Math and Yoga

Insights on creativity

I’ve been practicing Iyengar yoga for over a year now, and the journey so far has been extremely rewarding. I’ve discovered some fascinating parallels with the intellectual, creative realm of mathematics.

In any field, understanding first principles is valuable: what are the basic ideas that you know to be true?

In math, there are things like Euclid’s five axioms in geometry — but also more philosophical ideas of assuming as little as possible, demanding rigorous proof of theorems, and obtaining general theorems with broader scope rather than special cases (Taylor series over Maclaurin series, for example).

Likewise, as one remarkable polymath explained to me, “the main aim of yogasana is to keep the spine straight.” Another key, axiomatic insight in yoga is to focus on exhalation: if you exhale well, the inhalation will follow naturally.

Things become even more fascinating when you discover the idea of connectivities. My yoga instructor once commented, “paschim namaskarasana (reverse prayer pose) holds the key to mastering shirshasana (headstand).” I was intrigued: such connections come up all the time in math!

For instance, complex numbers are composed of a real and imaginary part, and can be plotted on an Argand diagram, which is simply a plane with the real part on the x-axis and the imaginary part on the y-axis. De Moivre’s theorem says that if you raise a complex number to the power of n, the modulus (distance of the point from the origin) is exponentiated and the argument (angle made with the x-axis) is multiplied by n. This can be understood in any number of ways — from Taylor series expansions to proof by induction — and leads to hyperbolic trigonometry and Euler’s identity.

Euler’s equation, often called the most beautiful equation in math

Just as there are many ways of proving a mathematical theorem — Pythagoras’ famous theorem has more than 350 different proofs! — there are multiple ways of observing an asana. For instance, one can enter adho mukha svanasana by jumping back from uttanasana, or by raising the knees upwards from adho mukha virasana. As with mathematical proofs, each such path reveals and illuminates a different facet of the asana, or theorem.

One interesting element I’ve noticed in classical, Iyengar yoga is that the asanas themselves are static — for instance uttanasana as opposed to, say, alternate toe touch — but they are dynamic in the sense that the sharpness and focus in the pose are supposed to enhance with every exhalation.

I also find that with both math and yoga, pedagogical style is crucial. Especially, does a learner adopt a static, fixed mindset or a dynamic, growth mindset?

There will be asanas one cannot yet perform and theorems one cannot yet prove; the key is that one must not look at others who can seemingly do these “effortlessly” and think of oneself as being somehow “not smart enough” — instead one must ask, what am I missing? How can I get better? Or, as Christopher Begg says, persistent incremental progress eternally repeated (PIPER).

Here are two quotes from Guruji BKS Iyengar, the first from Light on Yoga and the latter from Light on Life.

“By performing asanas, the sadhaka first gains health, which is not mere existence. It is not a commodity which can be purchased with money. It is an asset to be gained by sheer hard work.”

“Physical health is not a commodity to be bargained for. Nor can it be swallowed in the form of drugs and pills. It has to be earned through sweat. It is something that we must build up.”

Cal Newport’s wonderful podcast with Andrew Huberman has a debate on deliberate practice v/s flow — deliberate practice is required to attain hard new skills (0 to 1), while a flow state is experienced when repeating what one is already proficient at (1 to n). In both, math proofs as well as yoga asanas, once one has mastered a certain technique or insight, it is easy to repeat — but the acquisition of that expertise to begin with requires sustained deliberate practice, or sadhana.

As you sit comfortably on a chair, each and every pose, when, with effort, perfected, it is as comfortable as one sitting on a chair.

— Guruji BKS Iyengar in a 1976 demonstration

Let me end with this extract from a biography of Leonardo da Vinci.

What made Vitruvius’s work appealing to Leonardo and Francesco was that it gave concrete expression to an analogy that went back to Plato and the ancients, one that had become a defining metaphor of Renaissance humanism: the relationship between the microcosm of man and the macrocosm of the earth.

This analogy was a foundation for the treatise that Francesco was composing. “All the arts and all the world’s rules are derived from a well-composed and proportioned human body,” he wrote in the foreword to his fifth chapter. “Man, called a little world, contains in himself all the general perfections of the whole world.” Leonardo likewise embraced the analogy in both his art and his science. He famously wrote around this time, “The ancients called man a lesser world, and certainly the use of this name is well bestowed, because his body is an analog for the world.”

— Walter Isaacson, in Leonardo da Vinci

Feedback and reading recommendations are invited at malhar.manek@gmail.com