6
$\begingroup$

I don't know whether I should ask this question or not, but I am asking this:

(Before anything, this is an analysis of closed 2D figures.)

I was analyzing $\pi$ . It seems to be a constant, But being a constant is my curiosity. Can we say that $\pi$ is a property of circle which makes circle a circle. Same as there must be constant for each and every figure possible. So I suggested a new theory of $\pi$ or a new definition of $\pi$, which says -

$Figure_\pi = {L \over D}$

where, L = Length of the curve,

D = Maximum distance possible between any two points

So, $Circle_\pi = {Circumference \over Diameter} \implies Circle_\pi = \pi$

Also, $Square_\pi = {Perimeter \over Diagonal } \implies Square_\pi = {4a \over a\sqrt2} = {2\sqrt2}$

And, $Rectangle_\pi = {2(a+b) \over \sqrt{a^2+b^2}}$

Now I got stuck, as for rectangle I am not getting a constant, but I realized that this is obvious. Two squares or circles are always similar but two rectangles are not necessarily similar, because for two rectangles to be similar, their ratios of lengths and breadths must be constant. So I suggested types of figure on the basis of their variables.

As a circle or a square can be defined by a single variable, I called them single variable figures. For a rectangle, a rhombus, or an ellipse, we must have two variables, so I called them 2 variable figures. For random figure to be similar we must know by some means the length of the curve and maximum distance between any two points, and we will get a constant calling it $Figure_\pi$

My question is this: Am I thinking in the right direction or not, because I am finding different constants for different 2D figures as different variable figures?

  • 0
    On a tangential note, you might be interested in looking up the "shape factor" of various shapes.2012-08-25
  • 0
    Yes, exactly, and $\pi$ must be one of them.2012-08-25
  • 2
    It might interest you to know that [all parabolas are similar](http://mathworld.wolfram.com/UniversalParabolicConstant.html).2012-08-25
  • 0
    Thnks J.M., I didn't know that2012-08-25
  • 0
    You say this number should be a defining property of a circle, so I have to ask: is there any reason these numbers should be unique (up to scaling)?2012-08-25
  • 0
    @guy: I am not saying that this number should be unique, but I am saying that a figure must have a constant to define its shape. Two figures can have same constants, but by just knowing that constant you cant guess the figure. As per my assumption, It is a ratio which can be generated by set of numbers.2012-08-25
  • 1
    What would be a more interesting way (for me, at least) is to define it as the maximum of $\frac{4*Area}{Perimeter^2}$ over every closed figure where this quantity makes sense.2012-08-25

1 Answers 1