Results 1 to 2 of 2

Math Help - an odd PI from dividing circle diameters?

  1. #1
    Junior Member
    Joined
    May 2010
    Posts
    27

    an odd PI from dividing circle diameters?

    Question:
    I want to algebraically find the diameter of a larger circle that has contact with two smaller circles and is bound not to grow larger than x=y (see attached image)

    It seems for some reason that if I take 6*(smaller circle)/(larger circle), I get PI. (see attached image)

    If I get PI, why do I get PI? Is that normal?

    I have serious trouble solving the problem algebraic so I made a script that tries to find the solution. I'm not sure why, but the script is not very precis. I can barely squeeze out the result 3.141

    I don't see how I can solve this algebraically when the larger circles diameter is unknown and also position is unknow? how?

    I was inspired by the NowIsForevers post here at mathhelpforum:
    "A fractal made with circles?"
    http://www.mathhelpforum.com/math-he...e-circles.html

    here is the script (as3)
    [PHP]circl.x =48.0;
    circl.y =0;

    var h1:Number = 48.0;
    var k1:Number = 48.0;

    var stepsize:Number = 1.0;

    var R1 = h1;
    var R2;
    var radiusSum;
    var dist;
    var dir = true;

    this.addEventListener(Event.ENTER_FRAME,iter);

    function circleSize() {
    circl.height = circl.width = circl.x*Math.sqrt(2);
    }

    function iter(event:Event)
    {
    circl.x +=stepsize;
    circleSize();
    checkR1R2();
    }

    function checkR1R2()
    {
    R2 = circl.height/2;
    radiusSum = R1 + R2;
    var val = (h1-circl.x)*(h1-circl.x)+(k1)*(k1)

    dist = Math.sqrt(val);

    if(dist<radiusSum)
    {
    updateStepsize(true)
    }else{
    updateStepsize(false)
    }
    }

    function updateStepsize(val)
    {
    if(val==dir)
    {

    }else{
    stepsize *=-1;
    var rand = Math.random() /1000;
    stepsize /=(1.0001+rand);

    trace(stepsize);
    }
    dir=val;
    traceQuote();
    }

    function traceQuote()
    {
    var bigcirl = circl.height;
    var smallcircl = 2*(R1+R1*Math.sqrt(2));
    trace("q: " + 6*smallcircl/bigcirl);
    }[/PHP]
    Attached Thumbnails Attached Thumbnails an odd PI from dividing circle diameters?-screenshot_01.jpg   an odd PI from dividing circle diameters?-screenshot_02.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    May 2010
    Posts
    27
    It was close but no cigar... probably nested in some way, I don't know?

    ===========================
    Assumption that there exists radiuses such that:

     <br />
6\frac{r_2}{r_3} = \pi<br />

    <br />
r_2 = r_1 + \sqrt{2} <br />

    We set the smaller radius to one
    <br />
r_1 = 1<br />

    then:
    <br />
6\frac{\ 1 + \sqrt{2} }{r_3} = \pi<br />
     <br />
r_3 = 6\frac{\ 1 + \sqrt{2} }{\pi} <br />

    The larger radius is set such that:
     <br />
r_3 = \sqrt{\left (\frac{x_2}{2}\right)^2 + \left (\frac{x_2}{2}\right)^2}<br />

     <br />
x_2 =\sqrt{2\left(\frac{6\left(  \left( 1 + \sqrt{2} \right)\right)}{\pi}\right)^2}<br />

    Coordinates:
    <br />
r_1\left[x_1,y_1 \right] =\left[r_1, r_1 \right] <br />

    <br />
r_3\left[x_2,y_2 \right] =\left[x_2, 0 \right]<br />

    And the distance d between the coordinates are:\\*
    <br />
d = \sqrt{\left (x_1 - x_2\right)^2 + \left (y_1 - y_2\right)^2}<br />

    <br />
r_1 + r_3 = \sqrt{\left (1 - \sqrt{2\left(\frac{6\left(  \left( 1 + \sqrt{2} \right)\right)}{\pi}\right)^2}\right)^2 + \left (1 - 0\right)^2}<br />


     <br />
r_1 + r_3 = 5.610505374600274951670709091362115183698005023761  2053686454...<br />
     <br />
 r_3 = 4.610505374600274951670709091362115183698005023761  2053686454...<br />
    <br />
 r_2 = 2.414213562373095048801688724209698078569671875376  9480731766...<br />

    since:
    <br />
6*\frac{ r_2}{r_3} = 3.141799043124296556570853519060856595396815103571  4525865439...<br />

     <br />
6\frac{r_2}{r_3} \not = \pi<br />
    Last edited by ellensius; May 25th 2010 at 02:25 AM. Reason: fixed typo
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lines Parallel to Diameters
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: July 7th 2011, 07:33 AM
  2. Replies: 6
    Last Post: July 8th 2010, 06:39 PM
  3. Replies: 7
    Last Post: March 15th 2010, 05:10 PM
  4. Replies: 2
    Last Post: February 6th 2010, 09:31 AM
  5. diameters and radius
    Posted in the Geometry Forum
    Replies: 2
    Last Post: January 21st 2008, 11:44 AM

Search Tags


/mathhelpforum @mathhelpforum