Is π the Same in Every Universe?

The beginning bit of the physics answer is the answer.<p>Pi comes from logic, not nature.<p>This is a bit like when you&#x27;re trying to ask your kid what the area of the triangle is, and the kid tells you you&#x27;ve drawn the lines bent, or the corners aren&#x27;t sharp. It&#x27;s actually not terribly easy to explain to them that they&#x27;re supposed to understand the idealized entity, and what exactly those ideals are, because you also can&#x27;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&#x27;re breaking your own rules.

If you consider solutions to one of the simplest non-trival second-order differential equation, f&#x27;&#x27;(x) = -f(x), then you will find that the solutions all have period 2π, no geometry needed.

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.

If you define pi based on <i>physical</i> circles, then it isn&#x27;t even the &quot;same pi&quot; as the mathematical one in <i>this</i> universe. In fact, it changes locally from point to point. That&#x27;s literally what General Relativity says: spacetime is curved!

In some alternate universes they use τ instead and I would rather live in those universes.<p><a href="https:&#x2F;&#x2F;tauday.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;tauday.com&#x2F;</a>

It&#x27;s interesting to think about what mathematics might have been like if geometry hadn&#x27;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.<p>High school mathematics spends much time on plane geometry as a formal system for historical reasons, not because it&#x27;s a particularly interesting or useful formal system.

Richard Feynman had a satirical theory that any complex mathematical problem&#x2F;theory when stated in layman&#x27;s terms becomes self-evident. This is the perfect example of this.<p>There&#x27;s only one universe, QED.

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, <i>d</i> = √Δ<i>x</i>² + Δ<i>y</i>², but the problem here is that the neighborhoods you get are circles and we&#x27;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 <i>d</i> = max(Δ<i>x</i>, Δ<i>y</i>) 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).

&gt; 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.<p>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.

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.

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.

The difficulty with questions like this is that we don&#x27;t know what we don&#x27;t know.<p>To put a fine point on it, we don&#x27;t know if the concept of Pi is unique to us as humans.<p>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.<p>This idea gets explored a lot in science fiction. But we don&#x27;t have enough of a handle on it to say anything interesting. It&#x27;s just different levels of speculation and conjecture.

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

Consider the following alternate universe experiment:<p>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<p>How would observers in this universe recognize that this ratio is not correct?

Obligatory rant that tau is better than pi.<p>Tau (2 * pi) is a better constant. Especially for teaching people trigonometry. It makes radians so much more intuitive (90 degree angle becomes 1&#x2F;4th tau radians).<p>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.<p>This isn&#x27;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.<p>&#x2F;rant

I don&#x27;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.<p><a href="https:&#x2F;&#x2F;imgur.com&#x2F;a&#x2F;VBXLVA6" rel="nofollow">https:&#x2F;&#x2F;imgur.com&#x2F;a&#x2F;VBXLVA6</a>

Unsurprisingly, like most such questions, the effective answer is: &quot;Define π&quot;

Well this blog post ended precisely how I expected it to end. &quot;Maybe.&quot;

How many constants that exist in nature are also in logic?<p>Or is it just Pi and E?

Things in mathematics are definitionally true. That&#x27;s the whole system of axioms and proofs.<p>Pi is pi because of our axioms.<p>If you define things differently, pi doesn&#x27;t have to be discovered at all.

Pi comes from its own universe: a universe which contains a point in a plane, and some equidistant points.<p>When we work with Pi, we embed that universe into other universes.

Pi only exists in the context of euclidean space. We have never seen a real euclidean metric space in THIS universe. Pi literally isn&#x27;t a universal constant. Unlike physical constants (All the force constants, speed of light ect) apparent values of Pi will vary wildly if you can even find situations where it makes sense to apply it.<p>Concrete example: Assume you somehow magically can have a planet in a circular orbit around a star, surely you can use Pi to derive the circumference from the distance to the center of the star? Nope, the space in the system is distorted, Pi varies with radius and the mass of the star.

Seems a bit silly to ask a mathematical question from the perspective of a physicist, but I guess as the comments point out speculation is the point here.

Yes, because pi is not measured but mathematically defined. Same is true for all other constants, that are not measured.

Someone should do a calculation of Pi in the hexagonal coordinate, good for gaming in general.

No. Some universes use tau :^)