Is π the Same in Every Universe?(askamathematician.com)
askamathematician.com
Is π the Same in Every Universe?
https://www.askamathematician.com/2020/12/q-is-%cf%80-the-same-in-every-universe/
207 comments
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.
We do not yet know if there is some similar assumption hiding behind our ignorance of physics.
Euclidian geometry would be the same in any universe though, as it does not depend on objective physical reality.
It could be argued that the Euclidean plane is its own universe.
The Euclidean plane is a conceptual model also not based on physics.
It is based on observing (and needing to measure) plots of land.
The abstraction came later.
Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).
The abstraction came later.
Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).
> Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).
Do you mind elaborating on this, or pointing to some resources where I can read more?
Do you mind elaborating on this, or pointing to some resources where I can read more?
The map is not the territory. It was always an abstraction.
how is general relativity related to any prior experience of the 1920s?
The maths that underpin GR were developed before Einstein. Without the maths you have no ontology, no way to evaluate the truth of a scientific claim.
https://en.wikipedia.org/wiki/Tullio_Levi-Civita
https://en.wikipedia.org/wiki/Tullio_Levi-Civita
the math doesn't prove anything. we didn't get experimental results until decades later
Yes yes. But you can ask that question, and we can answer it, regardless of whether our space is curved or flat. PI is the same in a flat space in any universe where you ask the question, because we're doing a mental exercise, not measuring it locally.
> PI is the same in a flat space in any universe
pi is the same in _any_ space, flat or not.
Indeed, the space we are in is probably not flat, but pi is unchanged.
pi is the same in _any_ space, flat or not.
Indeed, the space we are in is probably not flat, but pi is unchanged.
True but Pi (mathematical constant Pi) isn't necessarily the ratio between the circumference and the diameter in those spaces.
Even in a non-Euclidean geometry, wouldn't π still emerge (or be approached in the limit) in "flat" areas?
With regular spherical and hyperbolic geometry, yes, it is flat-ish if you look at a small enough area. On the other hand, if you have a different distance metric, the ratio of a circle's diameter to circumference can be a different constant. For example, in taxicab geometry, it is 4
In Uber Geometry, it can vary from 5 to 14 during surge pricing.
In London geometry, the distance depends where you are in the one-way system.
I know this is sort of a low quality joke post, but I don’t think I’d ever thought about a directed taxi-cab distance before. I’m not yet convinced it’s a useful model outside GIS, but I may spend considerable time today thinking about it.
Thank you.
Thank you.
Happy to be of service, in filth it will be found!
TIL that Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes.
It depends if you define a circle as "constant distance from centre point" or "minimum (circumference/area)".
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.
>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.
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.
> Logic itself is a human construct that comes from observing nature - God's surely didn't hand it over to us.
I'm not sure that is true. If we estimate Pi by tossing needles onto a grid of lines[1] then we are obtaining the value by observing nature. But, while that's a fun exercise, it isn't how mathematicians actually determine the value of Pi. They use formulas[2] based on elementary numbers, such as this: Pi = Sqrt( 6 * Sum[k=1..infinity](1/k^2) ).
Maybe you believe that the numbers 1, 2, 3... are different in different universes. Personally, I can't fathom that. But if you accept that the basic counting numbers are fundamental to logic and reason instead of being derived from nature and somehow varying from one universe to another, then simply square those numbers, take the reciprocals, sum the series, multiply by 6 and take the square root -- that's Pi.
There ARE various geometries possible. But those consist of different definitions of what terms like "straight line" mean. The geometries all have the same value for the fundamental circle constant, Pi.
Whether we choose to talk about Pi or instead about Tau (where Tau = 2Pi) is a social convention. But what the value is (of both constants) is a fundamental property of mathematical reasoning independent of convention.
[1] https://www.sciencefriday.com/articles/estimate-pi-by-droppi...
[2] https://mathworld.wolfram.com/PiFormulas.html
I'm not sure that is true. If we estimate Pi by tossing needles onto a grid of lines[1] then we are obtaining the value by observing nature. But, while that's a fun exercise, it isn't how mathematicians actually determine the value of Pi. They use formulas[2] based on elementary numbers, such as this: Pi = Sqrt( 6 * Sum[k=1..infinity](1/k^2) ).
Maybe you believe that the numbers 1, 2, 3... are different in different universes. Personally, I can't fathom that. But if you accept that the basic counting numbers are fundamental to logic and reason instead of being derived from nature and somehow varying from one universe to another, then simply square those numbers, take the reciprocals, sum the series, multiply by 6 and take the square root -- that's Pi.
There ARE various geometries possible. But those consist of different definitions of what terms like "straight line" mean. The geometries all have the same value for the fundamental circle constant, Pi.
Whether we choose to talk about Pi or instead about Tau (where Tau = 2Pi) is a social convention. But what the value is (of both constants) is a fundamental property of mathematical reasoning independent of convention.
[1] https://www.sciencefriday.com/articles/estimate-pi-by-droppi...
[2] https://mathworld.wolfram.com/PiFormulas.html
>But, while that's a fun exercise, it isn't how mathematicians actually determine the value of Pi. They use formulas[2] based on elementary numbers, such as this: Pi = Sqrt( 6 Sum[k=1..infinity](1/k^2) ).*
Yes, but this didn't come from God (or the Platonic realm of Ideas) into mathematicians heads. Humans arrived at that by first observing natural laws / behavior (e.g. 1 apple + 1 apple = 2 apples, or observing that somebody can't be A and not A, and so on) and progressing from those into abstractions. Nature comes first, not abstraction.
Nor there is some plane where mathematical abstraction or logic exists outside of the universe. It exists only (a) explicitly on people's heads (which, unaraguably, are part of nature), (b) as implicitly observed on natural behavior by said people.
>Maybe you believe that the numbers 1, 2, 3... are different in different universes. Personally, I can't fathom that.
No, I believe that numbers don't exist. What would "1" even be, as an entity in our very universe? There is a rock or a planet or a tiger, but not an 1.
Countable things do (but not necessarily on another universe, where e.g. there might be just a single entity).
Also, we already have boolean algebra in this universe, which has different properties compared to regular algebra. In that sense a logic circuit is already a kind of universe where the algebra that governs it is different. One could well imagine a universe that only observes circuit board like behavior.
Yes, but this didn't come from God (or the Platonic realm of Ideas) into mathematicians heads. Humans arrived at that by first observing natural laws / behavior (e.g. 1 apple + 1 apple = 2 apples, or observing that somebody can't be A and not A, and so on) and progressing from those into abstractions. Nature comes first, not abstraction.
Nor there is some plane where mathematical abstraction or logic exists outside of the universe. It exists only (a) explicitly on people's heads (which, unaraguably, are part of nature), (b) as implicitly observed on natural behavior by said people.
>Maybe you believe that the numbers 1, 2, 3... are different in different universes. Personally, I can't fathom that.
No, I believe that numbers don't exist. What would "1" even be, as an entity in our very universe? There is a rock or a planet or a tiger, but not an 1.
Countable things do (but not necessarily on another universe, where e.g. there might be just a single entity).
Also, we already have boolean algebra in this universe, which has different properties compared to regular algebra. In that sense a logic circuit is already a kind of universe where the algebra that governs it is different. One could well imagine a universe that only observes circuit board like behavior.
Observing that something is either a or not a is not an observation in the empirical sense. Empirical observations suffer from the problem of induction, but neither of us believes that that statement might be disproven someday, in the way that observing a black swan would disprove the statement that all swans are white.
Similarly the conservation law of apples is not derived from nature, but from our definition of what an apple is. If you took a heap of sand and added another you would have one heap of sand.
Similarly the conservation law of apples is not derived from nature, but from our definition of what an apple is. If you took a heap of sand and added another you would have one heap of sand.
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.
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.
I think this is just Carl Sagan's definition of what he (as an atheist) thinks an Almighty Deity _should_ be able to do.
I mean, being able to write an arbitrary message in a logical constant like Pi sounds harder than just merely creating the physical universe, no?
Also, your dismay of the idea sounds like a kind of Platonism: "Obviously" the logical rational reality of mathematics just is, right?
I mean, being able to write an arbitrary message in a logical constant like Pi sounds harder than just merely creating the physical universe, no?
Also, your dismay of the idea sounds like a kind of Platonism: "Obviously" the logical rational reality of mathematics just is, right?
It's definitely a type of Platonism. The question isn't how hard it is, but if it's possible at all.
That probably sort of his point, I guess: a genuine God should be able to do the impossible and this is an interesting kind of impossible.
What part of this is "really bothering"?
We can make a simulation today, and have it obey such natural laws (of course programmed behavior) + logic (ditto), so that we could encode whatever we want inside a physical or geometrical constant as it appears inside the simulation.
We can make a simulation today, and have it obey such natural laws (of course programmed behavior) + logic (ditto), so that we could encode whatever we want inside a physical or geometrical constant as it appears inside the simulation.
We can't make a simulation that changes the result of infinite series. They were computing pi, not measuring it. This wasn't changing the curvature of space, it was changing math itself.
>We can't make a simulation that changes the result of infinite series.
Huh? It's trivial to do so. You just give the simulation the appropriate algebra that applies to all their measurements.
Not more difficult than setting 12=66 to programming languages that allow it (there are some, it can also be done in others like Python through some trickery [1]), and thus changing all the subsequent calculations done there.
[1] https://www.reddit.com/r/Python/comments/2441cv/can_you_chan...
Huh? It's trivial to do so. You just give the simulation the appropriate algebra that applies to all their measurements.
Not more difficult than setting 12=66 to programming languages that allow it (there are some, it can also be done in others like Python through some trickery [1]), and thus changing all the subsequent calculations done there.
[1] https://www.reddit.com/r/Python/comments/2441cv/can_you_chan...
I'm not talking about physical measurements. How do you make a simulation that makes the sum of 0.9 + 0.09 + 0.009 + ... not equal to 1.0? It really comes down to whether math is universal. Maybe it's not, but I certainly don't see a trivial way to simulate this.
I haven't read the book, but I would imagine a story where you'd have a simulation where the maker decides that whenever the humans try to find pi by squeezing a polygon between two circles, you mess with them in the small digits. Similarly for the inverse squares sum.
Yet somehow a case is missed out, and some day someone asks the VM for pi in some other way, a discrepancy is found, and there you go...
Yet somehow a case is missed out, and some day someone asks the VM for pi in some other way, a discrepancy is found, and there you go...
that's a common trope, though. Baxter did it the Xeelee Sequence: the monads selected our particular universe "seed" bc it was amicable to life, and the proto-Xeelee and others tweaked universal constants (via unexplained methods) in the first few millionths of a second after the Big Bang that allowed for, well, us. And there's too many to list of the whole "Ancients/Precursors" who basically made intelligent life or whatever.
> Pi comes from logic, not nature.
What if logic is different in different universes? Logic is universal... but is it multiversal?
What if logic is different in different universes? Logic is universal... but is it multiversal?
Can you conceive of a universe where the abstract quantity 1 added to another abstract quantity 1 does not equal the abstract quantity 2? How would that work?
Edit: added the word abstract to make it clear that the context of the question is the realm of logic, and not the boundaries of nature (physics).
Edit: added the word abstract to make it clear that the context of the question is the realm of logic, and not the boundaries of nature (physics).
There could be a universe where "thingness" doesn't work like in ours -- e.g. where things aren't portable, or where the natural world doesn't divide easily into things. Maybe quantity wouldn't mean anything in such a universe.
More concretely, there are lots of ways that the mathematics of a different universe could be so different from ours that π is at best a theoretical concept.
There could be a universe where spacetime is discrete, à la Conway's Game of Life. We might even live in such a universe, but the discretization in ours is too small to probe.
There could be a universe with a different distance metric, such as a L1 ("taxicab") or L∞ (max difference), so that "circles" look like squares.
Or spacetime, or even "thingness" could be modulo some number p -- if p=2, then 1+1 = 0. Space could still be infinite, though -- it could have more dimensions, or it could be based on (e.g.) a complete field of characteristic p.
Or the metric could be p-adic, where two numbers are close if their difference is divisible by many powers of p -- kind of like the US highway system, where highways 80 and 880 are nearby. These metrics have rules like the triangle equality: the longest two sides of a triangle are the same length.
More concretely, there are lots of ways that the mathematics of a different universe could be so different from ours that π is at best a theoretical concept.
There could be a universe where spacetime is discrete, à la Conway's Game of Life. We might even live in such a universe, but the discretization in ours is too small to probe.
There could be a universe with a different distance metric, such as a L1 ("taxicab") or L∞ (max difference), so that "circles" look like squares.
Or spacetime, or even "thingness" could be modulo some number p -- if p=2, then 1+1 = 0. Space could still be infinite, though -- it could have more dimensions, or it could be based on (e.g.) a complete field of characteristic p.
Or the metric could be p-adic, where two numbers are close if their difference is divisible by many powers of p -- kind of like the US highway system, where highways 80 and 880 are nearby. These metrics have rules like the triangle equality: the longest two sides of a triangle are the same length.
You've mostly described physics i.e. nature, not logic.
Logic doesn't exist except in 2 forms:
(a) as a theory in minds
(b) as a description of how we see the world (nature) work
(a) as a theory in minds
(b) as a description of how we see the world (nature) work
[deleted]
Those are just representations of logic that have developed via evolution in the physical world.
The logic itself doesn't have a physical presence, it's just the inescapable conclusions drawable from any number of starting assumptions. There doesn't need to exist a universe with thinking things for logic to be logic.
Part of the explanation problem is that we tend to say "if you were to assume..." naturally in our language. But if course this causes a problem because logic is not part of nature and that phrase trends to assume some kind of thinking being doing the logical thinking.
The logic itself doesn't have a physical presence, it's just the inescapable conclusions drawable from any number of starting assumptions. There doesn't need to exist a universe with thinking things for logic to be logic.
Part of the explanation problem is that we tend to say "if you were to assume..." naturally in our language. But if course this causes a problem because logic is not part of nature and that phrase trends to assume some kind of thinking being doing the logical thinking.
This is a very strong philosophical assumption.
Logic definitely exists as a method in human philosophy. You could argue that it's somehow "baked in" to the universe, whether by a creator or by some other cause, and that we only discovered it and didn't invent it. But the claim that it transcends our universe and would exist and be accessible from all other possible universes, even ones with different physics, different kinds of space and time (two temporal dimensions? who knows?) etc doesn't seem obvious in the slightest.
This is especially true when there are so many types of logic used in mathematics and philosophy to begin with: first-order or higher-order logic, constructive logic, Peano arithmetic, Zermelo-Fraenkel, ZF with Choice, etc.
Even once you have the axioms, they may support different models. Are just the axioms true universally, but models varying across universes? Is it possible that there are universes where all statements are true? Are there universes with Choice and others without it?
Logic definitely exists as a method in human philosophy. You could argue that it's somehow "baked in" to the universe, whether by a creator or by some other cause, and that we only discovered it and didn't invent it. But the claim that it transcends our universe and would exist and be accessible from all other possible universes, even ones with different physics, different kinds of space and time (two temporal dimensions? who knows?) etc doesn't seem obvious in the slightest.
This is especially true when there are so many types of logic used in mathematics and philosophy to begin with: first-order or higher-order logic, constructive logic, Peano arithmetic, Zermelo-Fraenkel, ZF with Choice, etc.
Even once you have the axioms, they may support different models. Are just the axioms true universally, but models varying across universes? Is it possible that there are universes where all statements are true? Are there universes with Choice and others without it?
Again, it might be best not to conflate logic and physics. Having different universes with different laws of physics is very conceivable to the mind. Logic is pure philosophy. It's an exercise in reasoning. No physical observation is required. The various kinds of logic you cite are different subsets that coexist without ambiguity under the same big umbrella. That is, it's still the same logic. If one law were to create ambiguity with another (which is the mechanism used in proofs), then at least one would be declared void, or some more work would be needed to explain the conditions that lead to the paradox. Logical truths have no order of precedence. Our minds simply use the ones it's more comfortable to reason about as a scaffold to uncover new ones. But they all exist together and are all equally true without ambiguity, including those we have not uncovered yet. If we posit that another universe has logical truths that differ from our own, they will not only be foreign to us, but they will also be totally inaccessible to our imagination in a way that makes any kind of sense (unlike alternate laws of physics), as being able to reason about them in our own universe then gives them validity and creates the aforementioned ambiguity. 1 + 1 must be equal to 2 when adding those two quantities. If you say it doesn't, you're either mistaken, or you're talking about a different subset of the same logic (bases, set theory, etc) where the same symbols are used to mean something different. That doesn't qualify as conflicting logic.
You may want to read up on fuzzy set theory. Most certainly there is a set of 1 that when added to itself results in the set of 3, 4 or really any arbitrary set to varying degrees.
Bivalant sets is only a special case subset of fuzzy sets where 1+1 must equal 2.
Bivalant sets is only a special case subset of fuzzy sets where 1+1 must equal 2.
>the quantity 1 added to another quantity 1 does not equal the quantity 2
Our universe already fits the description, as there are algebras where that is not the case (similar as we have geometrics where triangle angles add up to > 180 degrees -- in fact geometry calculations on a sphere surface like earth works like that).
There are also probably physical examples where that's not the case (e.g. unlike adding 1+1 apple, adding two bodies of water gives you one unified body of water).
Could it hold even more fundamentally? Sure why not. Our "not being able to conceive it" doesn't mean much regardless to whether it's possible.
Our universe already fits the description, as there are algebras where that is not the case (similar as we have geometrics where triangle angles add up to > 180 degrees -- in fact geometry calculations on a sphere surface like earth works like that).
There are also probably physical examples where that's not the case (e.g. unlike adding 1+1 apple, adding two bodies of water gives you one unified body of water).
Could it hold even more fundamentally? Sure why not. Our "not being able to conceive it" doesn't mean much regardless to whether it's possible.
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").
"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").
The question "Is pi the same in all universes?" is poorly defined.
You're talking about pi as "the circle constant for whatever universe I'm in", whereas the GP is talking about pi as "the circle constant for a very specific geometry (that we once accepted as representative of the universe we're in)"
You're talking about pi as "the circle constant for whatever universe I'm in", whereas the GP is talking about pi as "the circle constant for a very specific geometry (that we once accepted as representative of the universe we're in)"
None of the other metrics come from nature either, that's the point. "Logic" is more of a catchall for "not from nature" than a specific axiomatic system.
I'm no mathematician but it's my understanding that axiomatic set theory is considered foundational for both what you call mathematics and also for discrete mathematics (aka logic).
Axiomatic set theory can't very well be considered foundational for logic, as its axioms are formulas of first-order predicate logic.
Yeah that's fair. I was more thinking about the natural mapping of set operations to boolean logic operations (e.g. OR equivalence to union for given sets, de Morgan's laws applied to sets and boolean operations, etc.).
> "Logic" refers to the most basic set of logical operations (AND, OR, etc.) and their use.
That's not what mathematicians mean with "logic". It's a much broader field than just boolean operators. Those are what computer architects and perhaps electrical engineers mean with "logic".
That's not what mathematicians mean with "logic". It's a much broader field than just boolean operators. Those are what computer architects and perhaps electrical engineers mean with "logic".
I meant the whole field of logic; I was keeping it simple.
Pi is not "from logic" however broadly you construe it.
Unless you endorse logicism, but I think that's both highly implausible and somewhat incidental.
Pi is not "from logic" however broadly you construe it.
Unless you endorse logicism, but I think that's both highly implausible and somewhat incidental.
Logic refers to its philosophical branch of study. The dichotomy is apt. Math is pure logic, meaning that you can be locked in a bunker for generations and keep discovering new laws. You can't do the same with physics. A mathematical or logical circle is nothing but a thought, an idea, it's perfect and has no physical equivalent. Pi is bound to that idea in any universe. The metric system is irrelevant (it mathematically cancels out).
Mjburgess is referring to the failure of the logiscist program in research in the foundations of mathematics: https://plato.stanford.edu/entries/logicism/
The article doesn't make it entirely clear that the program is widely considered to be a failure, but it is.
A better way to express the claim that you're making about mathematics would be to say that mathematical truths can be known a priori.
The article doesn't make it entirely clear that the program is widely considered to be a failure, but it is.
A better way to express the claim that you're making about mathematics would be to say that mathematical truths can be known a priori.
Logic refers to the most basic set of logical operations, and the axioms which we can build on top of those fundamental operations. Mathematics builds on these fundamentals with non-logical axioms, postulates which are used as building blocks for a more complex system.
Euclidean geometry is a mathematical system which starts from a set of five basic postulates:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Through any given point not on a line there passes exactly one line parallel to that line in the same plane.
These, with corresponding idealized definitions of 'point' and 'straight' and 'indefinite' and 'circle', rather than physical constructions, concretely define pi as a ratio between the circumference of the circle and the length of the line segment. Any universe where a mathematician can imagine these five postulates and use them to calculate pi will result in the same value.
If you discard or modify one of these postulates, you can come up with a system like a hyperbolic or elliptical geometry where pi could have a different value. Euclidean geometry closely corresponds to a our macroscopic physical world, one can form something fairly close with a straightedge and compass.
If you're imagining other universes, you might as well imagine one where the universe itself has a different geometry and the basic axioms are different. In that universe, children drawing a line in the dirt with their fingers would fundamentally understand that there are infinitely many lines parallel to the first in the same plane and would be taught so in primary school, but if they grew up to be a mathematician and replaced that postulate with ours, independently inventing Euclidean geometry, he could discover our definition of pi.
Conveniently, typical algebraic postulates like those resulting in Euler's formula, differential equations, integration, power series, and other mathematical constructs also result in the same definition of pi as that derived from Euclidean geometry. This seems to indicate to me that it's not just a constant in our preferred flavor of geometry but perhaps more fundamental.
Euclidean geometry is a mathematical system which starts from a set of five basic postulates:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Through any given point not on a line there passes exactly one line parallel to that line in the same plane.
These, with corresponding idealized definitions of 'point' and 'straight' and 'indefinite' and 'circle', rather than physical constructions, concretely define pi as a ratio between the circumference of the circle and the length of the line segment. Any universe where a mathematician can imagine these five postulates and use them to calculate pi will result in the same value.
If you discard or modify one of these postulates, you can come up with a system like a hyperbolic or elliptical geometry where pi could have a different value. Euclidean geometry closely corresponds to a our macroscopic physical world, one can form something fairly close with a straightedge and compass.
If you're imagining other universes, you might as well imagine one where the universe itself has a different geometry and the basic axioms are different. In that universe, children drawing a line in the dirt with their fingers would fundamentally understand that there are infinitely many lines parallel to the first in the same plane and would be taught so in primary school, but if they grew up to be a mathematician and replaced that postulate with ours, independently inventing Euclidean geometry, he could discover our definition of pi.
Conveniently, typical algebraic postulates like those resulting in Euler's formula, differential equations, integration, power series, and other mathematical constructs also result in the same definition of pi as that derived from Euclidean geometry. This seems to indicate to me that it's not just a constant in our preferred flavor of geometry but perhaps more fundamental.
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.
Yeah it's pretty limited what you can describe in words. You have to sort of "get it" with the diagrams what is being said.
I figure it'd be different in any non-euclidean geometry.
But we can get (hypothesize) non-euclidean geometry right here in our Universe. So wouldn't that make the "euclidean" part of the answer orthogonal to the "universe" part?
No, in non-euclidean geometry we have a notion of tangent space, where we measure angles and that looks the same as usual flat space. That is, what is meant with smaller and smaller circles in the physics answer. So the difference between euclidean and non euclidean geometry vanishes at very small scales and you can just define pi as the value in the limit.
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.
I assume that you mean solutions where x is in R here
I do mean that in the sense that you only need ℝ. However, it is also the case that the solutions over the complex numbers also all have period 2π.
And in my view, `R` is "fundamentally" a geometrical construct.
It's the only complete ordered field, so fundamentally it's an algebraic structure (with some topology thrown in but basic enough that it can be phrased without referring to geometry).
"complete" here is a matter of analysis / topology much more than it is a matter of algebra.
In some sense, the algebraic numbers are indeed important in algebra as being the 'basic' field with infinite characteristic.
You get to `R` by demanding completeness (for others reading, the demand that every converging sequence has an existing limit). I would say that the idea of limits and completeness are fundamentally topological / analytical. Which is closer to geometry than to algebra.
I think the main conclusion here is that there is good reason that the field "algebraic geometry" exists.
In some sense, the algebraic numbers are indeed important in algebra as being the 'basic' field with infinite characteristic.
You get to `R` by demanding completeness (for others reading, the demand that every converging sequence has an existing limit). I would say that the idea of limits and completeness are fundamentally topological / analytical. Which is closer to geometry than to algebra.
I think the main conclusion here is that there is good reason that the field "algebraic geometry" exists.
General topology is a pretty far cry from geometry though. What you need for the reals isn't even directly related to the topology of R, since you just need that every set with an upper bound has a least upper bound, which is a property that many types of orders can have and which doesn't necessarily have anything to do with geometry.
It depends what you give "priority" to -- my view is a philosophical one.
I think we ought really see `R` as an algebraic stand-in for talking about geometry.
I think we ought really see `R` as an algebraic stand-in for talking about geometry.
Eh, if R is geometric then so is algebra. The only thing geometric about R is that you can define an absolute value (i.e. it is an ordered ring) which is a property it inherits from the fractions. The fractions in turn are a basic consequence of the definition of a field (the fractions have a unique image into all fields).
If you've got addition and multiplication then you can get the reals by just adding additive inverse, multiplicative inverses and completeness, none of which are inherently geometric properties. So any geometry must come from the distributive property, because there's not much else it could be.
If you've got addition and multiplication then you can get the reals by just adding additive inverse, multiplicative inverses and completeness, none of which are inherently geometric properties. So any geometry must come from the distributive property, because there's not much else it could be.
R is an analytical construct, it exists to make limits meaningful. You don't need all of R to do geometry, if you are Ancient Greek you don't need any field at all.
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.
Said differently: periodic things in some sense express motions on a circle.
I'd say periodic things express motions on a loop, in the topological sense of looping back to itself. But because topology is only defined up to deformations, thus you cannot really say that loops are circles.
However, there is indeed a connection to circles. One way of seeing this by looking at the solutions to this differential equation over the complex "plane" and looking at the exp(i x) and exp(-i x) solutions, and seeing how these functions wrap the real number line around in a circle.
The complex plane fundamentally has a Euclidean geometry to it. So much so that if you grew up in a world with a strong non-euclidean geometry, or even grew up as some sort of digital being with no real notion of space at all, so long as you are able perform moderately sophisticated mathematics you are going to wind up discovering the complex numbers, and things like their absolute value and multiplication and the exponential function and how they map values around. Perhaps you never mentally arrange the complex numbers into a "plane", but none the less, all your basic geometric concepts are embedded into the algebraic operations on these complex numbers, and you will wind up effectively doing Euclidean geometry, even if you never identify it as such.
However, there is indeed a connection to circles. One way of seeing this by looking at the solutions to this differential equation over the complex "plane" and looking at the exp(i x) and exp(-i x) solutions, and seeing how these functions wrap the real number line around in a circle.
The complex plane fundamentally has a Euclidean geometry to it. So much so that if you grew up in a world with a strong non-euclidean geometry, or even grew up as some sort of digital being with no real notion of space at all, so long as you are able perform moderately sophisticated mathematics you are going to wind up discovering the complex numbers, and things like their absolute value and multiplication and the exponential function and how they map values around. Perhaps you never mentally arrange the complex numbers into a "plane", but none the less, all your basic geometric concepts are embedded into the algebraic operations on these complex numbers, and you will wind up effectively doing Euclidean geometry, even if you never identify it as such.
In a 1-dimensional spring-mass system there are no circles, but π emerges nonetheless.
Edit: natural frequency ω = Sqrt(k/m). ω comes out in rad/s, to convert to more natural units (cycles per second) π is required.
Edit: natural frequency ω = Sqrt(k/m). ω comes out in rad/s, to convert to more natural units (cycles per second) π is required.
To some degree, isn't that essentially just "adding one more turtle"?
Now instead of space being Euclidean, you're saying space-time is? And effectively if space-time were no longer Euclidean, then the conversion to cycles/sec would use the "alternate pi" of this non-euclidean space time wouldn't it?
Now instead of space being Euclidean, you're saying space-time is? And effectively if space-time were no longer Euclidean, then the conversion to cycles/sec would use the "alternate pi" of this non-euclidean space time wouldn't it?
Well, you might say that the circle emerges from the dynamics of that system. Two such spring-masses out of phase draw a circle.
No. Trig functions can be defined independently of geometry. One way is by considering the differential equation above. Another is through the Taylor expansion of the sin(x), which is periodic as a completely emergent property.
Likewise, it's hard to imagine what would happen to that very famous equation e^iπ + 1 = 0 if π were something different.
Likewise, it's hard to imagine what would happen to that very famous equation e^iπ + 1 = 0 if π were something different.
I know. But they do also reflect geometry.
You can derive pi/4 from summing alternating reciprocals of odd numbers. No geometry. You do need limits (since pi is transcendental, there are no purely algebraic solutions. You can use special functions, but those have limits or other transcendentals "in them").
https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
Indeed, good point.
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?
The self-similarity in these two DEs is why Euler's formula works. Are there any other such DEs?
In general any such linear differential equation
Something like f''(t) = -f(t) can be split up as follows:
f'(t) = M f(t)
f(t) = e^Mt f(0)
Something like f''(t) = -f(t) can be split up as follows:
f(t) = h'(t)
h'(t) = -f(t)
M = [[ 0 1]
[-1 0]]
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.
I guess this must be true, but from the sidelines it seems that “the set of points 1 metre from this particular point” and “the solution to this particular second-order differential equation” are quite different things. Any competent student of mathematics can connect them for sure, but it’s still quite remarkable they’re related isn’t it?
Yeah, a circle is basically a particular exchange between the velocity in the x and velocity in the y direction, offset by a phase equal to half of the period. With different kind of exchanges, you can get squircles and all types of other pseudo-circular shapes.
Or just rely on Euler's formula: e^(iπ)=-1
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.
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.
I'm not saying enjoying his story is a bad thing.
Yeah, the invocation of quantum entanglement in the 3 body problem was ... a problem. The best scifi of any "hardness" is by authors that know the limits of their own understanding of science and physics and beyond that point they don't try to explain things. I can accept a premise that violates physics, I have a tough time accepting an explanation that violates physics, if that makes sense. "Assume that the society actually found tachyons and figured out how to use them to communicate FTL" is fine, but "quantum entanglement really does let you communicate FTL" is hard to get past.
Like you said, though, still great books and a great writer :)
Like you said, though, still great books and a great writer :)
Do you like how Star Trek handles this? Your transporter can't determine a particle's location and trajectory simultaneously? Add a Heisenberg compensator. Fixed. You can't accelerate to warp without soupifying the crew? Inertial dampeners. But never explain how either of these things work.
Generally speaking, yep! It works for Star Trek, although sometimes they get to elaborate in their descriptions or start to take the "science" too far. But generally they walk the line of science-y sounding explanation decently well. How about you?
What’s the name of this short story?
I think it is The Mirror
It's... it's not at all short.
Anathem by Neal Stephenson also touches on that topic.
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!
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)
(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)
"Locally flat" means flat only very close to the point of consideration. That is the mathematical definition, the "practical" definition is "flat only as long as you do not stray further away than some characteristic measure of distance defining your problem and accuracy requirements". So as long as your circle is very small then the " experimentally measured" pi and the math pi are the same. The moment your circle becomes big enough to pass through interesting (measurable) gravitational effects, the experimental value will be different. If anything, experimentally measuring that difference is one way to experimentally measure spacetime curvature.
That's why I asked about GR specifically, to have a common framework to work with. Without it you can say words like experimentally, practical and talking about differences between different pis whatever that means but this will only lead to misunderstandings at best and nonsense at worst
Not in accelerating frames of reference, which includes both rotating frames (such as spinning objects) and frames with high gravitational distortion.
Spinning black holes have a measured circumference that is not 2 x pi x r.
(Consequences of the accelerating frame of reference include the spaghettification in a black hole... We can't use a mathematical conversion to go to a frame of reference where the subject isn't being spaghettified).
Spinning black holes have a measured circumference that is not 2 x pi x r.
(Consequences of the accelerating frame of reference include the spaghettification in a black hole... We can't use a mathematical conversion to go to a frame of reference where the subject isn't being spaghettified).
Why not? (to the first sentence, locally is locally)
But I think I have found the source of my confusion in my original post, I was thinking about manifolds only in the topological sense and for some reason forgot that when there is a metric then we have to preserve distances. No 'mug and a donut' type of shenanigans.
But I think I have found the source of my confusion in my original post, I was thinking about manifolds only in the topological sense and for some reason forgot that when there is a metric then we have to preserve distances. No 'mug and a donut' type of shenanigans.
As distance to center of black hole approaches zero, curvature approaches infinity. There is no "locally" one can constrain small enough to observe no curvature upon approach to a singularity... Given two particles arbitrarily close together at arbitrary velocity (< c) relative to the center of a black hole, there is a distance from the center at which they will encounter unequal acceleration.
In Riemannian geometry (i.e. GR) each point has some local invariants associated to it, a famous one being curvature. "Local" here means that it only depends on the metric in an arbitrarily small disc/ball around the point. So the answer is no, in GR you can't always change coordinates so that space is locally flat. (This is in contrast to symplectic geometry for example, where each point does look locally like the model space).
Ok, yeah. We obviously can't get rid of a curvature. I think SE answer [1] explains what I assumed incorrectly to mean 'locally flat'. I meant it so that you can make a metric look like Minkowski metric at a point p, basically existence of normal coordinates. Of course second derivatives (and thus curvature) won't vanish in the neighborhood. Case closed, thank you.
[1] https://physics.stackexchange.com/a/157019
[1] https://physics.stackexchange.com/a/157019
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.
But you can define an experimentally measured pi. Define an experimental circle (the set of points experimentally measured to be at a given distance from a center) and then experimentally measure its circumference. The fact that this experimental measurement is different from the pi defined through an l2 norm in Euclidean space is certainly cool and interesting. It is basically a way to measure gravitational curvature.
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.
Not quite.
Observing an object with high relative velocity distorts the observed shape, but all observers can agree what the shape would be at zero relative velocity. For similar reasons astronomers say distant objects are redshifted rather than assuming they are the colors directly observed.
Observing an object with high relative velocity distorts the observed shape, but all observers can agree what the shape would be at zero relative velocity. For similar reasons astronomers say distant objects are redshifted rather than assuming they are the colors directly observed.
Although in the special case of a sphere, despite the length contraction, the observed behaviour is a rotation because light travel time perfectly balances length contraction: https://www.spacetimetravel.org/fussball/fussball.html
That would be very unusual (unorthodox) definition of Pi though.
Just because it's the same number doesn't mean it's the same relationship.
Any definition that relies on trig (sometimes disguised as trig expansion) assumes the existence of a flat Euclidean space - which is a convenient mathematical abstraction in real-world terms, but not required in the general case.
Most everyday physics assumes a flat space, so it's not surprising certain constants fall out of this.
There are more complex and interesting definitions which rely on relationships defined by group theory. I don't know enough about those to say if they're just more complex ways to operate in a flat space, or more complex ways to disguise simple trig expansions, or if they're completely independent, or if something else is going on.
Any definition that relies on trig (sometimes disguised as trig expansion) assumes the existence of a flat Euclidean space - which is a convenient mathematical abstraction in real-world terms, but not required in the general case.
Most everyday physics assumes a flat space, so it's not surprising certain constants fall out of this.
There are more complex and interesting definitions which rely on relationships defined by group theory. I don't know enough about those to say if they're just more complex ways to operate in a flat space, or more complex ways to disguise simple trig expansions, or if they're completely independent, or if something else is going on.
That's not fully true.
You can get trig functions without any kind of geometry.
You can get trig functions without any kind of geometry.
In some alternate universes they use τ instead and I would rather live in those universes.
https://tauday.com/
https://tauday.com/
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.
> Feet aren't really different from meters from a physical perspective
As a non American I can't help but think if you are doing anything that involves short and long distances feet must surley be painful. Either you use 5280 feet instead of a mile, or 0.000189 miles instead of a foot. And then you have yards...
Madness!
As a non American I can't help but think if you are doing anything that involves short and long distances feet must surley be painful. Either you use 5280 feet instead of a mile, or 0.000189 miles instead of a foot. And then you have yards...
Madness!
a) very few people work on the scale where miles and feet are interconverted.
b) The mistake was picking base-10 as the basis of the metric system. The Base-12/Reciprocal-halves (2^63/2^4) system of feet, inches, fractional inches, is hands-down the best way to engineer and build things on the human scale. It makes it super easy to scale measurements by N/2, N/3, N/4, N/6, N/8, N/12, N/2^m. Three weeks ago had to make 9 evenly spaced holes in a 24' board, with an inch of buffer on either side, which works out to 35-3/4". This results in really convenient interval of N3'-less-N/4" (35-3/4, 71-1/2, 107-1/4, 143"), so convenient I still remember them weeks later.
The yucky part is when you have to use a base(2*5) calculator. Now you get 35.75, 71.5, 107.25, 143", which is a much less obvious pattern.
Every day I'm in the shop, I yearn for a dozenal calculator and measuring tape.
One can dream.
http://www.dozenal.org/
Base 6 is also nice:
http://www.seximal.net
b) The mistake was picking base-10 as the basis of the metric system. The Base-12/Reciprocal-halves (2^63/2^4) system of feet, inches, fractional inches, is hands-down the best way to engineer and build things on the human scale. It makes it super easy to scale measurements by N/2, N/3, N/4, N/6, N/8, N/12, N/2^m. Three weeks ago had to make 9 evenly spaced holes in a 24' board, with an inch of buffer on either side, which works out to 35-3/4". This results in really convenient interval of N3'-less-N/4" (35-3/4, 71-1/2, 107-1/4, 143"), so convenient I still remember them weeks later.
The yucky part is when you have to use a base(2*5) calculator. Now you get 35.75, 71.5, 107.25, 143", which is a much less obvious pattern.
Every day I'm in the shop, I yearn for a dozenal calculator and measuring tape.
One can dream.
http://www.dozenal.org/
Base 6 is also nice:
http://www.seximal.net
As an American, I have some distant hope that we will switch .. but no sooner than the year 2050
You just switch to miles for anything longer than a few hundred feet. Big units for big things, small units for small things.
The flipside to this I’ve been encountering a lot is engineering drawings in metric for things that are several meters long and all the measurements are in millimeters! Why all the big numbers!
The flipside to this I’ve been encountering a lot is engineering drawings in metric for things that are several meters long and all the measurements are in millimeters! Why all the big numbers!
> measurements are in millimeters! Why all the big numbers!
Precision, usually if you are getting something fabricated the numbers are quoted in millimeters if the millimeters is under 10,000. So a desk might be 1800x800x700mm. I don't think I have ever seen something listed as 120000x10000x10000mm. At that point they would switch to meters.
Precision, usually if you are getting something fabricated the numbers are quoted in millimeters if the millimeters is under 10,000. So a desk might be 1800x800x700mm. I don't think I have ever seen something listed as 120000x10000x10000mm. At that point they would switch to meters.
I have so many piles of drawings at work for all sorts of European manufactured machines that are about 6m in length that have their layouts (where mm level precision is silly and pointless) drawn out and measured in mm. 6000mm x 2000mm x 10000mm etc etc. It’s like...why??
As for the precision, I suspect it’s to defeat the perils of floating point numbers in computers...if you only need mm precision, you rather your CNC computer do integer math and not floating point math...am I correct?
As for the precision, I suspect it’s to defeat the perils of floating point numbers in computers...if you only need mm precision, you rather your CNC computer do integer math and not floating point math...am I correct?
> As for the precision, I suspect it’s to defeat the perils of floating point numbers in computers...if you only need mm precision, you rather your CNC computer do integer math and not floating point math...am I correct?
I doubt that. It would be trivial to have the computer convert from 6.000m or 600.0cm to 6000mm internally.
I doubt that. It would be trivial to have the computer convert from 6.000m or 600.0cm to 6000mm internally.
Why not, though? Millimeters are trivial to convert to meters in your head, since you don't need to calculate anything; just move the decimal point.
Using millimeters gives you a "standard precision" which is a boon for interoperability.
Using millimeters gives you a "standard precision" which is a boon for interoperability.
>Why not, though?
Because all my tape measures are in inches ;(
Because all my tape measures are in inches ;(
You wouldn’t be helped by cm-s either :/
Because if everyone millimeters then there is no confusion. For that reason I rarely see anything non-consumer specified in cm.
> CNC computer do integer math and not floating point math...am I correct?
Good point.
> CNC computer do integer math and not floating point math...am I correct?
Good point.
For precision work using inches, decimal thousandths of an inch ('thou') are typically used - for what I assume are similar reasons.
No one ever uses miles and feet together. You just use decimal miles: 5.3 miles, etc.
You use feet for short distances, miles for long.
You use feet for short distances, miles for long.
Or more likely, fractional miles. In Chicago, a standard city block is 1/8 of a mile by 1/16 of a mile (usually the longer dimension is considered a "block" for describing distances since it corresponds to the grid's "hunnerts" for addresses¹, something which confuses my wife to no end, especially when a 3-block walk may cover more or less than 3 physical blocks).
---
1. There are a few areas where the hunnerts don't correspond to 1/8 mile blocks: The first mile from Madison to Roosevelt goes from 0 to 1200. Then from 16th to 22nd (Cermak) there are some missing hunnerts and again one missing hunnert between 26th and 31st. Some of these are a consequence of the replatting of the city at the beginning of the twentieth century and some of it, who knows?
---
1. There are a few areas where the hunnerts don't correspond to 1/8 mile blocks: The first mile from Madison to Roosevelt goes from 0 to 1200. Then from 16th to 22nd (Cermak) there are some missing hunnerts and again one missing hunnert between 26th and 31st. Some of these are a consequence of the replatting of the city at the beginning of the twentieth century and some of it, who knows?
Indeed, the claimed "elegance" of SI doesn't really hold up, with the SI base units now defined in terms of inelegant defining constants ("a metre is the length of the path travelled by light in a vacuum in 1/299 792 458 of a second") and base-10 being inelegant to work with on modern computers.
The whole point of a system of units is that the vast majority of people working with them don't need to deal with the fundamental definitions. For those who do, there's no system that's any less bad than SI anyway (except natural units, but those wouldn't work for day to day things).
It is quite amusing that the old Imperial system of using fractions everywhere is actually more compatible with binary computers (provided the denominators are a power of 2)
Have you ever worked with radians?
Suppose you were teaching radians. Which is an easier explanation of a 90 degree corner. One based on pi, or one based on tau?
The advantage of tau is largely didactic. It helps highschool students a lot more than it does phds.
Suppose you were teaching radians. Which is an easier explanation of a 90 degree corner. One based on pi, or one based on tau?
The advantage of tau is largely didactic. It helps highschool students a lot more than it does phds.
Programmers would never let this discrepancy exist for this long!
Unfortunately, mathematicians are too lazy to "refactor" their work ...
Unfortunately, mathematicians are too lazy to "refactor" their work ...
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).
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).
...and ½τr² makes sense because it comes from an integration.
> If there's a better way, they'd use it
Oh sweet summer child.
> If there's a better way, they'd use it
Oh sweet summer child.
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.
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.
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.
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
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.
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.
There's only one universe, QED.
There's only one directly observable universe*
There may be other universes created by whatever mechanism created ours. It is possible also that in some such mechanisms, indirect effects of those universes may be observable.
There may be other universes created by whatever mechanism created ours. It is possible also that in some such mechanisms, indirect effects of those universes may be observable.
There's a limit to how much we can hypothesize about the nature of other universes, given we don't know if they actually exist. It'd be foolish to claim "QED" either way.
Since we can't possibly know anything about these universes for all intents and purposes we can say that they don't exist. This QED is more than acceptable.
I actually think what your saying is foolish. There's no "other universes." As I stated there's only one universe. Seems obvious to me.
"further research may be required".
Ask any student of etymology and they'll give a similar answer.
You've made my day haha
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).
> 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.
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.
Sometimes I think the whole universe is a cost function iteratively optimizing those constants.
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.
For those wanting to read more about this this is exactly how the real numbers are defined in the lambda calculus.
https://cs.stackexchange.com/questions/2272/representing-neg...
https://cs.stackexchange.com/questions/2272/representing-neg...
[deleted]
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.
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.
Pi-the-number is usually lowercase (and frequently italic) pi, "𝜋" , almost never uppercase "Π", which is used for other things.
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?
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?
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
> 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?
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?
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
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
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
https://imgur.com/a/VBXLVA6
Unsurprisingly, like most such questions, the effective answer is: "Define π"
Well this blog post ended precisely how I expected it to end. "Maybe."
How many constants that exist in nature are also in logic?
Or is it just Pi and E?
Or is it just Pi and E?
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.
Pi is pi because of our axioms.
If you define things differently, pi doesn'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.
When we work with Pi, we embed that universe into other universes.
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'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.
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.
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.
as per the article, in this universe, and potentially others as long as they have some notion of smooth space, we infer an abstract ideal of flatness and from there derive Pi.
The notion of smooth space is an approximation, it doesn't exist. So yes, you could find a space with an "apparent pi" within a given delta of the ideal, but it won't be "ideal pi"
You either missed or ignored keithnz's point. There are plenty of spaces that are close enough to smooth that we can deduce the properties of a perfectly smooth space, and from that, the value of a perfect pi.
So we can't find a space that has that same perfect pi. So what?
So we can't find a space that has that same perfect pi. So what?
My point is that the argument is wrong. "ideal flat space" isn't a property of our universe. That "perfect pi" is a property of the mathematical abstraction of "smooth space". Our universe might asymptotically flirt with such a state but "flat space" isn't real not in the sense that the speed of light and force constants are descriptions of things that actually happen and actually exist.
As others have already mentioned, no geometry is required for the definition of pi. Real numbers are enough, where of course the problematic infinity does come up - but there do exist quite simple approximations that will calculate pi to any degree of precision regardless of physical space.
There is no real argument, essentially the article says well, "it depends" and basically we have no idea, but, given a few assumptions, is it reasonable that other universes would have denizens that conceive of circles and calculate the same value of Pi? and they postulate that if that also had a notion of flat space (not that the universe comports to that) then they too probably would come up with the same value of Pi
Space itself doesn't exist. As soon as you put something that exists in it, that part of it is no longer space.
I'd think the construct of "The shape of something where all points that are x distance away from a certain point" is something that would be universal. (Thinking of spheres rather than circles.)
AFAIK pi crops up in any dimensionality in any type of space for spheres.
AFAIK pi crops up in any dimensionality in any type of space for spheres.
You don't need any geometry to define pi. At uni, I had it defined as the smallest positive zero of sin(x). And sin(x) was defined axiomatically (in terms of real analysis) in conjunction with cos(x) (and then shown to be equal to its Maclaurin expansion).
Space is locally euclidian. You can easily define pi as a universal constant in our universe, in terms of local flat regions.
local euclidian-ness is an approximation. Nowhere is flat and it wont be until well after the heat death of the universe when every particle is outside the light cone of every other (making a lot of dubious assumptions about the eventual fate of the universe)
No, local euclidean-ness is (more or less) the equivalence principle. The locality could be very small though! But it would break GR if local flatness weren't possible.
Equivalence principal is about the fact we can't differentiate between a flat spacetime+force and a curved spacetime. It doesn't require actual flat spacetime to exist anywhere outside of a model. It only says you can't tell how curved it is based on the forces acting on you.
Different argument to same end: as long as the universe is expanding, nowhere can be flat. Any 2 things have an inherent repulsive force due to spatial expansion.
Different argument to same end: as long as the universe is expanding, nowhere can be flat. Any 2 things have an inherent repulsive force due to spatial expansion.
Pi comes out in the limit of flat spacetime.
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.
I meant in a hexagonal coordinate system, like the hexagon tiles in games.
It’s still exactly 3, assuming you’re defining it by π´ = c/2r and c = [count of number of hexes that require exactly r hex-boundary crossings to be reached from the starting cell]
6 cells at a distance of 1 hex, 12 at a distance of 2, 18 at a distance of 3, and the underlying growth pattern requires adding 6 on every discrete ring.
6 cells at a distance of 1 hex, 12 at a distance of 2, 18 at a distance of 3, and the underlying growth pattern requires adding 6 on every discrete ring.
Interesting. Good direction in thought with counting the whole cell as a unit. In that case shouldn't the diameter be 3 instead of 2? See the grid in [1]. The center tile counts as distance 1, the left adjacent tile counts as 1, and the right adjacent tile counts as 1, so the total diameter d is 3. Circumference c = 6. c/d = 6/3 = 2. Pi = 2.
Then the next ring has 12 tiles. c = 12. d = 5. c/d = 12/5 = 2.4. So Pi = 2.4 for the next ring!
[1] https://www.gamedev.net/articles/programming/general-and-gam...
Then the next ring has 12 tiles. c = 12. d = 5. c/d = 12/5 = 2.4. So Pi = 2.4 for the next ring!
[1] https://www.gamedev.net/articles/programming/general-and-gam...
I’d count the central cell as distance = 0 for the purposes of board game mechanics, and use c=2πr rather than c=πD, but I guess that would make the formula fail my definition of circumference.
If we go from center of the cell to center of the cell, the distance would be 2. I guest that would work.
That's a discrete space, no?
Yes, discrete space coordinate. Something like this with a visual grid, https://www.gamedev.net/articles/programming/general-and-gam....
No. Some universes use tau :^)
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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.