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π in Other Universes

azeemba.com

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Re: π in Other Universes

#21

Post by charlieyu1 »

azeemba wrote: post #11
TheOtherHobbes wrote: post #6

All 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.

There is no concept of the "background metric" here. Both the radius and the circumference are measured in the defined metric itself. Any metric that "pulls on the origin" compared to Euclidean distance will have to do the mapping in a continuous way. This will basically result in both the radius and circumference being expanded in that metric. Matter of fact, I linked an article that proves that for _all_ metrics, t…

How is circumference defined?

And I can think of a counterexample on a sphere, just using Euclidean distance on the surface. Consider a circle with centre at North Pole and radius being the distance from the North Pole to a point on the equator. For this circle it is easy to find out that pi=2

Re: π in Other Universes

#22

Post by daxfohl »

daxfohl wrote: post #20

The area of the circle in Manhattan distance comes out to 2 million, but pi * r^2 is 4 million. What am I doing wrong?

Oh, I was measuring the sides in Euclidean length. In Manhattan length they're 2000 each, so area is 4 million.

Re: π in Other Universes

#23

Post by defrost »

ajkjk wrote: post #19
defrost wrote: post #3

IF I've correctly assesed the zeitgeist of HN postings THEN it follows Terence Tao's Introduction to Measure Theory must be a bullet. https://news.ycombinator.com/item?id=38064211 But seriously, who's going to read|skim a free 260+ tract on measure theory? https://en.wikipedia.org/wiki/Measure_(mathematics)

> who's going to read|skim a free 260+ tract on measure theory? Why is that so hard to believe? People read 260 page books all the time. I'm not going to read this one, but only because it's not my area of interest. I'm busy reading 100+ page books on other subjects.

Take that as a tongue in cheek comment - I've read such things with close attention, I was studying measure theory back in the 1980s when I first met the author of this work here in Australia.

There is a subset of people on HN that do read and enjoy mathematical texts, they appear outnumbered by a larger group that seem to post and comment on anything Terence Tao without seeming to be that deep in the actual math, which is fine, but it has struck me as a HN trend of late.

Re: π in Other Universes

#24

Post by jacquesm »

lfnoise wrote: post #18

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's interesting is that if you manage to exceed the hull speed doing that you'll end up surfing on your own bow wave!

Re: π in Other Universes

#25

Post by Zanni »

lfnoise wrote: post #18

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.

I knew someone would make this comment. I love HN for this kind of pedantry when it's specific, accurate and doesn't dismiss the entire article for one inaccurate analogy.

Re: π in Other Universes

#26

Post by jws »

Zanni wrote: post #25
lfnoise wrote: post #18

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.

I knew someone would make this comment. I love HN for this kind of pedantry when it's specific, accurate and doesn't dismiss the entire article for one inaccurate analogy.

Also, "broad reach" would like a word with lfnoise. (It's complicated.)

https://physics.stackexchange.com/questions/186515/why-is-a-...

Re: π in Other Universes

#27

Post by cyclotron3k »

The boat analogy seems particularly poor.

a) Comparing a sailboat on a windy day to a sail boat on an [implied] non-windy day? Surely the boat with no wind wouldn't even have a circle.

b) I'm no boatologist, but if the wind is X knots, then the boat can travel downwind at a rate of X knots, but contrary to what the article states, the boat would be able to travel cross-winds at some multiple of X. So you would get something resembling an oval, but in the opposite orientation as depicted.

Also, it's worth pointing out that it's perfectly possible for a boat to travel "into" the wind via "tacking and jibing"

Re: π in Other Universes

#29

Post by passion__desire »

tzs wrote: post #16

Note 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.

would their π re-emerge if unit distance i.e distance between 2 and 3, 5 and 6 is defined by their metric. sort of like change in base in number systems.

Re: π in Other Universes

#30

Post by passion__desire »

skykooler wrote: post #28

The hexagonal metric at the end uses pi in its definition - is this our value of pi, or the value of 3 that that metric provides?

My belief, which could be wrong, if we change π and distance metric i.e. definition of unit distance between 1 and 2, 4 and 5, 10 and 11 to be their unit distance, all the equations involving numbers and pi would come out to be same. e.g. basel problem etc.
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