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.

  1. Begin with a square.
  2. Divide the square into nine equal squares.
  3. 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 Golden Ratio Symphony!

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!

enter image description here

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! enter image description here

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. enter image description here

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:

enter image description here

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:

enter image description here

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!

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    $\begingroup$ Is there any particular reason you're looking for golden ratio constructions? $\endgroup$ – Lanier Freeman May 28 '16 at 22:46
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    $\begingroup$ @LanierFreeman yes! I use it in a lot of my art and photography and thus became very curious about it! I am working on a couple art-oriented books, but as I have a Ph.D. in physics, I love the science if it too! Every high school student should study the golden ratio in depth! Even though I attended Princeton and majored in physics, nobody every mentioned it! Well, I hope to change that with a couple papers/books :) $\endgroup$ – Astrophysics Math May 28 '16 at 22:49
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    $\begingroup$ Thanks @LanierFreeman! You can see the golden ratio in a lot of my ballerina photography: facebook.com/media/set/… :) $\endgroup$ – Astrophysics Math May 28 '16 at 22:53
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    $\begingroup$ Why so many? In your images, there are quite a few pairs that are equal due to Thales' intercept theorem, or due to scaling and reflection symmetry. $\endgroup$ – TastyRomeo May 28 '16 at 23:02
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    $\begingroup$ I understand that you have new material that you want to share with the world, but that is not what edits are for -- they are for clarifying the question. Consider starting a blog and publishing your results there instead. $\endgroup$ – Rahul Jun 9 '16 at 5:37

(Converting comment to answer, as requested.)

If the side of each sub-square has length $2$, then the radius of the circle is $\sqrt{10} = \sqrt{2}\cdot\sqrt{5}$. Once you have $\sqrt{5}$ floating around, it's not too difficult to find $\phi$.

Edit. It seems you could use a geometry lesson to help put your "discoveries" into some perspective.

Let us first note this basic property of parallel lines:

Theorem. Three (or more) parallel lines cut any transversal into segments with a fixed proportion (such as $p:q$ in the figure).

enter image description here

And now, let us consider your circle-and-grid diagram, in which one readily shows that $a/b = \phi = 1.618\dots$. We'll call this the figure's primary instance of the golden ratio.

enter image description here

By the nature of $\phi$, an immediate (and thoroughly-unsurprising) corollary is that $b/c =\phi$, as well. We can write $$a:b:c \quad=\quad \phi : 1 : \frac{1}{\phi}$$

Adding-in some lines through the dividing-point of $b$ and $c$, parallel to the sides of the square, we can exploit the parallelism Theorem by effectively transferring the $a:b:c$ proportion to the edges of the square:

enter image description here

By symmetry, we get proportions all around the square:

enter image description here

It's important to note, though, that these additional instances of the golden ratio are unremarkable; each is merely an echo of the primary instance. Indeed, thanks to the parallelism Theorem, we can effortlessly construct countless more instances of the golden ratio; here are just a few (dozen):

enter image description here

Above, because a particular set of horizontal lines cut the sides of the square in the proportion $\phi:1$, those lines cut all those segments in the same way. We can get even more instances of $\phi$ by connecting other pairs of points, midpoints of those points, or whatever ... and let's not forget about the rotational and reflective symmetries in the figure. OMG! Golden ratios everywhere! And yet, as before, these instances of $\phi$ are unsurprising and unremarkable, either individually or collectively, because they're obvious follow-ons of obvious follow-ons of the original primary instance. That's not a symphony you hear; it's the accumulation of echoes of one single note.

You seem to be spending a lot of time and effort "discovering" various instances of $\phi$ by manually plotting points and measuring ratios in GeoGebra. Please do yourself (and the rest of us) a favor: pay attention to some of the dead-simple geometric principles at work here. You'll do far less measuring, and you'll get far fewer false-positives (and -negatives); most importantly, you'll learn to distinguish utter trivialities from whatever legitimately-interesting construction you might happen to come across, vastly improving the value of your questions here.

Oh ... As to whether your recent "green square", "red segment", and "orange circles" additions are "utter trivialities" or "legitimately-interesting constructions" ... well ... I believe the first two go directly into the "trivialities" box.

I'm not sure how you think the orange circles relate to each other. If you're suggesting that the smaller circle (with $2/3$ the diameter of the larger circle) is tangent to both the square and the bigger circle ... then we're going to need a third box: "bogus claims". The double-tangency requires that $$\frac{\text{smaller diameter}}{\text{larger diameter}} = \frac{1+2\phi^{-1}}{2\phi} = \frac{1}{2}(3 - \phi ) = 0.6909\dots \neq \frac{2}{3}$$

(I honestly don't understand how a Ph.D in physics lets such easily-caught computational errors slip by, but I guess that's just a failure of imagination on my part.)

Anyway ... I hope this discussion helps you in your journey. It's the last contribution I intend to make.

  • $\begingroup$ Thanks for the simple yet profound insight Blue!! :) That pretty much sums it up--it comes down to the ratio between the foundational characteristics of the circle and squares--the radius of the circle and the length of a side of a square or squares. $\endgroup$ – Astrophysics Math May 29 '16 at 5:47
  • $\begingroup$ Thanks so much @Blue! I corrected the orange circle treatment with the correct figure. I had confused it with an earlier, incorrect figure when I uploaded. Thanks again for all your time and input. I hope you feel you also gained some insights while expounding upon it! Your explanations are really great, and you have a great talent for tracing elements back to their foundations and explaining it. Thanks again for your talents & time! You'll enjoy Newton's mistake: nytimes.com/1987/06/11/us/… $\endgroup$ – Astrophysics Math Jun 9 '16 at 18:09
  • $\begingroup$ You'd enjoy this book @Blue : amazon.com/Einsteins-Mistakes-Human-Failings-Genius/dp/… "Although Einstein was the greatest genius of the twentieth century, many of his groundbreaking discoveries were blighted by mistakes, ranging from serious errors in mathematics to bad misconceptions in physics and failures to grasp the subtleties of his own creations." Einstein made numerous mistakes in his papers "proving" E=mc^2, but yet, E=mc^2 turned out to be true. :) More here: arxiv.org/ftp/arxiv/papers/0805/0805.1400.pdf "Einstein’s E = mc^2 mistakes." Enjoy! :) $\endgroup$ – Astrophysics Math Jun 9 '16 at 18:13

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