This is a corrected answer. In January 2008 David Radcliffe pointed out an error in my earlier response.
Ceri,
The isoperimetric quotient of a closed curve is defined as the ratio of the curve area to the area of a circle with same perimeter as the curve.
I want to call the Isometric Quotient IQ. If the area inside the curve is A and the perimeter is p then the expression for A divided by the area of a circle with perimeter (circumference) p simplifies to
IQ = ^{4 A}/_{p2}
I used an expression for the area of a regular polygon with n sides to simplify this further and I arrived at
IQ = ^{}/[n tan( ^{}/_{n})]
I then used my calculator to evaluate this expression for various values of n. For example
when n = 1000 I got IQ = 0.9999967103
when n = 10000 I got IQ = 0.9999999671
when n = 100000 I got IQ = 0.99999999967
when n = 1000000 I got IQ = 1
I expect You got very similar results. My calculator tries to give a numeric result that is valid to 11 digits.
Next I tried a computer program called Mathematica which can do calculations to as many digits as you want as long as you have enough computer memory and are willing to wait for the results. I asked Mathematica to calculate ^{}/[n tan( ^{}/_{n})] when n = 1000000 and to give me the correct answer to 20 digits. Mathematica gave me 0.99999999999671013187
Your conundrum does come from the fact that calculators can only do calculations to a certain finite number of digits. The IQ of a million sides regular polygon is not 1, but it's very close.
Penny
