CPS Geometry Part 5 – Ratios and

Similarity

23. The Mental-Experimental Method

24. From Spheres to Geometrical Points

25. Similarity Theorem in CPS

26. The Golden Section and the Fibonacci Sequence - 1

27. The Golden Section and the Fibonacci Sequence - 2

Nick Trif

Ottawa, Ontario, Canada – 2018

www.platonicstructures.com

CPS Geometry Part 5 – Ratios and Similarity – 26: The Golden Section and the Fibonacci Sequence - 1

YouTube: https://youtu.be/CwXYfob0vYg

Let’s restate the great quote from Johannes Kepler:

"Geometry has two great treasures;

one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio.

The first we may compare to a measure of gold; the second we may name a precious jewel."

Part 1: The Golden Ratio

A lot has been written over the years about the golden ratio, the golden section, or the division of a line into extreme and mean ratio

Let’s mention here few facts that will provide a new understanding of this ratio, and its connection to CPS.

There are at least three ways the golden ratio appears in classical geometry:

a. First, let’s look into the way Euclid introduces the Golden Section in his Elements.

Euclid, in the second Book of Elements, Problem 11 states:

“To divide a line such that the area of the rectangle made by the line with one segment to be equal with the area of the square of the second segment.”

Euclid, in the second Book of Elements, Problem 11 states:

“To divide a line such that the area of the rectangle made by the line with one segment to be equal with the area of the square of the second segment.”

Euclid, in the second Book of Elements, Problem 11 states:

“To divide a line such that the area of the rectangle made by the line with one segment to be equal with the area of the square of the second segment.”

This definition shows, in a profound and undeniable way, the direct connection between the Golden Section and the Surface, or the Area, as the domain, absolutely required, for generating the Golden Section.

This definition shows, in a profound and undeniable way, the direct connection between the Golden Section and the Surface, or the Area, as the domain, absolutely required, for generating the Golden Section.

In simpler words, the Golden Section is a relationship between line segments, but its full existence can only be seen and understood on a surface.

In simpler words, the Golden Section is a relationship between line segments, but its full existence can only be seen and understood on a surface.

The division of a line segment into the extreme and mean ratio is such that, the whole segment AB is to the greater segment AC, as the greater segment AC is to the smaller segment BC.

Let’s look next into a method of dividing a segment AB into the "extreme and mean" ratio.

Let’s look next into a method of dividing a segment AB into the "extreme and mean" ratio.

Let’s look next into a method of dividing a segment AB into the "extreme and mean" ratio.

First, we need to consider the segment AN, perpendicular on AB, and with the same length as AB.

Next, we determine the point M such that AM is half the segment AB.

And finally, we draw a circle, with the center in M, and the radius MN.

The intersection, between the circle and the segment AB, is the point C.

This point, divides the initial segment AB, into the extreme and mean ratio.

If we consider, the extension of the segment AB, then, this line intersects again the circle, in point D.

Like magic, the point D, also divides the initial segment, into the extreme and mean ratio.

Let’s look again into our geometrical construction.

As mentioned before, remember, the construction has to be done on a surface.

At this point you might ask: So, what is it so special about this ratio?

Why did Kepler say that the division of a segment into the extreme and mean ratio is the jewel of geometry?

At this point you might ask: So, what is it so special about this ratio?

Why did Kepler say that the division of a segment into the extreme and mean ratio is the jewel of geometry?

Let’s shed some light on this ratio. We will mention next few facts that make this division so special.

1. Every right triangle can be imagined as being inscribed in a circle, or more precisely into a semicircle.

If one considers the height of the triangle, than by looking at this construction, it becomes obvious, that the division intothe extreme and mean ratio is part of any right triangle.

A very simple manipulation, or transformation, of this ratio, or this relationship, is known as the Theorem of Pythagoras.

Or, in other words, the Theorem of Pythagoras is equivalent with the relationship, corresponding to the division of a segment, into the extreme and mean ratio.

2. Let’s extend the original segment in both directions indefinitely, and form a straight line.

2. Let’s extend the original segment in both directions indefinitely, and form a straight line.

Next, let’s extend the division into the extreme and mean ratio to the neighboring segments, on the right side of the original segment.

This can be done easily, by imagining multiple circles, of ever decreasing radius, intersecting the straight line.

This can be done easily, by imagining multiple circles, of ever decreasing radius, intersecting the straight line.

The intersection points form segments on the line.

It can be proven that all the ratios between any two neighboring segments are equal with the Golden Ratio.

3. Of course, using a similar approach, the division of the straight line, can be extended to the left side of the original segment.

By doing this, one can imagine the complete line, divided into segments, related to each other by the Golden Section ratio.

4. At this point, the magic of this division should become evident to everyone.

One can see here, in its simplest form, the process of the natural growth and development, observed everywhere in the Universe.

One can see here, in its simplest form, the process of the natural growth and development, observed everywhere in the Universe.

One can see here, in its simplest form, the process of the natural growth and development, observed everywhere in the Universe.

The same rules apply to the very small and very large.

Let's start by stating a well known fact:

A sphere can be completely surrounded by exactly twelve others identical spheres, in only one way.

5. A lot more has been said and can be said about this progression.

It is not our purpose here, to give examples from real life, where this principle of development manifest itself, and have been observed, and have been studied.

6. Instead, we will just concentrate on another aspect of this phenomenon, known today, as Fractal.

Let’s imagine that we start from a number of such segments, and resize them continuously. As expected, all these states are similar with each other.

Let’s imagine that we start from a number of such segments, and resize them continuously. As expected, all these states are similar with each other.

As we decrease the size of the initial segments, at one point, the first segment becomes equal with the second segment.

If we take the reduced version of the segments, and translate them to the right, somewhere they perfectly fit over the initial version.

It is important to realize first that this overlapping is possible.

Also, the final figure has an extra segment, to the right.

It is important to realize first that this overlapping is possible.

Also, the final figure has an extra segment, to the right.

The initial and reduced versions of the segments are in fact fractals.

The reduction is done by a factor equal with the golden ratio.

Even more, any reduction or expansion of the initial set of segments, with a factor equal with a power of the golden ratio, produces a fractal.

b. A better and more interesting encounter with the golden ratio, in the classical geometry, happens when one looks closely to the pentagon.

Let’s illustrate this, by examining, how one can construct a pentagon, inscribed in a circle.

To do this, we start from two perpendicular lines, and draw a circle, with the center defined by the intersection of the two lines.

Next, we determine the middle of a radius, and draw a circular sector.

Let’s illustrate this, by examining, how one can construct a pentagon, inscribed in a circle.

To do this, we start from two perpendicular lines, and draw a circle, with the center defined by the intersection of the two lines.

Next, we determine the middle of a radius, and draw a circular sector.

From now on, we continue in the same way, as done when determining the golden section.

The length of the segment EG, is equal with the length of the sides of the regular pentagon.

From now on, we continue in the same way, as done when determining the golden section.

The length of the segment EG, is equal with the length of the sides of the regular pentagon.

This new finding is so beautiful. One starts wondering: Is there a hidden reason or explanation for this amazing fact?

This new finding is so beautiful. One starts wondering: Is there a hidden reason or explanation for this amazing fact?

Buckle up! There are still some more surprises waiting for us inside the pentagon.

Let’s draw next, the diagonals of the regular pentagon.

Let’s draw next, the diagonals of the regular pentagon.

Like magic, this process creates a new, smaller pentagon, inside the initial pentagon.

Let’s draw next, the diagonals of the regular pentagon.

Like magic, this process creates a new, smaller pentagon, inside the initial pentagon.

It is worth having a closer look, at the length of the diagonal of the pentagon, and the length of a side of the initial pentagon.

By doing this, one discovers that the ratio between these two segments is again the Golden Ratio.

Even more, the ratio between the two segments, determined by the intersection of two diagonals, is also equal with the golden ratio.

But why stop here?

Why not continuing with drawing the diagonals of the smaller pentagon?

By doing this, we are no longer very surprised, when we get again, an even smaller pentagon inside.

The ratio, between the length of sides of this new pentagon, and the length of the diagonals, are again, our old friend, the Golden Section.

And of course, the diagonals intersect each other into segments, obeying the same ratio.

This process can be continued. Nested pentagons, of smaller and smaller sizes, can be generated indefinitely.

This process can be continued. Nested pentagons, of smaller and smaller sizes, can be generated indefinitely.

With a little bit of imagination, one can imagine the sequence of increasingly expanding pentagons, and obeying the same rules.

This process can be continued. Nested pentagons, of smaller and smaller sizes, can be generated indefinitely.

With a little bit of imagination, one can imagine the sequence of increasingly expanding pentagons, and obeying the same rules.

This process can be continued. Nested pentagons, of smaller and smaller sizes, can be generated indefinitely.

With a little bit of imagination, one can imagine the sequence of increasingly expanding pentagons, and obeying the same rules.

We also want to highlight, the Fractal nature of the sequence of pentagons.

As seen before, any reduction or expansion of a set of nested pentagons, with a factor equal with a power of the golden number, will produce a set of nested pentagons, which will fit perfectly over the initial set.

In this case, a rotation of 180 degrees is also required to get a perfect fit between two consecutive pentagons.

Let's start by stating a well known fact:

A sphere can be completely surrounded by exactly twelve others identical spheres, in only one way.

c. In classical geometry, the golden section is also mentioned, as a method of constructing the icosahedron.

A golden rectangle, is a rectangle, for which the ratio between the longer, and shorter side, is equal with the golden number.

This rectangle, has the property, that if we construct a square, on the smaller side, the smaller rectangle formed, is also a Golden rectangle, similar with the initial rectangle.

Also, in some very amazing and unexpected way, we can observe that, the vertices of three orthogonal golden rectangles, form the vertices of the icosahedron.

The Golden Section is closely related to the Fibonacci Sequence.

The CPS explanation of this relationship is covered in Part 2 of this presentation.

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Beauty makes beautiful things beautiful!