Propelled by Beams of Intellectual Light

One frustrating thing about being a mathematics teacher is the difficulty of conveying to young students the sometimes terrifying giddiness of plunging deep into mathematics. There is an awesome sort of thrilling vertigo associated with trying to understand, and work through, high level mathematics.

The cool thing about mathematics is that it is endlessly capable of providing such a thrill, no matter what your age or talent, no matter what level of ability you already have. There are also many different paths one can explore to get these adrenalin rushes.  Godel’s incompleteness theorems loosely suggest there is no end to the depths and heights of mathematical investigation.  There will always be a need for new distilled crystallized axioms that try to best express our most basic and unquestionable mathematical presumptions.  A possible future might even see multiple parallel universes of mathematics, pure imaginary worlds that can never collide because their alternative fundamental axioms will never be able to be proven to be across-world consistent, and yet which cannot be proven to be inconsistent.

One recent path I took was reading about some recent discoveries from the papers of Srinivasa Ramanujan. Ramanujan’s work is one of the most amazing collections in mathematical history. Not always the most applicable to modern technology (hardly any physicists have ever made use of Ramanujan’s results), but as pure abstract journeys of the mind Ramanujan’s work stands almost unparalleled in history.


Ramanujan’s manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. An equation expressing a near counter example to Fermat’s last theorem appears. Image courtesy Trinity College library. (From: )

The analogy I conjured up was that of climbing Mount Everest without ropes or oxygen. Getting deep into mathematics can be that terrifying. You constantly get the piercing anxiety of, “I will never understand this!” Everyone knows this feeling, because school mathematics is still compulsory in most countries. Everyone hits this barrier at some stage. No matter how good they are with mathematics. People only vary in when they get to such a wall.

People who hit this wall early probably bifurcate: they either haul themselves over the wall with gargantuan heart-pounding effort, and continue on to excel in mathematics or sciences, while the others get daunted by the height and cannot see the other side and give up.


An image I borrowed from blog, it seemed to capture the physical/intellectual interconnections in our minds, plus a sort of infinite mental expanse quite well.  I didn’t want to post a brain explosion image!  That might give the wrong impression.

This is, I know, too simplistic a picture, but I think it captures the psychological impact of mathematical learning for many people. Sometimes you can hit several walls, and maybe only the sixth or seventh seems insurmountable, and so you retire grand mathematics ambitions and turn to maybe history or teaching or a relatively safe branch of applied mathematics. But even going down those seemingly safe avenues there are walls of unimaginable height and beauty that can disturb you.

And this is part of the wonder and beauty of mathematics.

The other imagery that came to my mind was the primal fear the astronaut Chris Hadden spoke about when sitting confined in a Space Shuttle launch cabin and getting the giant or all kicks up the backside when the main rockets ignited and hurled him into orbit. (By the way; Hadden recorded a pitch-perfect cover of David Bowies’ “Major Tom” track while in orbit on the ISS, one Bowie himself described as, “the best cover he had heard.”) Going by recent history, there is only a 1 in 30 to 1 in 40 chance of surviving such rocket launches. So it is a fantastic gamble deciding to be an astronaut. It is amazing people still volunteer, considering robots are almost capable of performing most of the tasks needed in space missions. Why take the risk?

If you talk to Hadden and his colleagues I’m sure they will tell you it is dozens of times over worth the risk. Just watching the Earth slowly rotate underneath in the vastness of black space is something that seems to change the soul.

With surviving great overwhelming terror comes profound spiritual awareness.

The terror can be purely mental, it does not have to be physical. But there is a fascinating connection here in the human brain. Terror and other similar deep emotions like fear and envy, arise in the amphibian primitive centres of our brains, the amygdala and hippocampus, while the impact rises up to higher conscious brain functions and we can sometimes get an experience of an inner world of abstract delight and insight when these primitive regions are stimulated. (I know the mappings of brain regions to psychological states is not as simple as this, so neurologists please do not hassle me about this, q.v.  The Amygdala Is NOT the Brain’s Fear Center, by Joseph DeLoux, Psychology Today, 10 August 2015. The amygdala is more correctly merely a threat-response system, it is not a source of conscious fear, the amygdala merely contributes in small part to a more neocortex driven feeling of fear or fright.)

The flight-or-fight response originates primarily in the amygdala, and it is an unconscious response. The consciousness of being in sheer panic or rage filters up to higher brain regions only after a few seconds or moments, which is neurologically a fairly long time — at least a few dozen or hundred cycles of 40 Hz brain wave activity. But we are eventually consciously aware of our responses. What the conscious systems do with these feelings is then a complex matter. Some people are able to thrive on the fear or rage and go deeper into the rabbit hole. Others rebel and go for safety. So perhaps a whimsical caricatured “difference between” X-Game competitors and Wingsuit flying daredevils and a mathematical genius is only the type of stimulus they fly form or dive into. Get anxiety from heights or open spaces or hanging upside down then you might be more of a mathematician. Get anxiety from an undecipherable maze of symbols on paper that are demanding decryption, and feel ill at the hopelessness of untangling them, then you might be more of a skydiver or rock climber.

So I wonder if the act of doing mathematics has a tremendous amount of associated unconscious neural activity? I wonder if this translates into a thrill and adrenalin rush when some insight is gained and a forbidding intellectual wall appears to crumble and a new revelatory insight into the world of mathematics is unveiled? And I even wonder if this can be addictive?

Whatever your inclination, when you next hear about a mathematical or scientific breakthrough, spare a thought for those who made those endeavours possible. For every breakthrough there are hundreds or thousands of researcher’s who will never get the accolades and awards, but who daily put themselves through the anxiety-ridden turmoil of smashing their minds up against intellectual barriers and paralyzing laser grids of the mind, or who feel constantly like they are falling from infinite heights of mental anguish and never know when the fear will cease. But all they are doing is sitting or pacing around in their laboratory or study wondering desperately where the much needed inspiration will come from to rescue them from the impending calamity of intellectual loss.

When the magical insight arrives, if ever, then the risk of the depression becomes all worth it, because the thrill of insight and discovery in the invisible planes of abstract theory and intellectual monuments is like being driven across the cosmos on beams of light. The journey is an expansion of your mind, it receives new ideas, allows your brain to form new connections, and opens up fields of intellectual inquiry previously barred. The propulsion system is imagination, insight and, dare I say t, some sort of spiritual impulse. If you are a physicalist then I suppose there are neural correlates for all of this — and you may think of it all as non-miraculous if it makes you feel better —, but think deeply and you might realise there is something more. Doing truly insightful mathematics or science does, I think, at the very forefront and apex of human endeavour, bring something new into the physical universe. From where it comes is perhaps unknowable. You might admit, if you have been touched by real inspiration, that perhaps, just possibly, maybe even likely, there is a world of imagination and abstraction beyond our physical reality, perhaps even closer to us that the atoms constituting our body, perhaps more like the essence of our selves, an existence our body and brain are merely borrowing temporarily ‐ or the converse, depending on your point of view.

This is what is so hard to explain to young children and even mature students. To explain the feeling of experience of these intellectual thrills, and to even hope to remotely compare them to physical danger and excitement, is incredibly difficult for a teacher. These are in the realms of “you really have to experience it for yourself.”

To replicate such intellectual adventure in a classroom is one of the prime responsibilities of a teacher. Yet our schools suppress most attempts in these directions, sadly. I call upon call teachers to put away textbooks and exam-preparation sessions, and replace them with adventures into mathematical depths that offer no clear or easy chance of escape. How you motivate such exploration is up to you, all I can say is try it! Just give your students freedom to explore. Then be prepared to catch them with your firm gedanken safety rope when they cry out in terror!


The Man Who Knew Infinity“, by Robert Kanigel, Abacus Books, 1992 (See goodreads.)

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A Beautiful Folding — and the Rise of Transformers

If you want to treat your brain then try watching the MIT lectures by Professor Erik Demaine over at 6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2010). Not sure if that was the most recent year his course was offered, but I’m sure you can find the latest version. I will not update this post or any links in any of my blogs, so as always, just Google the key words and you are bound to find what I’m pointing you at.


Among many cool results, the two prompting me to write this brief post were:

  1. The universality result that there is a crease pattern from which any modular cuboid polyhedron can be folded.
  2. The self-folding paper construction: a crease pattern can be folded in any way by electrical current stimulation. So we have Origamistless origami.

Ergo: the age of Transformers is upon us! Hahahaha!

Too bad artificial consciousness is not a paper fold.


I dunno man. … you see Demaine and his Dad with huge smiles on their faces, glass-blowing. folding cured crease patterns and chatting with John Conway and other legends, and you have to almost cry at the beauty of it all. So much life, so much joy, such intense devotion to art and science.


Oh yeah, … how many mathematicians have their work on permanent collection at MOMA?

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