Hi Everyone, firstly, some quick bullets:
- The Learning Gardens Slack has ended. Big thanks to Soft Surplus & friends for hosting a farewell party last Friday. I'm excited to see where diffusion will lead, and am thankful for the people I've met along the way.
- If you are in the Bay Area, let's do something together! I'm going to see Kim Ip dance on Friday, and Theo Parrish et al. in April.
- If you're in Detroit area, check out the Cranbrook MFA show. Some of my first serious conversations about art were as a docent in this museum back in high school; it is unbelievably cool to know that Sam is showing there.
Now on to the regular stuff.
One of the reasons I chose to be an applied math major in college was that I felt mathematics could provide powerful metaphors for making sense of the world.
My focus area was in partial differential equations. I'm still doing this: when we do analytical fluid mechanics, we're dealing with partial differential equations (PDEs). PDEs can be tricky because there aren't well-structured techniques for finding solutions to them. As such, a lot of what mathematicians do is try to simplify them into known forms, or at least get ballpark estimates of what is going on.
One of the tools for doing such is nondimensionalization, a process of scaling. For equations derived from physical situations, primarily, this means taking parts of the equation that have units attached (i.e. the acceleration in Navier-Stokes Equations could have units of meters per second^2) and pulling the units out as separate coefficients. Essentially, one can "clump" all of the elements with units into a single term, which yields a nondimensional quantity.
One of the reasons I chose to be an applied math major in college was that I felt mathematics could provide powerful metaphors for making sense of the world.
My focus area was in partial differential equations. I'm still doing this: when we do analytical fluid mechanics, we're dealing with partial differential equations (PDEs). PDEs can be tricky because there aren't well-structured techniques for finding solutions to them. As such, a lot of what mathematicians do is try to simplify them into known forms, or at least get ballpark estimates of what is going on.
One of the tools for doing such is nondimensionalization, a process of scaling. For equations derived from physical situations, primarily, this means taking parts of the equation that have units attached (i.e. the acceleration in Navier-Stokes Equations could have units of meters per second^2) and pulling the units out as separate coefficients. Essentially, one can "clump" all of the elements with units into a single term, which yields a nondimensional quantity.
A famous example (in fluid mechanics) is the Reynolds Number, or density times velocity times length divided by viscosity. The units on all of these quantities cancel, so the number itself has no unit, or is "nondimensionalized" along with the rest of the equation.
What's the value in finding this nondimensionalized value? Firstly, it can allow us to neglect a term in the PDE, which may simplify things to the point where a solution is attainable. Secondly, it removes more arbitrary thresholds from the original system, making clear that adjustments in one parameter only matter relative to the others via how they affect the nondimensional number. This brings us closer to understanding how the scales of the system are intertwined.
For example, the Froude Number is useful when making scale boat models. A small model boat, run at a small velocity, will have a particular Froude number; a large boat with a large velocity might have the same Froude number, and with this information we can know that our model is accurate despite these changes in scale. Nondimensional values help us find inherent scales to the system at play. I use "Inherent Scale" to refer to either a value or a relationship between values values that is "baked in" to the nature of a system. These scales may not always be immediately apparent, so thinking in inherent scales means seeking out the "natural" parameters for a system and using them to structure otherwise arbitrary decisions.
(What's so cool about fractals? They have no inherent scale; there is no "correct" level of zoom. They obey a geometric power law that makes them self-similar or "scale free.")
One of my favorite things to discuss with my middle school math classes was about scale of biologic systems. Why aren't humans twice as tall? It boils down to the physics of heating/cooling, liquid pumping, the strength of bones: these are the scales that "control" the size of our design. Our being the "right size"—somewhere between 4 and 7ft tall—may be determinable via nondimensionalization of our metabolic and physical process.
Why does a park being open sunrise to sunset feel right? I think it speaks to the inherent timescale, imposed by the rotation of the earth, of outdoor recreation in the daylight. Anything else (10:00-17:00 except on holidays and every other Sunday, with some clause for daylight savings time) feels arbitrary in comparison.
Sometimes scales are imposed on us through standards or pre-existing conditions, and sometimes the scales need to be asserted to optimize for something else. But, when a system feels complex, looking to the inherent scales is a place to start noticing the structure therein.
Nondimensionalizing,
Lukas