This person is not a sailor. Sailing orthogonal to the wind, a "beam reach", is the fastest point of sail due to the lift of the sail.
What does the "circle" look like with correct assumptions?
π in Other Universes
61–70 of 113 posts
Re: π in Other Universes
#62Earlier quoted context omitted.
> Another place our pi will come up is in the exponential function. It's periodic with period 2πi. Isn't it the opposite? As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equa…
Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results. note 1: exp(x) can alternatively be defined by the exponential series, but that series…
As I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities:
f(0) = 1
f'(x) = i f(x)
So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*
Re: π in Other Universes
#63Earlier quoted context omitted.
> Another place our pi will come up is in the exponential function. It's periodic with period 2πi. Isn't it the opposite? As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equa…
> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2 Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix…
I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π.
If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for example, Euler's formula below would still hold:
e^ix = cos x + i sin x
Re: π in Other Universes
#64Note that even if another universe has a different π when it comes to geometry they are still going to also have an important constant that has the same value as our π. E.g., the zeros of the function defined by the series x - x^3/3! + x^5/5! - x^7/7! + ... are nπ where n is an integer and π is our π. Another place our pi will come up is in the exponential function. It's periodic with period 2πi.
Right. Also (just a few more concrete examples): • the sum of the series 4(1 - 1/3 + 1/5 - 1/7 + …) will still be our π: https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 • the sum of the series (1 + 1/4 + 1/9 + 1/16 + 1/25 + …) will still be π²/6: https://en.wikipedia.org/wiki/Basel_problem • (therefore) the probability that two numbers chosen uniformly at random from [1…N] are relatively prime will still app…
In a change from the normal refrain of 'there's an XKCD about that' - in this case there is an Saturday Morning Breakfast Cereal (SMBC) about it: https://www.smbc-comics.com/comic/pi-2?ref=refind
For those unwilling to click-through, it essentially posits an alternate history where infinite series were explored by mathematicians before geometry, so rather than being surprised that the 'circle constant' is found in many infinite series, we would instead be surprised that the 'infinite series constant' is found in the geometry of a circle.
Re: π in Other Universes
#65> Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable This is a really, really n…
A good example of that is how the Axiom of choice impacts the measure/probability theory.
It imply the existence of some sets that cannot be Lebesgue measured (which is an generalization of width, volume, etc for arbitrary sets, also generalization of probability for arbitrary sets)... but it's not possible to present a single example of those non measurable sets, only prove that they exist.
And it's possible to construct an alternative theory with the axiom of determinacy, then any subset of R is measurable.
* https://en.wikipedia.org/wiki/Axiom_of_choice * https://en.wikipedia.org/wiki/Axiom_of_determinacy * https://en.wikipedia.org/wiki/Lebesgue_measure
Re: π in Other Universes
#66All of these assume your background metric is Euclidean. If your background 2D metric is a projection of a warped 3D space, you can make π as big as you want by tugging on the centre of the circle.
It's not the background metric but the space geometry is assumed Euclidean - in non-Euclidean geometry the ratio of of the circumference of any circle to the diameter of that is not a constant, it depends on such diameter (so you simply cannot define 'pi' in that case)
Well.... One still obtains pi for the ratio of circumference vs diameter in the limit that the diameter goes to zero.
Re: π in Other Universes
#67Earlier quoted context omitted.
Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results. note 1: exp(x) can alternatively be defined by the exponential series, but that series…
As I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities: f(0) = 1 f'(x) = i f(x) So any function that satisfies these equalities can work as a "complex exponential" function which we denote as e^ix. So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*
The only function that satisfies those two equalities is e^ix which has period 2π.
Re: π in Other Universes
#68Didn't 3blue1brown have a video on exactly this?
Re: π in Other Universes
#69Earlier quoted context omitted.
> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2 Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix…
I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for exampl…
You seem to think that the 2pi is injected into the definition of e^ix somewhere, but actually it's the other way round, 2pi comes out as a theorem. I'll give the rough outline.
exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ...
(BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't be too hard to see because it's already a Tailor series.)
Purely from this you can prove that exp(ix) with real x is periodic with period 6.28...
It just so happens that this number is also the circumference of the unit circle.
Re: π in Other Universes
#70Earlier quoted context omitted.
> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2 Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix…
I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for exampl…
I guess radians are “magic” in that sin and cos can be defined by infinite series that look nice (and feel canonical). You have to manipulate those series to get 360 or even just revolutions.