The Golden Ratio and the
World
Instructions to Construct a Golden Rectangle Using Geometer’s Sketchpad
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Start by going to the Graph menu and first click on the Show Grid option, then go back
to the menu and click the Snap Objects option.
Now go to the Graph menu and click on the Plot Points option, now enter the following
four points (1,1),(1,14),(22,14),(22,1). Enter one point at a time and click on Plot. After
you have plotted the fourth point click on Done.
Now using the Selection Arrow Tool from the left-hand toolbar select the four points in
a clockwise or counter-clockwise order and then go to the Construct menu and select
Segments.
Now that we have an initial rectangle we want to divide it into two pieces so that one
piece is the largest square that we could possibly make inside of our rectangle. This
means that our width will be equal to our maximum height of 13 units, so count thirteen
squares over from our left most points and plot points at this x-value on both the upper
and lower edges of the rectangle.
Now connect these two points using the Construct menu and choose Segment.
Continue taking the largest square possible out of each of the new rectangles constructed,
until all that you have left is a one by one rectangle.
Problems for the Golden Ratio
Just like we can measure our fingers and find the “ratio” of the lengths, we can use GSP to
measure the lengths of the sides of our “golden” rectangles and calculate the ratios between the
lengths of the sides.
1.) Measure the length and width of each rectangle in your sketch. To do this you must
measure each segment one at a time by selecting a segment and go to the Measure
menu and click on Length. Calculate to find the ratio between the different sides of
a.) each golden rectangle. b.) successive (explain here) squares.
2.) What is this ratio? (1.618)
Now imagine an ant walking around the rectangles.
1.) How far would s/he walk if he went around the big rectangle once?
2.) How about the next smallest rectangle?
3.) What is the ratio of those two perimeters? (1.618)
What about the area?
1.) What is the area of the biggest rectangle?
2.) What is the area of the next to the smallest rectangle?
3.) What is the decimal representation of the ratio of those two areas (place the larger
area over the smaller area)? (1.618^2)
4.) Why is it not the same as perimeter? (because it is merely the golden ratio squared
because area is squared!).
In the previous activities we have constructed squares like those in Figure 1.
Figure 1
- note that the numbers represent the length of the sides (not the area) of the individual squares
1.) When you order the lengths of sides of these squares, from smallest to largest, what do you
get?
(1, 1, 2, 3, 5, 8, 13, ….)
2.) Is there a pattern in these numbers?
3.) If so, what is the next number in the sequence? (21)
4.) What are the next four numbers in the sequence? (21, 34, 55, 89)
Instructions to Construct a Spiral Using Geometer’s Sketchpad
Now we are going to construct a spiral by using all of the squares that we have just created.
 First select the bottom right vertex of our largest square, now select the bottom left vertex
of our largest square. With these selected go to the Construct menu and click on the
Circle by Center + Point option.
 Now with the circle already selected, select the top point where the circle intersects the
largest square and then the bottom point. Now go back to the Construct menu and select
the Arc on Circle option.
 Next be sure that only the circle you created is selected and go to the Display menu and
select the Hide Circle option.
 Now go ahead and repeat the last steps to create an arc within almost all of the squares
(don’t do this for the second and fourth squares) first create a circle then create an arc on
each circle within the edges of the squares. For the second and fourth square all you will
do is create the circle but not the arc.
 Now go back to the second circle, make sure that it is the only thing that is selected.
With this done go to Construct menu and select the Point on Circle option. Now select
just the three points on this circle and go to the Construct menu and select the Arc
through 3 points option.
 Repeat this last step for the fourth circle, once you have created both these arcs go ahead
and hide both of the circles as you did before.
 If you would like to print your spiral go over to the left-hand menu and select the Text
Tool, now just click and drag a box. After you have created your text box a blinking
curser will appear inside of it, just type your name and then click on the Selection Arrow
Tool.
 With this done go to the File menu and click on the Print Preview… option and then
click the Fit to Page button near the top of the window. Before printing be sure you print
preview shows your sketch only on one page.