Mastering Geometry is a Lost Art

I am trying to learn Quantum Field Theory the hard way: Alone and from text books. But there is something harder than the abstract math of advanced quantum physics:

You can aim at comprehending ancient texts on physics.

If you are an accomplished physicist, chemist or engineer – try to understand Sadi Carnot’s reasoning that was later called the effective discovery of the Second Law of Thermodynamics.

At Carnotcycle’s excellent blog on classical thermodynamics you can delve into thinking about well-known modern concepts in a new – or better: in an old – way. I found this article on the dawn of entropy a difficult ready, even though we can recognize some familiar symbols and concepts such as circular processes, and despite or because of the fact I was at the time of reading this article a heavy consumer of engineering thermodynamics textbooks. You have to translate now unused notions such as heat received and the expansive power into their modern counterparts. It is like reading a text in a foreign language by deciphering every single word instead of having developed a feeling for a language.

Stephen Hawking once published an anthology of the original works of the scientific giants of the past millennium: Corpernicus, Galieo, Kepler, Newton and Einstein: On the Shoulders of Giants. So just in case you googled for Hawkins – don’t expect your typical Hawking pop-sci bestseller with lost of artistic illustrations. This book is humbling. I found the so-called geometrical proofs most difficult and unfamiliar to follow. Actually, it is my difficulties in (not) taming that Pesky Triangle that motivated me to reflect on geometrical proofs.

I am used to proofs stacked upon proofs until you get to the real thing. In analysis lectures you get used to starting by proving that 1+1=2 (literally) until you learn about derivatives and slopes. However, Newton and his processor giants talk geometry all the way! I have learned a different language. Einstein is most familiar in the way he tackles problems though his physics is on principle the most non-intuitive.

This review is titled Now We Know why Geometry is Called the Queen of the Sciences and the reviewer perfectly nails it:

It is simply astounding how much mileage Copernicus, Galileo, Kepler, Newton, and Einstein got out of ordinary Euclidean geometry. In fact, it could be argued that Newton (along with Leibnitz) were forced to invent the calculus, otherwise they too presumably would have remained content to stick to Euclidean geometry.

Science writer Margaret Wertheim gives an account of a 20th century giant trying to recapture Isaac Newton’s original discovery of the law of gravitation in her book Physics on the Fringe (The main topic of the book are outsider physicists’ theories, I have blogged about the book at length here.).

This giant was Richard Feynman.

Today the gravitational force, gravitational potential and related acceleration objects in the gravitational fields are presented by means of calculus: The potential is equivalent to a rubber membrane model – the steeper the membrane, the higher the force. (However, this is not a geometrical proof – this is an illustration of underlying calculus.)

Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola

Model of the gravitational potential. An object trapped in these wells moves along similar trajectories as bodies in a gravitational field. Depending on initial conditions (initial position and velocity) you end up with elliptical, parabolic or hyperbolic orbits. (Wikimedia, Invent2HelpAll)

(Today) you start from the equation of motion for a object under the action of a force that weakens with the inverse square of the distance between two massive objects, and out pops Kepler’s law about elliptical orbits. It takes some pages of derivation, and you need to recognize conic sections in formulas – but nothing too difficult for an undergraduate student of science.

Newton actually had to invent calculus together with tinkering with the law of gravitation. In order to convince his peers he needed to use the geometrical language and the mental framework common back then. He uses all kinds of intricate theorems about triangles and intersecting lines (;-)) in order to say what we say today using the concise shortcuts of derivatives and differentials.

Wertheim states:

Feynman wasn’t doing this to advance the state of physics. He was doing it to experience the pleasure of building a law of the universe from scratch.

Feynman said to his students:

“For your entertainment and interest I want you to ride in a buggy for its elegance instead of a fancy automobile.”

But he underestimated the daunting nature of this task:

In the preparatory notes Feynman made for his lecture, he wrote: “Simple things have simple demonstrations.” Then, tellingly, he crossed out the second “simple” and replaced it with “elementary.” For it turns out there is nothing simple about Newton’s proof. Although it uses only rudimentary mathematical tools, it is a masterpiece of intricacy. So arcane is Newton’s proof that Feynman could not understand it.

Given the headache that even Corpernicus’ original proofs in the Shoulders of Giants gave me I can attest to:

… in the age of calculus, physicists no longer learn much Euclidean geometry, which, like stonemasonry, has become something of a dying art.

Richard Feynman has finally made up his own version of a geometrical proof to fully master Newton’s ideas, and Feynman’s version covered hundred typewritten pages, according to Wertheim.

Everybody who indulges gleefully in wooden technical prose and takes pride in plowing through mathematical ideas can relate to this:

For a man who would soon be granted the highest honor in science, it was a DIY triumph whose only value was the pride and joy that derive from being able to say, “I did it!”

Richard Feynman gave a lecture on the motion of the planets in 1964, that has later been called his Lost Lecture. In this lecture he presented his version of the geometrical proof which was simpler than Newton’s.

The proof presented in the lecture have been turned in a series of videos by Youtube user Gary Rubinstein. Feynman’s original lecture was 40 minutes long and confusing, according to Rubinstein – who turned it into 8 chunks of videos, 10 minutes each.

The rest of the post is concerned with what I believe that social media experts call curating. I am just trying to give an overview of the episodes of this video lecture. So my summaries do most likely not make a lot of sense if you don’t watch the videos. But even if you don’t watch the videos you might get an impression of what a geometrical proof actually is.

In Part I (embedded also below) Kepler’s laws are briefly introduced. The characteristic properties of an ellipse are shown – in the way used by gardeners to creating an elliptical with a cord and a pencil. An ellipse can also be created within a circle by starting from a random point, connecting it to the circumference and creating the perpendicular bisector:

Part II starts with emphasizing that the bisector is actually a tangent to the ellipse (this will become an important ingredient in the proof later). Then Rubinstein switches to physics and shows how a planet effectively ‘falls into the sun’ according to Newton, that is a deviation due to gravity is superimposed to its otherwise straight-lined motion.

Part III shows in detail why the triangles swept out by the radius vector need to stay the same. The way Newton defined the size of the force in terms of parallelogram attached to the otherwise undisturbed path (no inverse square law yet mentioned!) gives rise to constant areas of the triangles – no matter what the size of the force is!

In Part IV the inverse square law in introduced – the changing force is associated with one side of the parallelogram denoting the deviation from motion without force. Feynman has now introduced the velocity as distance over time which is equal to size of the tangential line segments over the areas of the triangles. He created a separate ‘velocity polygon’ of segments denoting velocities. Both polygons – for distances and for velocities – look elliptical at first glance, though the velocity polygon seems more circular (We will learn later that it has to be a circle).

In Part V Rubinstein expounds that the geometrical equivalent of the change in velocity being proportional to 1 over radius squared times time elapsed with time elapsed being equivalent to the size of the triangles (I silently translate back to dv = dt times acceleration). Now Feynman said that he was confused by Newton’s proof of the resulting polygon being an ellipse – and he proposed a different proof:
Newton started from what Rubinstein calls the sun ‘pulsing’ at the same intervals, that is: replacing the smooth path by a polygon, resulting in triangles of equal size swept out by the radius vector but in a changing velocity.  Feynman divided the spatial trajectory into parts to which triangles of varying area e are attached. These triangles are made up of radius vectors all at the same angles to each other. On trying to relate these triangles to each other by scaling them he needs to consider that the area of a triangle scales with the square of its height. This also holds for non-similar triangles having one angle in common.

Part VI: Since ‘Feynman’s triangles’ have one angle in common, their respective areas scale with the squares of the heights of their equivalent isosceles triangles, thus basically the distance of the planet to the sun. The force is proportional to one over distance squared, and time is proportional to distance squared (as per the scaling law for these triangles). Thus the change in velocity – being the product of both – is constant! This is what Rubinstein calls Feynman’s big insight. But not only are the changes in velocity constant, but also the angles between adjacent line segments denoting those changes. Thus the changes in velocities make up for a regular polygon (which seems to turn into a circle in the limiting case).

Part VII: The point used to build up the velocity polygon by attaching the velocity line segments to it is not the center of the polygon. If you draw connections from the center to the endpoints the angle corresponds to the angle the planet has travelled in space. The animations of the continuous motion of the planet in space – travelling along its elliptical orbit is put side-by-side with the corresponding velocity diagram. Then Feynman relates the two diagrams, actually merges them, in order to track down the position of the planet using the clues given by the velocity diagram.

In Part VIII (embedded also below) Rubinstein finally shows why the planet traverses an elliptical orbit. The way the position of the planet has finally found in Part VII is equivalent to the insights into the properties of an ellipse found at the beginning of this tutorial. The planet needs be on the ‘ray’, the direction determined by the velocity diagram. But it also needs to be on the perpendicular bisector of the velocity segment – as force cause a change in velocity perpendicular to the previous velocity segment and the velocity needs to correspond to a tangent to the path.

I neither Met Newton nor Einstein

I am just reading The Trouble with Physics by Lee Smolin. I am not familiar with string theory, quantum gravity and the related communities, so I cannot comment on Smolin’s main statement. But there is a section in the last chapter of the book that resonated with me. He describes his expectations and feelings when entering graduate school as follows: I was in awe of Einstein, Bohr, Heisenberg and Schrödinger and how they had changed physics though the force of their radical thinking. …. The great physicists I was rubbing shoulders with at Harvard were rather different from that. The atmosphere was not philosophical; it was harsh and aggressive…

This is actually what I had experienced, too – but would rather use the term competitive instead of harsh and aggressive for a good reason: Competitive is used to denote this type of personality or behaviour of so-called professionals in the corporate world and it has positive connotations there. Thus to make a long-winded blog post short and give you an executive summary: I see lots of similarities between the academic and the corporate world. I do not attempt to offer a thorough review but rather give you my personal thoughts that are not significant statistically.

As Smolin I had also bewildered and sometimes disappointed to find the typical physics professor or postdoc deviate so much from the idol I had in mind. Probably this was an idol I had simply made up or that has been implanted into my mind by too much movies and biased documentaries on science professors. First of all, it would be extremely interesting to time-travel to the beginning of the 20th century and talk to members of the physics community back then. Heisenberg and Schrödinger might have quite competitive as well and not only driven by the philosophical quest to unveil the mysteries of the universe. E.g. the historian of science, Peter E. Fischer mentions Heisenberg’s and Schrödinger’s ambitions at the beginning of this article (in German). BTW he states that great results with be achieved (in physics) if there is a central major person with unquestionable high ethical standards and personal integrity; regarding quantum mechanics this was Niels Bohr according to Fischer.

When I am musing about this today I put it into a different context: Comparing “academia” to the “corporate world” (and by this I mean the world of large global corporation) I so not see so much difference with respect to selection mechanisms, the importance of self-marketing, and competitiveness. Both are complex systems and governed by certain rules in how your  status and success is being measured, such as number of papers, grants, citations – in academia – and utilization (AKA billable hours) and customer satisfactions in the corporate world. The latter example is taken from consulting business. Probably scientists will find this comparison insulting. I just can tell you that as an individual I felt exactly the same way when one or the other system has inflicted its laws on me. I even found the corporate world a bit easier to deal with for a probably unexpected reason – at least unexpected to me: In the corporate world it was about money and beating the competitors – phrased in even military language that turned conferences and meetings into airlifts, boot camps and war rooms.

In academia the targets you had to meet were not nobler (e.g. by cutting your big paper into a lot of small papers to increase the number of publications) but the official goals were: It was about increasing the knowledge of humanity, improving the standards of living of millions of people … or whatever was on the agenda of the large research programmes funded by governmental agencies (AKA the tax payer). I hated the part of grant applications most that required me to make up the statements on “social and economic improvements”. The glaring contradiction between these noble goals and motivations of researchers (*) in real life was one of the reasons I left academia.

I feel obliged to add some disclaimers here: Yes, I thought about “staying in the system and changing it from within” and yes, this is a personal statement only true to myself and it might not be applicable to any other scientist’s life. And yes, I did really well with regards to benchmarks and numbers in any of the systems according to whose rules I felt forced to play.
(*) I do not all say these are the scientists’ true intentions in the same way as corporate goals typically do not reflect individuals’ motivations. But I still think it is quite hard to meet the targets and keep up your true motivations as sort of hobby. I noticed that some people can deal better than others with this kind of divide. I did not.

Most ironically, academia and the corporate world are converging in a way that I did not expect 20 years ago:

  • Also corporations officially turned into noble entities, all “social”, “green” and whatnot. As I said I can deal better with a so-called greedy capitalistic company that does not cloak its intentions by having the CEO or his ghost writers blog on corporate social responsibility. Thus the corporate world is exhibiting the same sort of divide between metrics and targets on the one hand and noble ambitions on the other hand. (And yes, I know that corporations are now also attempting to measure success in terms of the noble goals – but I believe this will turn these goals in just another number to be met. I am a bit obsessed with metrics because I was so good at meeting them, but I might get back to that in a future blog post.)
  • The infamous metrics invented in the corporate world trickled into academia via management consulting wisdom. Even similar tools are used to manage numbers in the academic and the corporate world. Think SAP implementations at universities. (My favourite reference is in German again – philosophy professor Konrad P. Liessmann on  misapprehension of “knowledge society”).

Theoretically I should add a conclusion – after all what is left today for ambitious persons of integrity who want to make a difference? Do I really dare say that both large corporations and academia are built on sort of dysfunctional metrics? No, of course I do not because they have been far too successful applying whatever metrics including fluffy, philosophical goals – so far. But I believe that obsession with metrics in terms of reporting, forecasting, benchmarking, quality management etc. is going to extreme levels now and we have just seen the beginning. I sometimes wonder how big corporations can still be productive on top of administration after all and I have heard and read a lot of people in academia complaining about ever-increasing administrative duties.

Then I am guilty as well: I been part of both and what I learned from both and my so-called track records are the basis of what I am trying to do now. I have neither been as consequent and enduring to avoid these systems right from the beginning or spend all my life in one of them and try to make a change as a fully accountable member of the community.

I can only add some preliminary considerations here: I believe  groupthink and alignment of members of communities have become too strong today – in whatever aspect of life. Smolin points out that this is specific to the String Theory community but I believe it is a ubiquitous phenomenon. Metrics thinking e.g. seems to be contagious and it spreads from one organization to the other and all managers or metrics consultants seem to get the same training (My theory is that management systems that are used to keep all the data required to create benchmarks are selected by asking other managers on the golf course – I found some anecdotal evidence for that). I have worked in one of those industry sectors that is typically fast-paced and I was member of a very strong sub-community. It’s weird but what Smolin says about the String Theory physicists could be applied perfectly to that tech community I have in mind. I like especially his references to the slightly cult-like nature of the communities.

Despite and because of the options we have to “connect with each other” (in social media marketing lingo) we should try – harder as ever – to keep up our independence. I know this sounds a bit vague. I might expound in detail what this means for me personally in this blog, for the time being it is important for me that I was able to really experience how it feels to be part some communities or big systems. Otherwise the grass might always be greener over there.

Many years ago I literally said good-bye to the specters of the great physicists by visiting the physics library for a last time before I left. I bemoaned the lack of career opportunities for young scientists in the speech I gave at my graduation ceremony. I said that I once had thoughts as a physicist I would ponder about how the results of my research would be used (I argued quite pathetically actually: fight cancer or design weapons), but then I had realized that most young scientists are happy if they work in a job loosely connected with science at all. And philosophical musings meant luxury that would be affordable only to the lucky few who have made it.

Having “made it” now in a more money-oriented world I am gradually realizing my personal vision of a science-oriented job and I am indulging in all kinds of philosophical speculations .  (No, I am not Mike Lazaridis and I am not founding a research center. And no, I am also not working on sort of Theory of Everything, I hope my personal crackpot index is zero, although I referred to Einstein in this post. My “science” or “research” topics are so menial I do not dare yet to mention, at least not in this post that started with a referer to a book on quantum gravity.).

So I have found some sort of solution and I will try to investigate if their might be solution that scales to a larger number of physicists. I have not made my mind up, but generally I do not like solutions that only work for the lucky few and that require serendipitous conincidences.