Imagine you are G.I.Taylor and its the start of The Cold War. Life magazine has just published an amazing high speed photographic sequence of the detonation of an atom bomb. The nation is enthralled by these images, but you see a bit further into the imagery.

The pictures give you the radius of the blast as a function of time.

All that could be important In determining the radius R as a function of time are the initial energy released, E, and the density of the undisturbed medium, Rho.

Thanks to your natural curiosity, you know the value of dimensional analysis and reason that the radius, with dimensions of length, must depend only on E, rho, and t, for which the correct expression can only be:

R(t) = (E/Rho t^2)^(1/5)

Since E/Rho has dimensions of (length)^5/time^2

A log-log plot of r vs t (imagine measuring the radius of the blast and carefully noting time stamps given in the picture captions) gives a slope of 2/5, checking the theory, and E/Rho could be obtained from extrapolation to the value of log R when log t= 0. Rho is known For air and thus E was determined to within a factor of order 1.

For the practitioner of dimensional analysis, the nations deepest secret had been published in Life magazine. (Goldstein States of Matter, chapter 6)

Ah, but how best to convey the findings? Surely the best way is to use a mathcad like, But open source tool which prints straight to latex!

Ah ...if only it could express fractional units...

GI Taylor prepares a letter to John wheeler, requesting a closed time like networked computing device so that the feature request can be made.

In all seriousness though, This project is awesome! Just kick in fractional unit support and it will surely be unstoppable.

Haha if only.

GI Taylor should be able to publish his findings by the end next week

Yeah they managed to find their way into circulation somehow. ;)

Not sure what I really had to add with that whole bit up there. I suppose it’s that sometimes intermediary units are a fun part of the story!

Anyway, very cool project.

The unit of volatility is conceptually [time^{-1/2}], most commonly 1/sqrt(year), but that is rarely written. (People speak of a stock with 30% annual vol, not with a sigma = 0.3 a^-0.5 :-)

Weird units are cool. I was pretty mindblown when I realized that radians are basically meters/meters and steradians are meters^2/meters^2

> I was pretty mindblown when I realized that radians are basically meters/meters

This is true in a dimensional analysis way, where e.g. arclength around a circle is equal to the subtended angle times the radius and, since the arclength and the radius have the same dimension, the angle must be dimensionless.

But units are more than that. You can't do anything in chemistry without running into quantities like molarity or ppm that appear to be dimensionless but make no sense if you think about them as coefficients rather than as combinations of semantically-distinct units that happen to have the same name.

Imagine some subtance that is 5% salt by mass and 8% salt by volume. ppm by mass is just kg / kg, right? And ppm by volume is m^3 / m^3. Those are the same unit -- in fact, no unit at all. Except that the implication of that non-fact is that ppm by volume and ppm by mass are the same thing, which they aren't.

Similarly, radians are best conceived of as percentages of a rotation, not as dimensionless coefficients. Then we calculate an arclength around a circle by saying that the ratio between the arclength and the full perimeter is equal to the ratio between the subtended angle and the full rotation. Characterized so, we have both units present twice, canceling each other out on each side of the equation, and we don't need to clean up by saying that the angle has no unit at all. We divided part of a perimeter by all of a perimeter -- producing a dimensionless quantity -- and we divided part of a rotation by all of a rotation -- also producing a dimensionless quantity -- and then we correctly compared two dimensionless quantities to each other.

(On this analysis, the equation s = rθ is actually incorrect; it should be written s = rθ/1 . You can extend this -- we all know the area under sin(x) from 0 to pi is 2. 2 whats? Well, the x axis is radians and the y axis is sine values. Sine values are computed by dividing the length of a line segment by the length of another line segment; they are legitimately dimensionless. So the area under sin(x) must be 2 radians.)

This has always bothered me, that there were dimensionless units that a dimension-like quality, like radians.

Seriously, radians aren't dimensionless. If you calculate arclengths using angles measured in degrees, the whole problem disappears: a 60° arc around a circle of radius 3cm has length equal to (2·pi·3cm·60°)/(360°) = (2·pi·3cm)/6 = pi centimeters. Looking at it that way makes it very plain that the angles have units attached, and those units don't appear in the result of "pi centimeters" because they canceled out during the computation.

This is overt when calculating in degrees because they are an inconvenient size. The size of a radian is chosen so that the constant terms (2·pi) in the numerator and (2·pi radians) in the denominator appear to cancel out, giving a (1 radian) in the denominator which is omitted from the formula because dividing by 1 is numerically pointless. But dividing by 1 radian is not dimensionally pointless; omitting the 1 from the formula leads to the illusion that θ has no unit attached.