Rectangle in xy plane?

The ratio BC/AB is the positive root of the equation

z^4 – z^3 – z – 1 = 0.

The above equation has only 2 real roots.

BC/AB = 1.61803398875

Edit1:

After looking at the Vikram’s answer

z^4 – z^3 – z – 1 = 0

(z^2 – 1)(z^2 + 1) – z(z^2 + 1) = 0

(z^2 + 1)(z^2 – z – 1) = 0

The positive real root is really φ=(1+√5)/2.

http://i299.photobucket.com/albums/mm286/rozeta53/…

Edit2:

Let r1 < r2 < r3 < r4

All circles are inscribed in a similar right triangles (each one with an acute angle α) ==>

u/r1 = v/r2 = x/r3 = y/r4

and u/v = v/x = x/y = 1/φ ==>

r1/r2 = r2/r3 = r3/r4 = 1/φ

It’s hard to tell from your question or the picture but I guess the radius of the smallest circle is added to every circle to get the circle one step larger.

Quadrants:

2 | 1

—–

3 | 4

I’ll let the circle in quadrant 1 have radius r. So quadrant 2 has radius 2r, quadrant 3: 3r, quadrant 4: 4r.

The height, h, of the rectangle (shorter side) will be the same as the diameter of the 4th quadrant circle.

h = 2*4r = 8r

Finding the width, w, seems to be the tricky part. I’ll post what I have so far and edit it if I can figure out the full answer.

EDIT:

I’m still thinking about how to find the width but I thought I’d mention that if the points A, B, C and D are known finding the dimensions of the rectangle would just involve finding the distance between the points.

EDIT 2:

Oops! After looking at Vikram P’s diagram and closely at the original image I realized that the radius of the circles is not doubling but increasing by the radius of the smallest circle for each bigger one.

I corrected the equations above but the problem doesn’t seem to change in difficulty.

I’m not convinced that this rectangle is well-defined, even in terms of ratios of sides. For example, we can quite easily embed four circles into the [-2, 2] x [-2, 2] square, simply by putting four radius 1 circles centred at (1, 1), (-1, 1), (1, -1), and (-1, -1). This would define a side ratio of 1:1, but the rectangle in your picture clearly does not have such a side ratio.

Perhaps if we had an angle or a gradient for a side, then it might be possible (if very difficult).

51

Yeah it is really hard to believe that it is Φ

Here is the image of it when radius of the smallest circle is 1

http://i233.photobucket.com/albums/ee201/vcpandya/…

Let me think more on this

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Rozeta is correct with her equation, interestingly ratio of radius of circles is also Φ.

Rozeta can you prove this too?

More over if instead of it being rectangle if we have a square then also ratio of radius of circles follows a pattern. Can you find it?

Really interesting!!!

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Thanks Rozeta for proving that other part.

To generalize when circles are within a rectangle, radius of circles are in GP with common ratio = Φ and when circles are within a square, radius of circles are in AP.

philippinogirl43852: the only reason you would be able to desire to be attentive to that they are actually not parallel to the axes is so which you do not have an undefined slope. The made from 2 slopes that are perpeduclar = -a million Slope (AB) * Slope (BC) = -a million Slope (BC) * Slope (CD) = -a million Slope (CD) * Slope (DA) = -a million Slope (DA) * Slope (AB) = -a million So the made from each and every of the slopes = -a million * -a million *-a million * -a million = a million