Originally posted by Ollies930
Interesting. I am only familiar with simple circles as defined in Euclidean geometry, where Pi is a constant. Please give me a detailed mathematical example where Pi is other than conventional value.
Ok, distance depends on how you measure distance. There is a famous example in mathematics called the "Manhattan distance" or "taxi distance." This measures the distance between two street corners on Manhattan when you travel by taxi. The point is, if you travel from 46th St, 5th Ave to 50th St 6th Ave, you have to travel 1 block in the "x" direction (avenues, or east-west) and 4 blocks in the "y" direction (north-south). The total distance travelled is simply the sum of the two, 1+4=5.
This is unlike the Euclidean notinon of distance which is the way the crow flies. (Naturally, that is not permitted by taxi on Manhattan.) In such a case, you would travel a distance equal to (according to Pythagoras) sqrt(4^2+1^2) = sqrt(17) = 4.123 approximately.
The interesting thing is that the Manhattan or taxi distance satisfies all axioms of a distance in mathematics, so it is a valid notion of a distance, and in fact, it is widely used in mathematics too.
What does a "circle" look like according to the Manhattan notion of distance? Well, a circle of radius 1 consists of all points you can reach exactly 1 block from the street corner you're at.
Let's say you're at the New York Public Library, at 42nd St, 5th Ave. The "circle" then reaches to 4th Ave, 41st, 42nd and 43rd St, and to 6th Ave, 41st, 42nd and 43rd St, right? (Disregard all New York anomalies such as unidirectional streets and named avenues.)
Draw a little map of that "circle".
Now, what is the circumference of that circle? Travelling the full circle, you have to travel from 4th Ave 41st St to 4th Ave 43rd St. That's two blocks.
Then you have to travel from 4th Ave 43rd St to 6th Ave 43rd St. That's two blocks. Now you have travelled four blocks total.
Then you have to travel 6th Ave 43rd St to 6th Ave 41st St. That's two blocks. So total distance is now six blocks.
Finally, you have to travel 6th Ave 41st St to 4th Ave 41st St to complete your "circle." That's another two blocks and now you have travelled 8 blocks total.
Now let's have a look at the "diameter" of that circle. It's simply the distance from side to side of that circle, or 2 blocks.
So there you are: circumference 8 blocks; diameter 2 blocks; hence pi = 8/2=4.
This is not a joke but serious mathematics. Of course, it is set in a day-to.day context, but in fact it is an important way of calculating distances, and it is often used in professional mathematics.
As you point out, in the Euclidean geometry, pi is certainly 3.1415926535 or so, but this is not the only way of measuring distances, and therefore not the only possible value of pi. Although, pi = 3.14 is, of course, the only famous value with any respectable claim to fame.