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.


ALMA telescope catches live planet-forming action for the first time

The ALMA telescope in Chile has, for the first time, observed a star system that might be in the early stages of planet formation. The picture has astronomers drooling over it because the study of the origins of planets has until now been limited to simulated computer models and observations of planets made after they formed.

ALMA image of the protoplanetary disc around HL Tauri.
ALMA image of the protoplanetary disc around HL Tauri. Image: ALMA (ESO/NAOJ/NRAO)

According to a statement put out by the European Southern Observatory (ESO), the observation was one of the first made with the ALMA, which opened in September 2014 for a ‘Long Baseline Campaign’ (ESO is the institution through which European countries fund the telescope). ALMA uses a technique called very-long baseline interferometry to achieve high resolutions that lets it observe objects hundreds of light-years away in fine detail. It makes these observations in the millimeter/sub-millimeter range of wavelengths; hence its name: Atacama Large Millimeter/sub-millimeter Array.

The image shows a disc of gas, dust and other debris orbiting the star HL Tauri, located about 450 light-years from Earth. A system like this is originally a large cloud of gas and dust. At some point, the cloud collapses under its own gravitation and starts to form a star, further accruing matter from the cloud and growing in size. The remaining matter in the cloud then settles into a disc formation over millions of years around the young star.

In the disc, the gas and dust continue to clump, this time into rocky lumps like planets and asteroids. This is why the disc is called a proto-planetary disc. As a planet forms and its gravitational pull gets stronger, it starts to clear a space in the disc of matter by either sucking it for itself or knocking it out. The gaps that are formed as a result are good indicators of planet formation.

According to the ESO statement, “HL Tauri’s disc appears much more developed than would be expected from the age of the system [less than 100,000 years]. Thus, the ALMA image also suggests that the planet-formation process may be faster than previously thought.”

An annotated image showing the protoplanetary disc surrounding the young star HL Tauri.
An annotated image showing the protoplanetary disc surrounding the young star HL Tauri. Image: ALMA (ESO/NAOJ/NRAO)

In the Solar System, similar gaps exist called Kirkwood gaps. They represent matter cleared by Jupiter, whose prodigious gravitational pull has been pushing and pulling the orbits of asteroids around the Sun into certain locations. In fact, Jupiter’s movement within the Solar System – first moving away, then toward, and then away once more from the Sun – has been used to explain why the material composition of some asteroids between Mars and Jupiter is similar to those of Kuiper Belt objects situated beyond the present orbit of Neptune. Jupiter’s migration mixed them up.

Similarly, the gaps forming around HL Tauri, though they may represent planetesimals, may not result in planets in the exact same orbits as they could move around under the influence of subsequent gravitational disruptions. They could acquire unexpectedly eccentric orbits if their star system comes too close to another, as was found in the nearby binary star system HK Tauri in July 2014. Or, the gaps are probably being emptied by the gravitational effects of an object in another gap.

However, astronomers think the presence of multiple gaps is likely evidence of planet formation more than anything else.

At the same time, the resolution in the image is 7 AU (little more than Jupiter’s distance from the Sun), which means the gaps are very large and represent stronger gravitational effects.

Astronomers will use this and other details as they continue their investigation into the HL Tauri system and how planets – at least planets in this system – form. The Long Baseline Campaign, which corresponds to the long-baseline configuration of the ALMA telescope that enabled this observation, will continue into December.