Fibonacci/Golden Ratio Lab
- Press this button to bring up GeoGebra.
- Draw a small square near the top left of your screen with the "regular polygon" tool: Select the tool, then click one point, then another relatively close by. It will ask how many sides, defaulting to 4. Take that default. You should see a small square. Keeping the Regular Polygon tool (RPT), click the top right vertex of the square and the bottom right vertex. You should see a square pop out to the right. Now you should have two squares next to each other, making a longish rectangle.
- Still using the RPT, pick the bottom right point and then the bottom left point of the rectangle. Accepting the default, 4, creates a new rectangle.
- Now we have a tall rectangle. Using the RPT, click the top right and then bottom right vertices of the rectangle to create a new square to the right. That creates a still bigger rectangle.
- Go a few more steps, alternately drawing bigger and bigger squares below and to the right, making bigger and bigger rectangles. You should end up with something like this. Use the VCR buttons to move back and forth in the construction.
- Assuming the side of the smallest square is 1, going smallest to largest, what are the sizes of the squares? Do you recognize that sequence? What is the relationship betwee a number in the sequence and the previous two numbers?
- What is the ratio of the side of the largest square to the side of the next-largest square? (Ask Geogebra by typing
distance[A,B]/distance[B,C]if A and B are vertices of the largest square and B and C are vertices of the second-largest square. As you add more squares, this ratio approaches a number we'll look at soon. - Bring up this GeoGebra window:
- Note square ABCD. Use the midpoint tool to find the midpoint, E, of BC, then do it again to find the midpoint, F, of AD. Use the "segment between two points" tool to join E and F, and again to join E and D. Now use the "circle with center through point" tool to draw a circle with center E and point D. Now use the ray tool to create ray BC, and again to create ray AD. Use the new point tool to create a point, G, at the intersection of ray BC and the circle. Use the perpendicular line tool to create a perpendicular to line BG through G. Use the new point tool to create point H at the intersection of ray AD and the new perpendicular. You have just created rectangle ABGH, whose sides are in the golden ratio. Type
distance[B,G]/distance[A,B]to see the value of that ratio. What is it, approximately? - We can calculate that ratio exactly.
- Assuming a side of the square ABCD measures 2 units, what is the measure of DC?
- What is the measure of EC?
- Remembering your Pythagorean theorem, what is the measure of DE?
- Remembering the fundamental fact about radii of a circle, what is the measure of EG?
- What is the measure of the long side of the rectangle, BG?
- What is the measure of the short side, AB?
- What is the ratio of long to short?
- That is the Golden Ratio.
- But why is it golden?
- Using the RPT, click on G and then on B to create a square below segment BG.
- What do you notice about rectangle ABGH compared with rectangle AHIJ?
- Let's demonstrate that the sides of AHIJ are also in the same golden ratio. We'll do a similar thing to this bigger rectangle. Use the midpoint tool to find the midpoint K of GJ. (We're skipping a few steps compared with last time but don't worry.) Use the segment tool to create segment KB. Use the circle tool to create a circle centered on K containing B. Notice that point H seems to lie on circle K much like point G lies on circle E. Calculate the ratio of HJ to IJ.
- How does HJ/IJ compare with BG/AB? That's what makes it the Golden Ratio: no other rectangles do that!
- How close is that to the ratio of those two Fibonacci numbers?
- Note by the way this isn't a real proof because, though it seems as if H lies on circle K, we haven't proven that.
- If you get lost, here's what it should look like. Use the VCR buttons to go back and forth in the construction.