Have you ever wondered...

[mendhak]

What would the shape of a circle be like if PI were exactly three?

[Memnoch1207]

um...an oval? no...an ellipse? a frisbee? no wait that's round....how about one of my saturday morning pancakes! haha come and try to calculate the radius and circumference on one of those bad boys.

[twanvl]

What would the shape of a circle be like if PI were exactly three?

Pi is never 3 (in euclidian space). That is like asking "What would a ball look like if it were square", it just doesn't make any sense.

[bugzpodder]

pi is 30 in base pi/3

[mendhak]

I said if PI were 3... what would a circle look like???

Pi is never 3 (in euclidian space). That is like asking "What would a ball look like if it were square", it just doesn't make any sense.

I'm assuming you to be better at mathematics than me, can you explain that? How is PI never 3 in Euclidean space? What is Euclidian space, btw???

[twanvl]

Euclidean space (sorry mada a typo) is the normal space (Rⁿ), you have an x axis and a y axis (in 2D) There are also other spaces with strange properties. For example there is is space where every two lines intersect at 2 points instead of the usual 1. In a lot of spaces pi simply doesn't make sense, but in euclidian space it does.
Pi is defined as "0.5(circumference of circle/diameter)"

from mathworld.wolfram.com:
A real number denoted which is defined as the ratio of a circle's circumference C to its diameter (d=2r)

This number is always 3.14159265..., for all circles. Saying "if pi were 3" is like saying "if the surface area of a square wasn't side²" or "what if batchlors were married" or "what if blue were green".

Some people have tried though...

[RAEsquivelC]

Yeah, why not try, "If Pi were 1, or 2, or 3, or 4, or ...". Why not, simply, "If PI <> pi", or, "If 3.14159 <> 3.14159.

I think it's all the result of our non-sense moderen culture, where a spade is no longer called a spade! It's something like, "If 1=2, and 2=3, what would 3 equal?" When I promptly answered, "Four!", my big brother corrected me by saying, "Wrong!, I never said that three was anything else than three. " He showed me, like Humpty Dumpty told Alice, "A word means exactly what I want it to mean!" So, are we to be Humpty Dumptys, or Alices?

[RAEsquivelC]

Well, now, come to think of it, in modern math, I guess you could define a "space" called Space3, where Pi3 = 3, so that, the circle3 would have its circumference3 = 2*Pi3*r. For r = 1, C3 = 2*3*1 = 6. Yep, and, a Circle3 in geometrical Space3 would look just like a hexagon in our normal space, SpacePiN, in which PiN = Pi = 3.14159...

Does the above make sense?

And, for Space2, Pi2 = 2, and C2 = 2*Pi2*r, and, for r = 1, we would get, C3 = 2*2*1 = 4, and, a Circle2 would look just like our usual square!

Hey, mabe the OP has hit on a new, exciting, umpteen-dimensional spaces Newer-than-New-Math. Who will be its conquerer, expanding the math to its ultimate, and, then, finding some good applications for it in SpacePi?

[Raptor]

But isn't the definition of ac circle that all points of the circumference have the same distance to the center of the circle (= the radius).

But obviously this isn't the case in an hexagon or a square - and what would be the radius of a square or a hexagon? The distance from center to a corner or to the middle of a side?

So if it isn't a circle you can't calculate the circumference with 2*pi*r...

But I think it is just a problem of our imagination. Saying pi=3 would mean that a circle has a greater circumference but still the same radius and that seems just impossible because we cannot imagine that - it's just like saying what would a world look like that has four dimensions of space instead of three ... it's theoretically possible but we can't imagine it!

Edit:

one thing just came to my mind:

for PIx < pi: just imagine a circle with the center of the circle not lying within the same layer than the points of the circumference (so the radius gets longer)...

[nishantp]

Originally posted by RAEsquivelC
It's something like, "If 1=2, and 2=3, what would 3 equal?" When I promptly answered, "Four!", my big brother corrected me by saying, "Wrong!, I never said that three was anything else than three. "

If 1=2, then how can 2=3? 2 has to = 1, just like you just said. If 2=3, then 3 = 2. So you're brother was screwing with you worse than you thought

[Guv]

A circle is defined as the locus of all points equidistant from a given point called the center.

Given the above definition, a circle would look like a circle for any value of Pi. Changing Pi would merely change the ratio between the radius and the circumference.

If a circle is drawn on a rotating plane, the circumference is viewed as shorter by an observer not on the rotating plane, but the radius is viewed as the same length. This is due to General Relativity distance distortions. For the stationary observer, the ratio between the circumference and the radius is less than Pi.

I wonder how fast the object must rotate to make the ratio three.

The locus of all points on a perfect sphere and equidistant from the North Pole conforms to the definition. For such a circle, Pi does not have it usual value. On a very large sphere, a circle with a small radius should have a ratio close to the normal value of Pi. For circles farther from the North Pole, the ratio gets smaller. There is likely to be a distance for which the ratio is three.

BTW: In spherical geometry, the interior angles of a triangle add up to more than 180 degrees. I think the excess over 180 is proportional to the area divided by the radius of the sphere, approaching 180 degrees as the area decreases.

Another BTW: On a cylinder or cone (and certain other curved surfaces), the intrinsic geometry is Euclidean. The sum of the interior angles of a triangle is 180, the Pythagorean Theorem is valid, Pi has its usual value, et cetera.