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π in Other Universes

azeemba.com

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Re: π in Other Universes

#11

Post by azeemba »

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, the value of π is always between 3 and 4 (inclusive). Unfortunately the article might have gotten the hug of death so here is an alternative link: https://www.researchgate.net/publication/353330827_Extremal_...

Re: π in Other Universes

#12

Post by SamBam »

dclowd9901 wrote: post #5

I must be missing something. In the example of using a sailboat with constant wind and distance, wouldn’t sailing against the wind (let’s call it any constant oppositional force), cause us to get a circle, just shifted from the origin? Not an ellipsis?

I think you're right. Just apply a transform equal to the speed and direction of the wind.

Re: π in Other Universes

#15

Post by quickthrower2 »

cvoss wrote: post #9
tedunangst wrote: post #4

Not sure about your universe, but here on earth, pi is 2. The length of the equator is 4 times the distance from the pole. (Approx.)

Why stop there? Take the circle with its center at one pole, its radius running an entire meridian, and its perimeter making an infinitesimally tight loop around the other pole. That exhibits a pi that's zero.

You can have any Pi_Earth you like where 0 < Pi_Earth < Pi_Euclidian

Re: π in Other Universes

#16

Post by tzs »

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.

Re: π in Other Universes

#19

Post by ajkjk »

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.

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