the π theorem

Yes, it’s π day (3.14) and so I am obliged to write a short something on that topic.

Most people are celebrating the free or $3.14 pizza today. My mother is a little ticked that I didn’t tell her until the evening.

Academic types are geeking out over the proximity of the base-10 decimal number to today’s Gregorian date. Some might even throw a party at 1:59 local time.

I’m more excited about Buckingham-π

Yes, that is lovely. It’d take that over more math any day. BTW, do you think they serve Buckingham-Pi at Buckingham-Palace ? I’ll take two stones for 3.14 pounds please. [source]

Unfortunately, no. It is a math thing, again! It has awesome science and engineering dimensional-analysis applications (just using units!):

π-groups are sets comprised of physical variables (with units) raised to some powers so that the group product is devoid of dimensions. (Does not depend on length, time, mass, temperature, etc.)

This is used to construct an invariant vector space of all physically-consistent variable permutations within the system — hinting at possible underlying relationships or mechanisms.

Buckingham-π is recommended in multi-dimensional regression studies on instrumented data where there isn’t a clear relationship between measurable physical variables. These groupings imply similar solution space if relative relationships are enforced (asserting analogous conditions). This allows us to do test miniature bridges and wings in the lab, and infer fundamental relationships between dimensional things like like E=mc^2 .

Shamelessly stolen from MIT notes; the source for everything you don’t know.

We do this all the time in our heads but less rigorously. We know not to add length and time, but can divide them to make the units of speed. We know you can only add energy to energy, etc. Energy divided by distance has units of force — one of innumerable dimensional truisms.

As an example, the aspect ratio of your computer screen is a π-group, as width divided by height is dimensionless. n=2 variables and k=1 dimension type (length), so j = 2-1 = 1 possible π-group.

Non-dimensional parameters such as Reynolds number comprise the π-group for density, velocity, length, and viscosity. We expect self-similar relationships (and system behavior) if the Re is held constant :

In terms of SI (kg-m-s) units, this equates to dimensional nullity:

The above has n=4 variables and k=3 dimensions, so j = (n-k) = 1 means this also is the only possible combination for this group. If we had more variables, we would have more possible π-groups and exponential coefficients to determine.

While integer exponents are expected on dimensional variables, non-dimensional π-groups are technically allowed to have irrational (non-integer) exponents.

Since 1 to any power is itself, π-groups are invariant under power — the constant exponent relating weightings between multiple π-groups is constant for a specific configuration; coefficients are evaluated using an experiment test matrix for each independent variable.

Usually, integer exponents can be defined for all by multiplying each exponent by the lowest common exponent as they are often easily divided. In highly-empirical regressions, odd irregular decimals such as 0.37 are not uncommon as they are not factorable.

Read more about Buckingham-π on Wikipedia