Scientific computing with confidence using typed dimensions
laurentrdc.xyz
laurentrdc.xyz
On the Haskell Discourse [0], someone posted about another Haskell library, units [1], which allows to define unit systems NOT limited to the 7 physical dimensions. For example, adding a base dimension for currency would be nice to model electricity prices in currency per energy.
[0]: https://discourse.haskell.org/t/blog-post-scientific-computi...
(And I too changed an implementation to add units of measure to a system. It took a few days, and caused some grief with downstream system developers. But we also avoided the Mars Climate Orbiter experience in production )
I think a pretty well thought out approach to this is the mp-units library in c++ that uses the ISQ(International System of Quantities) [1].
There are some great blog posts on their site about modeling units systems and dimensions. Fingers crossed it may even be added to the standard library in C++29.
The illusion of having the same dimensions has been caused by a vote of the General Conference on Weights and Measures, which has been exactly as stupid as deciding by vote that 2 + 2 = 5. It is really a shame that something like this has happened.
According to that vote, some quantities have been declared as adimensional, even if arbitrary units of measurements must be chosen for them, exactly like for any other physical quantity and their units enter in the definitions of many other units, so changing any such choice of a unit changes the values of many other SI units, which implies that any version of SI that does not explicitly include the fundamental units chosen for "adimensional" quantities is incomplete.
The system of units of measurement required for the physical quantities includes fundamental units a.k.a. base units, which are chosen arbitrarily, without any constraints, and also derived units, whose values are determined in such a way as to eliminate all constants from the formulae that relate the physical quantity for which the derived unit is determined to other physical quantities for which units have already been determined.
The fundamental units are of 3 kinds, which I shall call arithmetic units, geometric units and dynamic units.
The arithmetic units are for discrete quantities, i.e. quantities that are expressed in integer numbers, so they have natural units.
There are a huge number of such fundamental units for quantities that can be obtained by counting, e.g. the number of atoms of a certain kind, the number of ions of certain kind, the number of molecules of a certain kind, the number of objects of a certain kind, the number of apples, the number of oranges, the number of humans and so on.
While "1" is the natural unit for all such quantities, in physics and chemistry there are many such quantities for which the conventional SI unit is not "1", but the number of Avogadro, a.k.a. one mole of substance.
Besides numbers of various entities, if we neglect nuclear/"elementary" particle physics, there is one more discrete physical quantity, which is the electric charge a.k.a. the quantity of electricity. The natural unit for the electric charge is the elementary electric charge and the conventional SI unit, the coulomb, is defined as an exact multiple of the elementary electric charge.
Besides the units for discrete quantities, there are 6 fundamental units for continuous quantities, 3 geometric units & 3 dynamic units. The 3 geometric units are the units of logarithmic ratio, plane angle and solid angle (which are respectively associated with 1-dimensional, 2-dimensional and 3-dimensional spaces).
The 3 fundamental geometric units are chosen by a mathematical convention, but they are exactly as arbitrary as the fundamental dynamic units. This is proven by the fact that in practice many different choices are used for these 3 fundamental units. Like length can be measured in meters or in feet, logarithmic ratios can be measured in nepers or in decibels, plane angles can be measured in degrees or in right angles or in cycles or in radians, while solid angles can be measured in orthogonal trihedral angles, in whole spheres or in steradians. Changing the choice for any of these fundamental units requires value conversions for all physical quantities that have units derived from them, in the same way like when replacing Imperial units with metric units.
There are many physical quantities in SI that appear to have the same dimensional formulae, but this happens only because the units of logarithmic ratio or plane angle or solid angle are omitted from the formulae. This omission is especially frequent for the unit of plane angle, which enters in a much greater number of dimensional formulae than it is obvious. For instance, the magnetic flux also includes in its dimensional formula the unit of plane angle, though you will almost never see this taken into account.
The SI includes a greater number of conventional fundamental dynamic units than necessary, for historical reasons, which results in a greater number of "universal constants" than for an optimal system of units, which requires only 3 fundamental dynamic units.
While at the end of the 18th century, when the metric system has been designed, it was convenient to choose as fundamental dynamic units the units of length, time and mass, for which distinct physical devices were chosen to materialize the units, the current SI system is equivalent with choosing a single physical device to materialize all 3 fundamental dynamic units, which are the units of length, time and electric voltage (yes, now the unit of mass is derived from the unit of electric voltage, not vice-versa, despite confusing definitions that can be encountered in many texts).
The single physical device that can materialize all 3 fundamental dynamic units is an atomic clock. The electromagnetic wave corresponding to its output signal provides the unit of length in its wavelength, the unit of time in its wave period, while the Josephson voltage corresponding to its frequency provides the unit of electric voltage. The conventional units of SI are obtained from the 3 units provided by an atomic clock by multiplication with conventional exact constants, in order to ensure continuity with the older units of measurement.
A few of the so-called base units of SI are no longer true base units, but they are derived from the true base units by inserting exact conventional constants in the formulae relating physical quantities, e.g. the units of temperature and luminous intensity.
When you use correct dimensional formulae, most of these cases disappear, e.g. torque and energy do not have the same dimensions (the cases that remain are for quantities of the same kind that measure different things, like the average and the peak value of some quantity, for which some arithmetic operations that combine them may be meaningless). The problem is that there are plenty of textbooks with incomplete formulae, so one may need to analyze them, instead of having faith that they are correct.
Moreover, for the discrete quantities obtained by counting one must define different types depending on what is counted, i.e. one must not use the same type for counting horses and for counting boxes (or more realistically for the kind of quantities that can appear in complicated expressions, a molar quantity of some chemical substance must not have the same type as a molar quantity of a different chemical substance, or, when simulating semiconductors, the concentrations of electrons, holes, donors and acceptors must have different types).
Do you have a reference for any undergrad text that does this correctly?
That being said, you can absolutely criticize any given unit system, just as you can criticize any given type system. And some of your criticisms* of SI are indeed valid. (*Out of those that I even understand at all.) It just means that when we compare unit systems, we should be focusing on factors like: what kinds of errors do they catch/prevent? How understandable are they? How reliable? You do that in some places, such as your excellent point about torque vs. energy. But it's less clear why we should be splitting out the 3 kinds of units or only ever specifying number of apples as integers (I can certainly have half an apple) -- how does this actually help us? I would also argue that if you're trying to say something like "1 Apple + 1 Orange != 2", then I'd argue for the same reason that "1 Apple + 1 Apple != 2", because different apples are different sizes, shapes, value, etc. Really I'm just not clear on almost any of your points about "discrete units".
And for the geometric units, why stop at 3? We obviously could have 4-dimensional angles as well (possibly not physically useful, but maybe for spacetime calculations ... and why should our system of units only apply to our particular universe?). You could have n^2 and n^3 units similar to logarithmic units (e.g. 1 squart = 1 foot, 2 squart = 4 feet, 3 squart = 9 feet). I'm not claiming any of those are useful, but I am claiming it shows that your examples are not "fundamental".
> The electromagnetic wave corresponding to its output signal provides the unit of length in its wavelength, the unit of time in its wave period.
But aren't the wavelength and period of an electromagnetic wave linearly correlated by the speed of light? Doesn't this make your 3 fundamental dynamic units only 2?
Doing it within an existing type system is more trouble than it’s worth: the algebra of units is simple, but it doesn’t apply to other types without stretching the compiler. It is far easier to have a distinct subsystem with measure types specifically identified.
There is even native support for reading dimensioned values from input files, so your user can specify "speed 12.7 [ft/s]" irrespective of what units you use internally in the code processing that input file. It just gets converted (or throws an error).
See e.g. this, from 4.2.6 onwards: https://doc.cfd.direct/openfoam/user-guide-v12/basic-file-fo...
But why not both? A number system with dimensions and which automatically propagates measurement uncertainty through calculations (a la Taylor's train wreck book) doesn't seem totally infeasible?
I would particularly like this for expressing cad models as code instead of brep.
It is my goal to provide quantities with uncertainty AND typed dimensions one day.
julia> using Unitful: m, s
julia> using Measurements
julia> Δx = (3.0 ± 0.1)m
3.0 ± 0.1 m
julia> t = (1.0 ± 0.2)s
1.0 ± 0.2 s
julia> v = Δx / t
3.0 ± 0.61 m s^-1Haskell has arrays and matrices of homogeneous types, so it wouldn't work by default.
If you needed this kind of functionality, you would have to create your own Matrix type which would look very similar to a 9-tuple, and then define matrix multiplication. It would then be possible to encode the dimensional constraints in the type signature, but it's already quite involved for 2x2 matrices:
data Mat2 a b c d
= MkMatrix a b c d
matmul :: ???? -- good luck writing this type signature
matmul (MkMatrix a11 a12 a21 a22) (MkMatrix b11 b12 b21 b22)
= MkMatrix ((a11 * b11) + (a12 * b21))
((a11 * b12) + (a12 * b22))
((a21 * b11) + (a22 * b21))
((a21 * b12) + (a22 * b22))
I wonder if there is any plausible approach that would work closer to how units are treated in physics calculations - not as types, but as special numerical values, more or less (values that you can multiply/divide by any other unit, but can only add/subtract with themselves).
What's so cool about `dimensional` is that the types work the way you describe!
Normally, in Haskell, the addition and multiplication functions are typed like so:
(+) :: Double -> Double -> Double
(*) :: Double -> Double -> Double
(+) :: Quantity d Double -> Quantity d Double -> Quantity d Double
(*) :: Quantity d1 Double -> Quantity d2 Double -> Quantity (d1 * d2) Double
This means that `dimensional` isn't only about 'statically checking' dimensions and units, but also inferring them.
You can see this interactively. What is the type of 1 meter divided by 1 second:
ghci> :t (1 *~ meter) / (1 *~ second)
Quantity (Dim Pos1 Zero Neg1 Zero Zero Zero Zero) Double
Here is an example:
mat1 = Table[
RandomChoice[{
Quantity[RandomReal[], "Meters"],
Quantity[RandomReal[], "Seconds"],
Quantity[RandomReal[], "Meters/Seconds"],
Quantity[RandomReal[], "Meters/Seconds^2"]
}] , 3, 3];
mat1 // MatrixForm
mat1 . Transpose[mat1]
use Physics::Measure :ALL;
sub infix:<·>(@x1, @x2) {
die "Incompatible dimensions."
unless @x1 == @x2[0] && @x1[0] == @x2;
[for ^@x1 -> $m {
[for ^@x1 -> $n {
[+] @x1[$m;*] >>*<< @x2[*;$n]
}]
}]
}
my @m = [[1m,2m],[3m,4m]];
say @m · [Z] @m; #[[5m^2 11m^2] [11m^2 25m^2]]
Two quantities with dimensions That’s not true though. Some units cannot be added together, for example all intensive units like speed, density, temperature etc.
from astropy import units as u
density = np.linspace(0, 1, 10)*u.g/u.cm**3
mass = (density*1*u.m**3).to(u.kg)
While Raku is less popular than Python, it does have deep roots in Haskell and strong types (the first Raku / Perl6 parser - PUGS - was written in Haskell and all the early devs were encouraged to learn Haskell first).
Similar concepts are used in these Raku modules... which provide dimensional analysis and marry types to dimensions.
- https://raku.land/zef:librasteve/Physics::Measure
- https://raku.land/zef:librasteve/Physics::Unit
I had some fun with making this example of using these Raku modules with Jupyter https://rakujourney.wordpress.com/wp-content/uploads/2023/08...
[disclaimer: I am the author of these modules]
Raku is also good at Slangs (Sublanguages) and unicode, so these tools can be used in a very intuitive way:
#!/usr/bin/env raku
use Physics::Constants;
use Physics::Measure :ALL;
say ~ℏ; #1.054571817e-34 J.s
my \λ = 2.5nm;
say "Wavelength of photon (λ) is " ~λ;
my \ν = c / λ;
say "Frequency of photon (ν) is " ~ν.in('petahertz');
my \Ep = ℎ * ν;
say "Energy of photon (Ep) is " ~Ep.in('attojoules');
Wavelength of photon (λ) is 2.5nm
Frequency of photon (ν) is 119.92PHz
Energy of photon (Ep) is 79.46aJSee: https://reference.wolfram.com/language/guide/Units.html
> The units framework [of Wolfram Language] integrates seamlessly with visualization, numeric and algebraic computation functions. It also supports dimensional analysis, as well as purely symbolic operations on quantities.
It is true that I (and others) avoid using Wolfram Language (WL) units. Or try to get the computation expressions units-free as quickly as possible. But that is also a normal procedure when crafting computational workflows.
Like if I have a cartesian coordinates in one reference frame, can I use them in a type-safe way? E.g. not add two vectors in two reference frames? Same for polar coordinates?
Etc.
newtype Framed f a = MkFramed (Vector a)
add :: Framed f a -> Framed f a -> Framed f a
add (MkFramed vec1) (MkFramed vec2) = ... add vec1 and vec2
data ReferenceFrame1
data ReferenceFrame2
v1 :: Framed ReferenceFrame1 Double
v1 = ...
v2 :: Framed ReferenceFrame2 Double
v2 = ...
main :: IO ()
main = print $ v1 `add` v2 -- will not type check
A similar thing I have been thinking about doing in robotics contexts is to annotate vectors and transformation matrices with their reference frames. So you can only matrix-vector multiply between compatible types: `object_in_world = camera_to_world @ object_in_camera`. It can be done somewhat in C++.
At least you can check the input of functions at runtime [1].
[0]: https://github.com/hgrecco/pint
[1]: https://pint.readthedocs.io/en/stable/advanced/wrapping.html
https://docs.sympy.org/latest/modules/physics/units/index.ht...