RIPE: Reconstructing Reality and Material Structure

In Plato’s allegory of the cave, he imagines chained prisoners facing a blank wall. A raging fire burns far in the background, behind their backs. The prisoners watch shadows cast by objects passing in front of the fire on the wall ahead of them. They form ideas about the true nature of the objects, from these shadows.

One can only guess the true nature of the objects from these shadows. The 2D shadows that 3D objects cast, project some information, and conceal the rest.

For example, a sphere and a cylinder can both cast a circular shadow.

Perspective is everything (google giraffe or elephant illusion if video below doesn't work).

This problem bedevils characterization of materials as well. A particular measurement or test (chromatography, rheology, thermal tests, scattering, microscopy etc.) gives us but one look at an unknown sample.

If we have a fairly good idea of what the material might be, to begin with, then perhaps this is enough.

More often than not, especially with new materials, each new test narrows the scope of the possible.

The fable of the six blind men and the elephant (which is perhaps a more pedestrian retelling of the Plato's allegory) provides a potential path out. In the story six blind men touch different parts of an elephant and form radically different notions of what an elephant is.

The only way (for the blind men) to come up with a more realistic notion of what an elephant looks like is to find ways of combining their knowledge to come up with a consensus view.

Analogy: Matrix As a Tree

When many of us first encounter matrices, we see them as a glorified table of numbers stuffed into special square brackets.

This is analogous to looking at a tree as a collection of biological cells.

It is not incorrect; but it is perhaps a tad too granular.

One can ask, "what is the function or application of a tree or a matrix?".

For a tree, the answer may be fruits, lumber, home for birds and animals etc.

For a matrix, it may be to flip a vector, stretch it, rotate it, or some combination thereof. Here we are thinking of the tree or the matrix in terms of its relationship with other things.

One could ask what the main structural parts are.

For a tree, it may be roots, trunk, branches, and leaves.

For a matrix, it may be useful to think of it in terms of a collection of column or row vectors, and the vector spaces that they define.

One could then ask a category question: "What kind of a tree is this?"

Is it an oak tree, a fruit tree, an evergreen, a gymnosperm, etc focusing on its distinguishing properties, and the categories that it belongs to.

Similarly, one can identify different types of matrices. Is a matrix symmetric? Invertible? Triangular? Sparse? What other trees and matrices are they related to.

More Thoughts on Modeling

Let me relate a story to set up my thinking for this post:

During my PhD at Michigan, I worked on some approximate models for polymer dynamics called slip-link (SL) models. This (SL) model was more accurate than the standard theoretical model. As expected, it took greater computational resources to numerically simulate my model (a few hours) than the standard theory (a few seconds).

There are some applications where this trade-off between accuracy and speed is desirable.

Of course, there are other richer models, which are more accurate and even more computationally expensive than mine.

One of the people on my committee asked me: "If computational speed were infinite, would anyone care about your model?" I don't remember exactly how I responded; my guess: some mumbled gibberish.

But this is indeed a profound question that touches upon what I said recently about seeking too much accuracy in models. If I can numerically compute the most accurate model available for something, should I waste my time with alternatives?

Let's set aside the hypothetical "if computer speed were infinite" part of the question, and work under the premise that such computers were indeed available.

Should we then simply use ab initio quantum mechanics, or perhaps, the standard model of physics (whatever that is!) to study everything?

But seriously, if you want to study migration of birds, or mechanical properties of a spaceship, or the ups and downs of a business cycle, would you really want to study it in terms of quarks?

As Douglas Adams pointed out in the Hitchhikers Guide, our computer model may give us the "Answer to the Ultimate Question of Life, the Universe, and Everything", and yet we may be unable to comprehend it.

Misquoting Richard Hamming, "the purpose of modeling is usually insight, not numbers."

Thoughts on Modeling

The best material model of a cat is another, or preferably the same, cat.
Norbert Wiener

Generally speaking, accurate models are a good thing. Unless, they are useless! Allow me to explain.

Consider my fair state of Florida. As Wiener claims, the most accurate model of Florida is Florida itself. But is it the most useful?

If I am a tourist visiting the state, the best model may be a Rand McNally road map.

If I am a hydrologist, the best model may be a 3D map where the waterways, the aquifers, and sinkholes are accurately represented.

If I am a political strategist, the best model may be a county/congressional district map colored in red, blue and purple.

If I am a epidemiologist, the best model may be a population density map.

A useful model for something successfully captures the key elements of that something, and de-emphasizes the rest. Judgment of the quality of a model cannot be unhooked from its intended purpose. In other words, the pursuit of accuracy for accuracy's sake is a classic beginner's mistake.

Good models separate the essential from the nonessential. Approximating or neglecting the nonessential is a skill. Approximation is a fancy way of saying, I am trading accuracy for clarity, insight or tractability.

As an aside, good cartoonists do this routinely: they exaggerate one or two key features, and and push the rest into the background. 

Essentially, all models are wrong, but some are useful.
Box and Draper