The thing about π

Consider the following setup, from the game ‘Factorio’, the game about factory management and automation:

There are two factories visible in this image – the two rectangular, green-walled buildings. Take the one on the left: it’s manufacturing electric furnaces, with steel plates, stone bricks and advanced circuits as ingredients. These three resources are visible on conveyor belts leading up to the factory (top, left, bottom resp.), terminated by blue and green inserters that move the objects from the belts to the factory floor.

In order to maintain a steady supply of electric furnaces, I need to keep the ‘resource pressure’ up. Think of it like a strong wind blowing against your window: even if you opened the window just a little, there’s enough air pressing on that side of the wall for a lot of it to flow into your room. Similarly, I need to make sure sufficient quantities of steel, stone bricks and advanced circuits are available whenever the factory needs it. And within Factorio, as in the real world I imagine, maintaining this resource pressure isn’t easy.

Even if we assume that all the raw materials for these ingredients are available in infinite quantities, the time taken to transport each resource, manufacture the required parts and then move them to the factory takes a different amount of time. And in the factory itself, each electric furnace consumes different quantities of each ingredient: 10 steel plates, 10 stone bricks and five advanced circuits). As a result, for example, if I maintain all three resources with equal pressure on the factory, I will still run out of steel plates and stone bricks faster than I will run out of advanced circuits.

In fact, I will run out of steel plates first because its crafting time is 32 seconds, versus 3.2 seconds for one stone brick. So the proper pressure to maintain here is P for advanced circuits, 2P for stone bricks and 20P for steel plates. (I’m ignoring the crafting time for advanced circuits to keep the example simple.) If I don’t keep up these proportions, I won’t have a steady supply of electric furnaces. Instead, I’ll run out of steel plates first, and by the time more plates are available, stone bricks will have run out, and by the time stone bricks are available, advanced circuits will have run out. And so on and on in a continuous cycle.

The concept of orbital resonance is somewhat similar. Did you know that for everyone two orbits Pluto completes around the Sun, Neptune completes three? This is the 2:3 resonance. And it’s comparable to the Factorio example in that the ratio between the two periodic activities – Neptune’s and Pluto’s revolution and the rate of repetitive consumption of stone bricks and advanced circuits – is a rational number. ‘Rational’ here means the number can be expressed as the ratio of two integers.

Animation of planets in a 2:1 resonance. Credit: Amitchell125/Wikimedia Commons, CC BY-SA 4.0
Animation of planets in a 2:1 resonance. Credit: Amitchell125/Wikimedia Commons, CC BY-SA 4.0

With Pluto and Neptune, it’s 2/3 of course, but in a more intuitive sense, the implication is that if you wait for long enough, you will be able to count off the number of times the orbital resonance plays out – i.e. the number of times both planets are back to their starting positions at the same time, which would be once every two Plutonian revolutions or once every three Neptunian revolutions.

Similarly, the resource-pressure resonance plays out once every 10 stone bricks or once every five advanced circuits are consumed.

This meta-periodicity, a term I’m using here to refer to the combined periodicity of two separately periodic motions, allows us a unique opportunity to understand how bizarre the number known as π (pi) is. π is an irrational number: there’s no way to express it as the ratio of two integers. (The following portion also applies to e and other irrational numbers.)

In ‘Factorio’, all resources are integral, which means there can only be 1, 2, 3, … stone bricks, and never 1.5, 2.25, 3.75, etc.; the same constraint applies to advanced circuits as well. So there is no way for me – no matter how I align my resource extraction and processing chains – to ensure that for every advanced circuit, an integer-times-π number of stone bricks are consumed as well. I can alter the length of the supply lines, increase or decrease the ‘normal’ processing time, even use faster/slower conveyor belts and inserters for different ingredients, but I will never succeed. So long as the quantities in play remain integers, there’s no way for me to achieve a resonance such that the ratio of its terms is π.

This is what makes π so beautiful and maddening at once. It exists on terms that no two integers can recreate by themselves.

There’s another way to look at it. Say two planets begin orbiting their common host star from the 12 o’clock position in their respective orbits. If they are in a π:1 resonance, they will never be exactly at the 12 o’clock at the same time ever again. It doesn’t matter if you wait a century, an epoch or forever.

This example offers to my mind an uncommon opportunity to understand the difference between attributes of π and ∞. There’s the oft-quoted and frankly too prosaic statement that π’s decimal places extend infinitely. I prefer the more poetic: that efforts using simple mathematical combinations of integers will never create π. Even if a combination operates recursively, and each cycle produces a closer approximation of π, it can run for ∞ time and still not get here.

Like there’s an immutable barrier between two forms of unattainability.

Life notes Op-eds

Where the Indian infiniteness?

I didn’t know Kenneth Wilson had died on June 15 until an obituary appeared in Nature on August 1. He was a Nobel Prize winning physicist and mathematician whose contribution to science was and is great. He gave scientists the tools to imagine the laws of physics at different scales — large and small — and to translate causes and effects from one scale into another. Without him, we’d struggle not only to solve physics problems at cosmological and nuclear distances at the same time but also to comprehend the universe from the dimensionless to the infinite.

Wilson won his Nobel Prize in physics in 1982 for his work with phase transitions — when substances go from solid to liquid or liquid to gas, etc. Specifically, he extended its study to include particle physics as well, and was able to derive precise results that agreed with experiment. At the heart of this approach lay inclusivity: to think that events not just at this scale but at extremely large and extremely small scales, too, were affecting the system. It was the same approach that has enabled many physicists and mathematicians take stock of infinity.

The idea of infinity

As physicist Leo Kadanoff’s obituary in Nature begins, “Before Kenneth Wilson’s work, calculations in particle physics were plagued by infinities.” Many great scientists had struggled to confine the ‘innumerable number’ into a form that would sit quietly within their theories and equations. They eventually resorting to an alternative called renormalisation. With this technique, scientists would form relationships between equations that worked at large scales and those that worked at small ones, and then solve the problem.

Even Dirac, renormalisation’s originator, called the technique “dirty”. And Wilson’s biggest contribution came when he reformulated renormalisation in the 1970s, and proved its newfound effectiveness using experiments in condensed matter physics. Like Wilson’s work, the idea was interdisciplinary. But how original was it?

The incalculable number

Kenneth Wilson did not come up with inclusivity. Yes, he found a way to use it in the problems that were prevalent in mid-20th century physics. But in the Mahavaipulya Buddhavatamsaka Sutra, an influential text of Mahayana Buddhism written in the third or fourth century AD, lies a treatment of very large numbers centered on the struggle to comprehend divinity. The largest titled meaningful number in this work appears to be the bodhisattva(10^37218383881977644441306597687849648128) and the largest titled number as such, thejyotiba (10^80000 infinities).

The jyotiba may not make much sense today, but it represents the early days of a centuries-old tradition that felt such numbers had to exist, a tradition that acknowledged and included the upper-limits of human comprehension while on its quest to deciphering the true nature of ‘god’.

Avatamsaka Sutra, vol. 12: frontispiece in gold and silver text on indigo blue paper, from the Ho-Am Art Museum. Photo: Wikimedia Commons
Avatamsaka Sutra, vol. 12: frontispiece in gold and silver text on indigo blue paper, from the Ho-Am Art Museum. Photo: Wikimedia Commons

The Mahavaipulya Buddhavatamsaka Sutra itself, also known as the Avatamsaka Sutra, also contains a description of an “incalculable” number divined to describe the innumerable names and forms of the principal deities Vishnu and Siva. By definition, it had to lie outside the bounds of human calculability. This number, known as the asamkhyeya, owes its value to one of three arrived at because of an ambiguity in the sutraAsamkhyeya is defined as a particular power of a laksha, but there is no indication of how much a laksha is!

One translation, from Sanskrit to the Chinese by Shikshananda, says one asamkhyeya is equal to 10 to the power of 7.1-times 10-to-the-power-of-31. Another translation, to English by Thomas Cleary, says it is 10 to the power of 2.03-times 10-to-the-power-of-32. The third, by Buddhabhadra to the Chinese again, says it is 10 to the power of 5.07-times 10-to-the-power-of-31. If they have recognisable values, you ask, why the title “incalculable”?

Lesser infinities

For this, turn to the Jain text Surya Prajnapati, dated c. 400 BC, which records how people knew even at that time that some kinds of infinities are, somehow, larger than others (e.g., countable and uncountable infinities). In fact, this is an idea that Galileo more famously wrote of in 1638 in his On two New Sciences:

So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes equal,’ ‘greater,’ and ‘less,’ are not applicable to infinite, but only to finite, quantities.

Archimedes, whose Syracusani Arenarius & Dimensio Circuli predated the Avatamsaka Sutra by about 300-400 years, adopted a more rationalist approach that employed the myriad, or ten thousand, to derive higher multiples of itself, such as the myriad-myriad. However, he didn’t venture far: he stopped at 10^64 for lack of a name! The father of algebra (disputed), Diophantus of Alexandria, and the noted astronomer Apollonius of Perga, who lived around Archimedes’ time, also stopped themselves with powers of a myriad, venturing no further.

Unlike the efforts recorded in the Avatamsaka Sutra, however, Archimedes’ work was mathematical. He wasn’t looking for a greater meaning of anything. His questions were of numbers and their values, simply.

In comparison — and only in an effort to establish the origin of the idea of infinity — 10^64 is a number only two orders of magnitude higher than one that appears in Vedic literature, 10^62, dated 1000-1500 BC. In fact, in the Isa Upanishad of the Yajurveda (1000-600 BC: Mauryan times), a famous incancation first appears: “purnam-adah purnnam-idam purnat purnam-udacyate purnashya purnamadaaya puram-eva-avashisyate“. It translates: “From fullness comes fullness, and removing fullness from fullness, what remains is also fullness”.

If this isn’t infinity, what is?

In search of meaning

Importantly, the Indian “proclamation” of infinity was not mathematical in nature but — even if by being invoked as a representation of godliness — rooted in pagan realism. It existed together with a number system, one conceived to keep track of the sun and the moon, of the changing seasons, of the rise and fall of tides and the coming and going of droughts and floods. There is a calming universality to the idea — a calming inclusivity, rather — akin to what a particle physicists might call naturalness. Inifinity was a human attempt make a divine being all-inclusive. The infinity of modern mathematics, on the other hand, is contrarily so removed from the human condition, its nature seemingly alien.

Even though the number as such is not understood today as much as ignored for its recalcitrance, infinity has lost its nebulous character — as a cloud of ideas always on the verge of coalescing into comprehension — that for once was necessary to understand it. Infinity, rather infiniteness, is an entity that transcends the character typical of the inbetweens, the positive numbers and the rational numbers. If zero is nothingness, an absence, a void, then infinity, at the other end is… what? “Everythingness”? How does one get there?

(There is a related problem here in physics, similar to the paradox of Zeno’s arrow: if a point is defined as being dimensionless and a one-dimensional line as being a collection of points, how and when did dimension come into being? Incidentally, the earliest recorded incidence of infinities in Early Greek mathematics is attributed to Zeno.)

The lemniscate

As it so happened, the same people who first recorded the notion of infiniteness were also those who recorded the notion of a positional numbering system, i.e. the number line, which quickly consigned infinity to an extremum, out of sight, out of mind. In 1655, it suffered another blow to its inconfinable nature: John Wallis accorded it the symbol of a lemniscate, reducing its communication to an horizontal figure-of-eight rather than sustaining a tradition of recounting its character through words and sense-based descriptions. We were quick to understand that it saved time, but slow to care for what it chopped off in the process.

Of course, none of this has much to do with Wilson, who by his heyday must have been looking at a universe through a lens intricately carved out of quantum mechanics, particle physics and the like. What I wonder is why did an Indian scientific tradition that was conceived with the idea of infinity firmly lodged in its canons struggle to make the leap from theoretical to practical problem-solving? There are answers aplenty, of course: wars, empires, scientific and cultural revolutions, industrialisation, etc.

Remembering too much

Wilson’s demise was an opportunity for me to dig up the origins of infinity — and I wasn’t surprised that it was firmly rooted in the early days of Indian philosophy. The Isa Upanishad incancation was firmly implanted in my head while I was growing up: the Brahminical household remembers. I was also taught while growing up that by the seventh century AD, Indians knew that infinity and division-by-zero were equatable.

It’d be immensely difficult, if not altogether stupid, to attempt to replace modern mathematical tools with Vedic ones today. At this stage, modern tools save time — they do have the advantage of being necessitated by a system that it helped create. Instead, the Vedic philosophies must be preserved — not just the incantations but how they were conceived, what is their empirical basis, etc. Yes, the household remembers, but it remembers too specifically. What it preserves has only historical value.

The Indian introspective and dialectic tradition has not given us just liturgy but an insight into the modes of introspection. If we’d preserved such knowledge better, the epiphany of perspectives that Wilson inspired in the late 1970s wouldn’t be so few nor so far between.


This piece was first published in The Copernican science blog on August 6, 2013.