Is π the Same in Every Universe?

[lordnacho]

The beginning bit of the physics answer is the answer.

Pi comes from logic, not nature.

This is a bit like when you're trying to ask your kid what the area of the triangle is, and the kid tells you you've drawn the lines bent, or the corners aren't sharp. It's actually not terribly easy to explain to them that they're supposed to understand the idealized entity, and what exactly those ideals are, because you also can't actually draw a triangle with no width and then expect them to appreciate that the qualities you want to expose are somehow exposed when you're breaking your own rules.

[fooker]

Now draw the same triangle on a curved surface. No matter how the idea of area comes from logic, the fundamental assumption for the commonly known formula to calculate area is that we are doing euclidean geometry.

We do not yet know if there is some similar assumption hiding behind our ignorance of physics.

[d0mine]

We live in a world that is well described by Euclidien geometry for many applications. It doesn't mean it is the only possible useful geometry. If we define Pi as a ratio C/D, it can be different. Look at the example in the article.

[coldtea]

>Pi comes from logic, not nature.

Logic itself is a human construct that comes from observing nature - God's surely didn't hand it over to us. It is also not necessarily a singular thing (same way there are different logics as well as different geometries, and so on).

A universe with different natural laws might also obey a different logic.

In fact, even in this universe, even basic logic propositions are said to be uncertain/untrue under specific scales or physical environments (quantum micro scales, inside a singularity, and so on).

>It's actually not terribly easy to explain to them that they're supposed to understand the idealized entity

There's no singular "idealized entity". For example there are geometries where triangle angles add up to > 180 degrees.

[pacman128]

This brings up the thing that really bothered me about Carl Sagan's novel _Contact_. A book that I really liked otherwise.

Warning, this might be considered a spoiler:

A pattern found deep in the computed binary digits of pi are used to "prove" that the universe was created by an intelligence.

[throwaaskjdfh]

> Pi comes from logic, not nature.

What if logic is different in different universes? Logic is universal... but is it multiversal?

[mjburgess]

I assume you're diving things into "from logic" or "from nature". However that's a highly misleading (if not false) dichotomy.

"Logic" refers to the most basic set of logical operations (AND, OR, etc.) and their use. Pi does not come from this.

It comes from Mathematics. If we define World to be "a set of true claims" then we can define worlds without any mathematical claims in; however the process of defining the world likely requires logic. (And so Mathematics != Logic).

In any case, as the article shows, the definition of Pi is relative to a distance metric. "Pi = Circ/Diam" which is ill-defined.

One can construct a World (as defined above) which admits only claims relative to a given distance metric ("A Taxi-Cab World"). At this world, "Pi" has a different value.

It isn't also clear that the value of "Pi" here is given purely by mathematics. As its value is relative to a distance metric, and we choose that metric for empirical reasons ("it applies to our world").

[makerofthings]

That’s exactly what my son does and I hadn’t really thought about why until this post. Really interesting comment. I might try asking him to imagine the shapes instead of looking at my diagrams, I doubt he would imagine my terrible art skills.

[anothernewdude]

I figure it'd be different in any non-euclidean geometry.

[rssoconnor]

If you consider solutions to one of the simplest non-trival second-order differential equation, f''(x) = -f(x), then you will find that the solutions all have period 2π, no geometry needed.

[beebmam]

I assume that you mean solutions where x is in R here

[gspr]

Oh I don't know about that. The solution involves trigonometric functions. Are they not geometric in nature?

Said differently: periodic things in some sense express motions on a circle.

[hyperpallium2]

That's really freaky, though no more so than that for DE f'(x) = f(x), the solution is e^x

The self-similarity in these two DEs is why Euler's formula works. Are there any other such DEs?

[hatsunearu]

Well, the thing about a circle is that the very core of the definition has the same sort of "information" as whatever abstract "information" f'' = -f contains.

[0-_-0]

Or just rely on Euler's formula: e^(iπ)=-1

[CapmCrackaWaka]

Reminds me of a short story by Liu Cixin, the author of the Three Body Problem. In it, a character discovers that, as a result of 11 dimensional string space, there is a finite number of possible initial configurations to the Big Bang, each one resulting in a deterministic universe with its own distinct combination of fundamental constants (speed of light, Coulomb’s constant, pi, etc.). He explores this quite a bit. It’s pure sci-fi, but fun to read.

[Koiwai]

I read a lot (if not all) of his work, I'd like to say just ignore his physics, he's really not good at that.

I'm not saying enjoying his story is a bad thing.

[legobmw99]

What’s the name of this short story?

[stillbourne]

Anathem by Neal Stephenson also touches on that topic.

[jiggawatts]

If you define pi based on physical circles, then it isn't even the "same pi" as the mathematical one in this universe. In fact, it changes locally from point to point. That's literally what General Relativity says: spacetime is curved!

[ithinkso]

I don't think this works in GR the way you think it works (regardless of what 'physical circle' means), aren't you always able to change coordinates for the space to be locally flat?

(This always confuses me, can someone that knows GR shed some light on this? I would greatly appreciate it. I've been trying to teach myself GR a while back from the lectures of Frederic Schuller and Alex Flournoy that are available on youtube but it's hard without being able to ask questions)

[wyager]

Pi is defined over a flat space with an L2 norm. It has nothing to do with physical circles except that physical space is locally flat and has an L2 norm.

[enkid]

Something being a circle or an oval would depend on your relative velocity, meaning the idea of a physical circle doesn't really make sense.

[aj3]

That would be very unusual (unorthodox) definition of Pi though.

[fooqux]

But wouldn't the measuring device also be curved?

[labster]

In some alternate universes they use τ instead and I would rather live in those universes.

https://tauday.com/

[wyager]

Never once while studying physics have I been bothered by constant factors, especially nice integer ones like 2. I don't really care if I have to multiply by 2. Feet aren't really different from meters from a physical perspective. The only choice of unit that really has any solid claim to superiority is the natural units.

[amelius]

Programmers would never let this discrepancy exist for this long!

Unfortunately, mathematicians are too lazy to "refactor" their work ...

[dhosek]

Bah, you're just relocating your constants to different formulae. What's worse is that you end up introducing fractions in your quest for “simplicity”. For every c = τr, there's an A = ½τr². It gets worse when you get into n-spheres where there are powers of π at play. For a 3-sphere, for example, the hypervolume would become 1/8τ²r⁴ and the surface volume ½τ²r³.

I'm reminded of something my double bass teacher told me, that people have been working with these things for hundreds of years (or thousands in the case of π). If there's a better way, they'd use it (and it's unlikely to come from someone who dismissed special relativity, quantum mechanics and natural selection).

[Animats]

It's interesting to think about what mathematics might have been like if geometry hadn't been developed so early. If algebra or number theory had come first, the history of mathematics might have been quite different. Alternatively, you could start from the Peano axioms and grind your way up to number theory.

High school mathematics spends much time on plane geometry as a formal system for historical reasons, not because it's a particularly interesting or useful formal system.

[linspace]

I think the reasons for using plane geometry are pedagogical. Also formulating mathematically real life objects is a nice skill to have. I would never teach mathematics axiomatically, it's extremely dull and I usually find the axioms as an approximation to the real thing and not the source of truth.

[raducu]

I sucked at algebra, and the math of physics until 7 or 8th grade, when geometry was introduced; I excelled at geometry and that completely changed my life, I later loved math.

[slongfield]

Some algebra was around very early--there are Cuneiform tablets inscribed with multiplication tables from ~2000 BC: https://www.metmuseum.org/art/collection/search/324587

[aj3]

You're definitely commenting on your own experience in high school and it's not universal. We were mostly doing analysis in high school with some analytical geometry thrown in.

[question000]

Richard Feynman had a satirical theory that any complex mathematical problem/theory when stated in layman's terms becomes self-evident. This is the perfect example of this.

There's only one universe, QED.

[dhosek]

I actually have used alternate metrics in production code: The problem was to be able to cluster points on a map. The naïve approach is to use Euclidean distance, d = √Δx² + Δy², but the problem here is that the neighborhoods you get are circles and we're looking at the map through a rectangular viewport. Clicking on a cluster to zoom in on it will not give an optimal zoom (and may include stray pins from other clusters as well). It turned out using a box metric of d = max(Δx, Δy) gives better results for our needs since the neighborhoods are now squares instead of rectangles (and has the added bonus of being easier to calculate).

[xyzal]

> The vast, vast, vast majority of the possible combinations of values of these physical constants produce boring universes, like a single huge black hole or just diffuse clouds of hydrogen without stars.

I wonder if there any universes possible which do not end in a big collapse or heat death. That is, which would be able to support life for eternity.

[ginko]

For a while I wanted to write some basic 3d simulation app where the pi value used for all kinds of calculations (including inside standard libs) was configurable with a slider in real-time. Would be interesting to see how things would break.

[ludston]

An interesting idea once presented to me, was that pi is not a number, but a function for producing a number to an arbitrary precision. The same could be said for any fraction that produces an irrational.

[RichardCA]

The difficulty with questions like this is that we don't know what we don't know.

To put a fine point on it, we don't know if the concept of Pi is unique to us as humans.

If we accept the premise that we are not the only species to evolve science and technology, we are still left with the question of how other sentient species might get their maths differently from how we did it.

This idea gets explored a lot in science fiction. But we don't have enough of a handle on it to say anything interesting. It's just different levels of speculation and conjecture.

[buescher]

Pi-the-number is usually lowercase (and frequently italic) pi, "𝜋" , almost never uppercase "Π", which is used for other things.

[osigurdson]

Consider the following alternate universe experiment:

Wrap a piece of string around a circular object.Straighten it out and measure its length.Measure the distance between the center and the edge of the circular object.Then observe that the ratio between these two measurements is 12

How would observers in this universe recognize that this ratio is not correct?

[mike_hock]

> Then observe that the ratio between these two measurements is 12

Then their space isn't locally approximated by R^3, so what's a circular object? What does it mean to be straight? What's length?

[eckmLJE]

A universe in which string cannot exist and objects freely pass through each other and through the fabric of spacetime, stretching through regions millions of light years apart

[rocqua]

Obligatory rant that tau is better than pi.

Tau (2 * pi) is a better constant. Especially for teaching people trigonometry. It makes radians so much more intuitive (90 degree angle becomes 1/4th tau radians).

In general, mathematics characterizes circles by their radius, not their diameter. Tau codifies this practice by giving us the relation between a circles radius, and its circumference.

This isn't a change that will make academic mathematics better. Its a change that will make high-school mathematics better. If anything, that makes it more important.

/rant

[DonHopkins]

I don't know, but I do wish π was the same in every font, so I could distinguish it from n, or a square missing its bottom edge.

https://imgur.com/a/VBXLVA6

[tpoacher]

Unsurprisingly, like most such questions, the effective answer is: "Define π"

[historyloop]

Well this blog post ended precisely how I expected it to end. "Maybe."

[matthewfelgate]

How many constants that exist in nature are also in logic?

Or is it just Pi and E?

[qqtt]

Things in mathematics are definitionally true. That's the whole system of axioms and proofs.

Pi is pi because of our axioms.

If you define things differently, pi doesn't have to be discovered at all.