Note: this construction is a vastly expanded version of my earlier construction here: Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or 1.6180.. exactly
Below please find a relatively simple golden ratio construction.
- Begin with a square.
- Divide the square into nine equal squares.
- Draw a circle so that it passes through the points shared by any two squares on the outer edges of the original greater square. (the circle could be defined by two squares alone--say the upper two squares on the left, where the circle passes through the three outer points as illustrated.)
A symphony of golden ratio harmonies then emerges. The question is, why so many golden ratios in such a simple construct? Is there a general proof that provides insight as to why so many golden ratios naturally emerge in this construction?
The golden ratio PHI 1.6180...(and its inverse .6180...) appears in the following illustrated places (as well as many more places!)
and on and on!
Why so many golden ratio harmonies throughout?
And here are some more implied constructions naturally found within, using both pre-existing points form the pre-existing grid, the pre-existing circle, and pre-existing golden ratio constructions all shown above:
By using pre-existing points on the pre-existing grid and pre-existing circle to draw the green square, we can again marvel at the golden ratio presented between a and b resulting from the naturally-implied 2x2 green square and the pre-existing circle!
By adding the red segment between pre-existing points on the pre-existing grid and pre-existing circle, we can again marvel at the golden ratio between a and b within the realm of the naturally-implied isosceles triangle and pre-existing circle!
By using pre-existing points and pre-existing golden ratio constructs to define a smaller orange circle illustrated in the figure below, we can again marvel at the pre-existing golden ratio construct in c:d! Simple geometry immediately shows us that the radius of the smaller orange circle is 2/3 that of the larger pre-existing circle found in the original "golden ratio symphony" diagram.
Indeed, the golden ratio symphony, despite its simplicity, is a most powerful and far-ranging diagram!
Again, beginning with the original golden ratio symphony figure presented above we can draw two blue circles defined by pre-existing points on the grid as shown below, with the top blue circle having the radius of a single square, and the bottom circle having a radius of the pre-existing segment t, while being centered on a point on the grid two squares below the top blue circle:
We can then again marvel at the pre-existing golden ratio found in a:b! :)
We can then add two yellow equilateral triangles whose sides are equal to the pre-existing segment t where we recall that the radius of the lower blue circle is also defined by the pre-existing segment t, which is based on pre-existing points and pre-existing golden ratio constructs defined in the original golden ratio figure above, and the addition of the two reflected equilateral triangles appears as thus:
We can then again celebrate the pre-existing golden ratio of a:b found in the original Golden Ratio Symphony figure as we have done numerous times!