝟎
2²3⋅5 = 60.0

Constant{N} where N

Truncated Irrational constant N with known value at compile time.

julia> golden # φ
φ = 1.618033988749895

julia> eulergamma # γ
γ = 0.5772156649015329

julia> exp # ℯ
ℯ = 2.718281828459045

julia> pi # π
3.141592653589793

julia> tau # τ
τ = 6.283185307179586

Physical dimension Constant represented by Group element D.

F, M, L, T, Q, Θ, N, J, A, R, C

Operations on Constant are closed (*, /, +, -, ^).

julia> force(Unified)
F [ħ⁵𝘤⁻⁸μ₀⁻¹mₑ⁻⁶Kcd⁻¹ϕ⁵λ⁻¹g₀⁵] Unified

julia> mass(Unified)
M [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] Unified

julia> length(Unified)
L [ħ⁻⁵ᐟ²𝘤¹¹ᐟ²μ₀¹ᐟ²mₑ⁴Kcd⋅ϕ⁻⁵ᐟ²λ¹ᐟ²g₀⁻³] Unified

julia> time(Unified)
T [ħ⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] Unified

julia> charge(Unified)
Q [ħ⁻³ᐟ²𝘤⁷ᐟ²mₑ⁵ᐟ²Kcd¹ᐟ²ϕ⁻²λ⁻¹g₀⁻²] Unified

julia> temperature(Unified)
Θ [ħ⁷ᐟ²𝘤⁻¹¹ᐟ²μ₀⁻¹ᐟ²mₑ⁻⁴ϕ⁷ᐟ²λ⁻¹ᐟ²g₀⁴] Unified

julia> molaramount(Unified)
N [kB⋅ħ¹ᐟ²𝘤⁻⁵ᐟ²μ₀⁻¹ᐟ²mₑ⁻¹ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹g₀] Unified

julia> luminousflux(Unified)
J [ħ⁶𝘤⁻⁹μ₀⁻¹mₑ⁻⁶Mᵤ⁻¹Kcd⁻¹ϕ⁶λ⁻¹g₀⁶] Unified

julia> angle(Unified)
A [ħ⁻¹𝘤⁴mₑ²Kcd⋅ϕ⁻¹g₀⁻²] Unified

julia> rationalization(Unified)
R [ϕ] Unified

julia> lorentz(Unified)
C⁻¹ [λ] Unified

Derived dimension can be obtained from multiplicative base of 11 fundamental dimension symbols corresponding to force, mass, length, time, charge, temperature, molaramount, luminousflux, angle, demagnetizingfactor, and a nonstandard dimension.

Mechanics: angle, angle, solidangle, time, angulartime, length, angularlength, area, angulararea, volume, wavenumber, angularwavenumber, fuelefficiency, numberdensity, frequency, angularfrequency, frequencydrift, stagnance, speed, acceleration, jerk, snap, crackle, pop, volumeflow, etendue, photonintensity, photonirradiance, photonradiance, inertia, mass, massflow, lineardensity, areadensity, density, specificweight, specificvolume, force, specificforce, gravityforce, pressure, compressibility, viscosity, diffusivity, rotationalinertia, impulse, momentum, angularmomentum, yank, energy, specificenergy, action, fluence, power, powerdensity, irradiance, radiance, radiantintensity, spectralflux, spectralexposure, soundexposure, impedance, specificimpedance, admittance, compliance, inertance; Electromagnetics: charge, chargedensity, linearchargedensity, exposure, mobility, current, currentdensity, resistance, conductance, resistivity, conductivity, capacitance, inductance, reluctance, permeance, permittivity, permeability, susceptibility, specificsusceptibility, demagnetizingfactor, vectorpotential, electricpotential, magneticpotential, electricfield, magneticfield, electricflux, magneticflux, electricdisplacement, magneticfluxdensity, electricdipolemoment, magneticdipolemoment, electricpolarizability, magneticpolarizability, magneticmoment, specificmagnetization, polestrength; Thermodynamics: temperature, entropy, specificentropy, volumeheatcapacity, thermalconductivity, thermalconductance, thermalresistivity, thermalresistance, thermalexpansion, lapserate, molarmass, molality, molaramount, molarity, molarvolume, molarentropy, molarenergy, molarconductivity, molarsusceptibility, catalysis, specificity, diffusionflux, luminousflux, luminousintensity, luminance, illuminance, luminousenergy, luminousexposure, luminousefficacy.

(U::UnitSystem)(v::Number, d) ↦ Quantity(U,v,d) = Quantity{U}(v,d)

Numerical Quantity having value v with dimension d specified in U::UnitSystem.

julia> Metric(1,energy)
1 [J] Metric

julia> English(1,energy)
1 [lbf⋅ft] English

An alternate syntax Quantity(U::UnitSystem, v::Number, d) is also available as standard syntax. When using UnitSystems instead of using Similitude, this same syntax can be written so that code doesn't need to be changed while the output is generated.

(D::Group{:USQ})(U::UnitSystem,S::UnitSystem) = ConvertUnit{U,S}(D)

Constant for unit conversion for D::Group{:USQ} from U::UnitSystem to S::UnitSystem.

julia> energy(Metric,CGS)
2⁷5⁷ = 1.0e7 [erg]/[J] Metric -> Gauss

julia> energy(Metric,English)
g₀⁻¹ft⁻¹lb⁻¹ = 0.7375621492772653 [lbf⋅ft]/[J] Metric -> English

There still exists further opportunity to expand on the implementation of ConvertUnit.

Unified = UnitSystem(...) # Unified System of Quantities (USQ)
F, M, L, T, Q, Θ, N, J, A, R, C # fundamental base dimensions

Standard Unified system of Quantities (USQ) in terms of UnitSystem basis, transformed from the basis of force, mass, length, time, charge, temperature, molaramount, luminousflux, angle, rationalization, and a nonstandard dimension.

julia> boltzmann(Unified) # entropy
FLΘ⁻¹ [ħ⁻¹𝘤³mₑ²ϕ⁻¹g₀⁻²] Unified

julia> planckreduced(Unified) # angularmomentum
FLTA⁻¹ [mₑ] Unified

julia> lightspeed(Unified) # speed
LT⁻¹ [ħ⋅𝘤⁻¹mₑ⁻¹ϕ⋅g₀] Unified

julia> vacuumpermeability(Unified) # permeability
FT²Q⁻²R⁻¹C² [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

julia> electronmass(Unified) # mass
M [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] Unified

julia> molarmass(Unified) # molarmass
MN⁻¹ [kB⁻¹𝘤²mₑ⋅g₀⁻¹] Unified

julia> luminousefficacy(Unified) # luminousefficacy
F⁻¹L⁻¹TJ [mₑ⋅Mᵤ⁻¹] Unified

julia> radian(Unified) # angle
A [ħ⁻¹𝘤⁴mₑ²Kcd⋅ϕ⁻¹g₀⁻²] Unified

julia> rationalization(Unified) # demagnetizingfactor
R [ϕ] Unified

julia> lorentz(Unified) # nonstandard
C⁻¹ [λ] Unified

julia> gravity(Unified) # gravityforce
F⁻¹MLT⁻² [αL⁻¹] Unified

@unitgroup(U::UnitSystem,S::UnitSystem) -> (u::typeof(normal(U)))(d::Group) = normal(S)(d)

Implements Group homomorphism for U in terms of existing specification from S.

@unitdim(U::UnitSystem,F,M,L,T,Q,Θ,N,J="lm",A="rad")

Specify the print output for each base dimension of U::UnitSystem with String input arguments force, mass, length, time, charge, temperature, molaramount, luminousflux, angle.

@unitdim Gauss "gf" "g" "cm" "s" "C" "K" "mol"
@unitdim Metric "kgf" "kg" "m" "s" "C" "K" "mol"
@unitdim British "lb" "slug" "ft" "s" "C" "°R" "slug-mol"
@unitdim IPS "lb" "slinch" "in" "s" "C" "°R" "slinch-mol"
@unitdim FPS "pdl" "lb" "ft" "s" "C" "°R" "lb-mol"
@unitdim English "lbf" "lbm" "ft" "s" "C" "°R" "lb-mol"
@unitdim IAU☉ "M☉f" "M☉" "au" "D" "C" "K" "mol"

These standard examples are some of the built-in defaults.

@unitdim(U::UnitSystem,S::UnitSystem) -> dimtext(::typeof(normal(U))) = dimtext(normal(S))

Specify the print output for each base dimension of U upon prior existing S data.

@unitdim EMU Gauss
@unitdim ESU Gauss
@unitdim LorentzHeaviside Gauss
@unitdim SI2019 Metric
@unitdim SI1976 Metric
@unitdim CODATA Metric
@unitdim Conventional Metric
@unitdim International Metric
@unitdim InternationalMean Metric
@unitdim Survey English

These standard examples are some of the built-in defaults.

@unitdim(D,U,S,L)

Specify print S::String and LaTeX L::String for derived dimension D in U::UnitSystem.

@unitdim magneticflux Gauss "Mx" "\text{Mx}"
@unitdim magneticfluxdensity Gauss "G" "\text{G}"
@unitdim magneticfield Gauss "Oe" "\text{Oe}"
@unitdim frequency Metric "Hz" "\text{Hz}"
@unitdim force Metric "N" "\text{N}"
@unitdim pressure Metric "Pa" "\text{Pa}"
@unitdim energy Metric "J" "\text{J}"
@unitdim power Metric "W" "\text{W}"
@unitdim mass British "slug" "\text{slug}"
@unitdim force FPS "pdl" "\text{pdl}"

These standard examples are some of the built-in defaults.

MetricSystem(Mu=Mᵤ,μ0=μ₀,Ru=Rᵤ,g0=𝟏,θ=𝟏,h=𝘩,me=R∞*𝟐*h/𝘤/α^2)

Constructs new UnitSystem from molarmass constant, vacuumpermeability, molargas constant, gravity force reference, angle scale, and planck constant.

UnitSystem(Ru*me/Mu/μₑᵤ/g0,h/τ/g0/θ,𝘤,μ0,me,Mu,Kcd*(mₑ/me)^2*(h/𝘩)*g0,θ,𝟏,𝟏,g0)

Examples include SI2019, SI1976, Metric, Engineering, MetricTurn, MetricSpatian, MetricGradian, MetricDegree, MetricArcminute, MetricArcsecond. In addition, the ConventionalSystem constructor further builds on MetricSystem, resulting in variations.

Other derived UnitSystem instances such as British or English or IAU are derived from an existing Metric specification generated by MetricSystem. The constructor MetricSystem incorporates several standard common numerical values and exposes variable arguments which can be substituted for customization, yielding the capability to generate historical variations having a common Universe. Derivative constructors are EntropySystem, ElectricSystem, GaussSystem, RankineSystem, and AstronomicalSystem.

ConventionalSystem(RK,KJ,Ru=Rᵤ,g0=𝟏) = MetricSystem(milli,𝟐*RK/𝘤*α,Ru,g0,𝟐^2/RK/KJ^2)

Constructs new UnitSystem from von klitzing constant and josephson constant, with an optional specification of universal gas constant and gravity reference constant.

Examples include Conventional (based on 1990) and CODATA (based on 2014).

ElectricSystem(U::UnitSystem,Ω,V) = EntropySystem(U,𝟏,𝟏,V^2/Ω,𝟏,vacuumpermeability(U)/Ω)

Constructs new UnitSystem from U with mass rescaled by electricpotential and resistance. In the International system, Ωᵢₜ and Vᵢₜ are used as definitions from the more recent United States results, while in InternationalMean an earlier estimate based on other nations was used.

GaussSystem(U::UnitSystem,μ0,λ,αL=𝟏,l=centi,m=milli,g0=gravity(U))

Constructs new UnitSystem from U rescaled for CGS with electromagnetic options. The first three options are to set the values for vacuumpermeability, rationalization, and lorentz constants. The following two parameters are scaling for length and mass, while the last is an option to change the gravity reference.

Examples include EMU, ESU, Gauss, LorentzHeaviside, and Kennelly.

EntropySystem(U::UnitSystem,t,l,m,θ=𝟏)
EntropySystem(U::UnitSystem,t,l,m,θ,μ0,Mu=molarmass(U)/m,g0=gravity(U))

Constructs new UnitSystem from U rescaled along time, length, mass, and temperature by the first four parameters. Additional optional parameters allow for customization of the vacuumpermeability, molarmass, and gravity constants.

Examples of this type include Nautical, Meridian, Gravitational, MTS, KKH, MPH, IAU☉, IAUE, IAUJ, Hubble, Cosmological, CosmologicalQuantum. However, most other constructors for UnitSystem derivations are based on internally calling EntropySystem, such as AstronomicalSystem, ElectricSystem, GaussSystem, and RankineSystem. This means EntropySystem also constructs the examples listed there.

RankineSystem(U::UnitSystem,l,m,g0=𝟏)

Constructs new UnitSystem from U rescaled along length and mass with optional gravity reference constant used to define technical and engineering units.

EntropySystem(U,𝟏,l,m,°R,vacuumpermeability(U)/m/l/g0,kilo*molarmass(U),g0)

Examples: FPS, British, IPS, English, Survey.

AstronomicalSystem(U::UnitSystem,t,l,m)

Constructs new UnitSystem from U rescaled along time, length, mass, and dimensionless boltzmann and molarmass constants. Examples are Hubble, Cosmological, CosmologicalQuantum.