The Secret to Powerful Math is Often a Simple Diagram

By Keith Devlin @profkeithdevlin

This month’s essay continues a thread commenced with last month’s post.

I gave a guest lecture at a local retirement community recently, my first in-person speaking gig since the start of the pandemic. Located in the heart of Silicon Valley, my audience was, as I had initially assumed (and was confirmed during the initial exchange of emails with the organizer), a highly educated group; indeed, some of them were former professors from Stanford and elsewhere. But for the most part, they were not particularly knowledgeable about mathematics. That’s not unlike the non-science-majors classes I had taught at Stanford and Princeton. Knowing what to expect, I had picked a topic I’d used often in my college-teaching career: the simple — indeed simplistic — picture that often lies behind powerful mathematics.

After my talk was over, most of the audience stayed around for some time to chat. One of the group (a former Humanities professor, I forget which subject) remarked that he’d always thought of mathematics as making things complicated, but after my talk appreciated that the goal is indeed to make complicated things simpler. (With the pursuit of our narrow career goals behind us, we retirees can spend much more of our time looking at other disciplines. Many of us try to do so throughout our careers, of course, but the drug of pushing one’s own disciplinary boundaries is hard to resist, particularly in elite research universities where that is the primary research goal.)

Fig 1. The Bohr Model of the atom, proposed by the Danish physicist Nils Bohr in 1913.

That retired Humanities professor’s conclusion was right. The modus operandi of mathematics always was, and continues to be, to view aspects of the world in a highly simplistic way; a way that permits fine-grained, precise analysis that can (among other things) provide a framework and a tool for great science and engineering. This was in many ways one of the main points of my post last month.

A great example is the high-school picture of the atom, shown in Figure 1. Presented by physicist Nils Bohr in 1913, this image — or more precisely the conceptual model it represents (usually referred to as the Bohr model) — views the atom as a system consisting of a small, dense nucleus surrounded by orbiting electrons, similar to the structure of the Solar System, but with attraction provided by electrostatic forces in place of gravity.

Given that one of mathematics’ most powerful tools, Differential Calculus, was developed in large part to make a precise analysis of the motion of the planets, it should come as no surprise that the Bohr model proved to be an extremely valuable way to think about atoms. It is the image of the atom that science teachers still present in schools around the world, and to this day, it remains the image that comes to most laypersons’ minds when they think of atomic structure. It’s also featured in the logo of the International Atomic Energy Agency (Figure 2), an entity that knows full well the atom isn’t remotely like that.

Fig 2. The Bohr Model of the atom is the logo of the IAEA, despite the fact that it is not a correct image of the structure of an atom.

Yes, you read that right. Whatever the atom looks like (it’s not clear what exactly such a statement would even mean), it is certainly not a minature Solar System. In fact, the life of the Bohr model as the way physicists think of the atom lasted only a handful of years, an incredibly fleeting moment in the development of modern science, its transiency captured insightfully, elegantly, and wittily by physicist Banesh Hoffman in his superb 1965 Penguin Book The Strange Story Of The Quantum, where he presents two consecutive chapters titled The Atom of Niels Bohr and The Atom of Bohr Kneels.

Before the Bohr model, as chemists and physicists gathered more and more evidence to show that matter must have an atomic structure, physicists and mathematicians proposed a number of other mental models on which to develop a mathematical atomic theory: Joseph Larmor’s solar system model (1897), Gilbert N. Lewis’s cubical model (1902), the Hantaro Nagaoka Saturnian model (1904), J. J. Thompson’s plum pudding model (1904), Arthur Haas’s quantum model (1910), the Rutherford model (1911), and John William Nicholson’s nuclear quantum  model (1912).

With theoretical and experimental results of Rutherford and others coming thick and fast during that time, each new model struggled (and in short order was shown to fail) to account for new observational data. (Bohr’s model improved Rutherford’s model by allowing for the quantum physical interpretation introduced by Haas and Nicholson.) The Wikipedia article on this exciting period provides a good overview, including descriptions of each of the models listed in this essay.

When problems quickly arose with the Bohr model, further modifications were proposed, most notably Arnold Sommerfield’s suggestion that the data was better accounted for if the electrons were assumed to follow elliptical orbits around the nucleus, rather than the circular paths in Bohr’s model, but it too soon failed.

And in fact, all attempts to develop a mechanical, “Solar System” inspired approach were, in just a few years, abandoned, in favor of a picture of the atom as a spherical cloud of probability, which grows denser near the nucleus. Atomic physics entered “the Quantum Age,” where probabilities rule; a development that led Albert Einstein, in 1926, to protest in frustration, “God does not pay dice with the universe.”

Well, it turns out that, in Einstein’s terms, “God does play dice”. At least, the Quantum Theory of matter turns out to be better than the Bohr Theory, as illustrated by the solid state memories in our mobile phones and various other technologies built using that theory, that are essential components of today’s societies.

Yet, of all the different models that came and went, it was Bohr’s that lived on, and continues to do so. The reason is that it still provides a good mental image on which usefully to understand phenomena and to guide thinking, in chemistry, physics, engineering, and biology. Yes, it’s not “correct”; but it is (extremely) useful. Just as the planetary theory-based Newton’s mechanics is still useful in space exploration, even though Einstein’s work showed it too is “wrong.”

The point is, a good model is usually worth it’s weight in gold  (or the far heavier Uranium and Plutonium in the case of the Bohr model), and frequently turns out to be the gift that keeps on giving. It’s a gateway that connects the simple-pattern-recognizing, sense-making, conscious human mind to the complexity of the world we live in.

And that, from my present perspective, is the point I want to make. Faced with new theoretical (i.e., mathematical) developments and experimental results emerging from physics laboratories (mostly in the UK), scientists trying to understand the nature of matter strove to come up with a simple picture; a picture that provided a way to understand the new results from experiments and which could serve as a basis on which to develop new experiments to advance their new Atomic Theory still further.

The Bohr model lives on because its particular simplicity (the features it includes and the ones it omits) makes it extremely useful. Knowing its limitations and how to overcome them, as we now do, simply increases its utility. Starting with a simple ­— indeed simplistic — model (a mental picture), deep, complex, powerful, and useful mathematics can be developed. You just have to find the “right” model for that to follow.

Though I was not present at the time, and have not read any reports to this effect, I am sure that part of the reason those early-Twentieth-Century physicists looked at Solar-System-inspired models was their expectation that doing so would enable them to use the most powerful, relevant mathematics known at the time: calculus.

Fig 3. The Mathematics Cycle. See corresponding text for link.

As I discussed in last month’s essay, mathematics gets called upon when humans try to understand various features of the world in a manner that allows high precision and accuracy. Getting to the top right node of the Mathematics Cycle (Figure 3, discussed in that essay) requires knowledge of, and familiarity with, the available mathematical tools, or at least some idea of new mathematical tools that could be developed. Hence the adoption of a Solar System framework for the second, “Technical Description” node of the cycle, made with an eye to being able to use the powerful tools of calculus.

An analogous trip round the Mathematics Cycle was made many centuries before the Bohr model, when the mathematicians of Ancient Greek made an earlier attempt to explain the structure of the universe using the mathematics available back then: Euclidean Geometry.

Inspired by the famous five Platonic solids of Ancient Greek Geometry, Plato suggested that all matter in the Universe consists of four elements: Air, Earth, Water and Fire, and that each of those elements corresponds to one of the five Platonic solids, with the fifth solid representing the universe as a whole. (Figure 4.)

Fig 4. Plato’s model of the universe, based on the five “Platonic solids”. The images are from Johannes Kepler’s book Harmonices Mundi, published in 1619.

The tetrahedron, with its sharp points and edges represented the element fire; while the cube, with its four-square regularity represented the earth. The octahedron and the icosahedron, concocted from triangles, represented air and water, respectively. The dodecahedron, with 12 pentagonal faces, represented the heavens with its 12 constellations.

From today’s perspective, it all seems very fanciful, but in fact is no more fanciful for the time than the Bohr model that came two millennia later. Both are attempts to use simple, orderly, and easily understood mathematical structures, as a way to make sense of the world around us.

Indeed, throughout history, whenever mathematicians have introduced a new mathematical structure, even if that structure was not directly inspired by some scientific question, scientists would, sooner or later, look to see if it provides a useful way to think about (i.e. to model) some aspect of our world.

For instance, in 1771, Alexandre-Théophile Vandermonde set the stage for the modern mathematical theory of knots by recognizing that their topological properties provided a systematic way to classify them. Building on that insight, in the 19th Century, Gauss defined the linking integral, a numerical invariant that represents the number of times a knot winds around itself. That provided the basis for developing a mathematical classification of knots.

Sure enough, soon after Gauss’s contribution, the British physicist William Thomson (later Lord Kelvin) proposed his Vortex Theory of the atom. The theory assumed that the basic building blocks of matter were knots in the ether, a hypothetical substance that permeated space.

According to the theory, every element (hydrogen, oxygen, gold, and so on) was made from a different kind of knot. That assumption fulfilled some key requirements for a system of building blocks for matter: many types of knots can exist, comprising many elements, and knots could be linked to form molecules. (Recent discoveries by chemists had shown that the hypothesized “atoms” of matter came in only a small number of varieties but in very large numbers.)

Fig 5. A table of all prime knots with 7 or fewer crossings (excluding mirror images).

In 1885, a Scottish physicist called Peter Guthrie Tait initiated the creation of knot tables (Figure 5) to provide a complete classification, publishing his own table of knots with up to ten crossings.

He created, in essence, the basis for a periodic table of knots in which hydrogen would be the "unknot" — the unknotted circle — and heavier elements would be knots whose filaments crossed over themselves an ever-higher number of times.

Like the Solar System models that came later, the Vortex Theory did not last long, as the discovery of more elements and the periodicity of their characteristics established captured by the Periodic Table made it clear that classification of knots would not support an adequate model. Yet, between 1870 and 1890, around 25 prominent British scientists published some 60 or so scientific papers on the theory, before it was completely abandoned.

Interestingly enough, physicists turned again to knot theory in their attempt to understand matter, this time much more recently, as a key tool in contemporary String Theory. Knot Theory has also found contemporary use in biology, chemistry, and several other disciplines. 

Still another famous example of how a simple picture can guide the development and use of deep, complicated mathematics, this one from subatomic physics, is the Feynman diagram, a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. (Figure 6.)

Fig 6. A generic Feynman diagram. See text for details.

The representation is named after the famous physicist Richard Feynman, who introduced it in 1948. In many, varied instances, a Feynman diagram can provides a simple visualization of a highly abstract formula developed by advanced theory.

Developed to be applied to quantum field theory, they can also be used in other fields, such as solid-state theory. The physicist Frank Wilczek noted that the intricate calculations that won him the 2004 Nobel Prize in Physics “would have been literally unthinkable without Feynman diagrams, as would [his] calculations that established a route to production and observation of the Higgs particle.”

The point I am making is that outsiders who see only the highly polished, mathematically-heavy developments of scientific theories often have no idea that the thinking that originally led to those theories rested on highly simplistic mental images.

Those mental images were in fact the driving core for the deep thinking those scientific pioneers engaged in. The essence of a successful scientific theory is what features of the phenomenon of interest the theory should focus on, and what should be, if not ignored, then at least in the early stages, put to one side. Once you have a good grasp of how the chosen model (mental picture) relates to the actual phenomenon, you can start to develop your theory (story). That part can rapidly become deep, abstract, and complex. The underlying model prevents you from getting lost in the complexity of the fine details.

To finish on a personal note, when I read the Banesh Hoffman book The Strange Story of the Quantum (cited earlier) as a physics-mad, new mathematics undergraduate, when the book was first published in 1965, I was inspired by the torrent of collective human creativity in the early decades of the Twentieth Century that it described. (I had chosen King’s College London for my mathematics degree in large part because the famous theoretical physicist Sir Herman Bondi was a Professor of Mathematics there.) I little knew that many decades later I would be involved in an analogous struggle to develop a simple picture on which to base a mathematical theory of another aspect of human life whose time had come: information.

That story will be the focus of next month’s post.

to be continued . . .