The animation begins by presenting a series of numbers. This is a very famous and recognized sequence since many centuries ago in the Western World thanks to *Leonardo of Pisa*, a thirteenth century Italian mathematician, also called *Fibonacci*. So it is known as ** Fibonacci Sequence**, even although it had been described much earlier by Indian mathematicians.

This is an infinite sequence of natural numbers where the first value is **0**, the next is **1** and, from there, each amount is obtained by **adding the previous two**.

The values of this sequence have been appearing in numerous applications, but one of the most recognized is the ** Fibonacci Spiral**, which has always been used as an approximation to the

*Golden Spiral*(a type of

*logarithmic spiral*) because it is easier to represent with help of a simple drawing compass.

This is the next thing to be shown on the animation, appearing just after the first values on the succession: the process of building one of these spirals.

We will create first a few squares that correspond to each value on the sequence: 1x1 - 1x1 - 2x2 - 3x3 - 5x5 - 8x8, etc. And they are arranged in the way how we see in the diagram at left.

Then we draw a quarter circle arc (90°) within each little square and we can easily see how it builds step by step the *Fibonacci Spiral*, looking at right graphic.

I have introduced a small optical correction in the animation in order to get the resulting curve more like a true *Golden Spiral*(more harmonious and balanced), as explained on this plate. It's something similar to what happens when we try to approach to an ellipse by drawing an oval using circular segments: the result is not the same as a true ellipse. And it shows.

**IMPORTANT NOTE:** while watching the animation conveys the idea that the *Fibonacci spiral* (or the *Golden Spiral*, it doesn't matter) is on the origin of the shape of a Nautilus, this isn't absolutely right.

It's funny because if you perform this search at Google Images: “spiral + nautilus” you will see how many images suggest that this shell is really based on the construction system described above.

But this isn't correct, as it's outlined on this other page.

The truth is that this is something I discovered when I had completely finished the screenplay for this project and I was too lazy to change. Therefore I must confess that I did a kind of cheat with this animation. Or you could explain in a more "genteel" way, saying that I have taken an *artistic license* ;-)

Once it has appeared the Nautilus we advance to the second part of the animation. It introduces the concept of * Golden Ratio* by constructing a

*Golden Rectangle*. We start from a simple square to get that and use a classic method that requires only a ruler and drawing compass. See the complete process on the following series of illustrations:

This is very special rectangle known since ancient times. It fulfills this ratio, also known as the *Golden Ratio* or *Divine Proportion*: the ratio of the sum of the quantities (**a+b**) to the larger quantity (**a**) is equal to the ratio of the larger quantity (**a**) to the smaller one (**b**).

The result of this ratio (ie the division of **a** by **b**) is an irrational number known as **Phi** —not to be confused with *Pi*— and an approximate value of **1.61803399…**Formerly was not conceived as a true "unit" but as a simple relationship of proportionality between two segments. And we find in many works created by the mankind in art and architecture, from the Babylonian and Assyrian civilizations to our days, passing through ancient Greece or the Renaissance.

**JUST A CURIOSITY:** it isn't evident on the animation, but there is a deep connection between the *Fibonacci Sequence* and *Golden Ratio*.

You have an example at right (we will see another one): if we divide each value in the*Fibonacci Series* by the previous, the result**tends to Phi**. The higher the value, the greater the approximation (consider that Phi, like any irrational number, has infinite decimals).

We are going one step further on the animation by introducing a new concept, maybe less known but equally important, the ** Golden Angle**. That is, the angular proportional relationship between two circular segments:

These two circular segments are accomplishing too with the same golden proportionality, but on this case the value of the angle formed by the smallest of them is another irrational number, we can simplify and round it as **137.5 º**And this value is deeply present in nature. This is the next concept we see on the animation: how to configure the structure formed by the

**sunflower seeds**.

Look at the figures below:

- We add a first red seed.
- Turn 137.5º
- Add a second green color seed and make the previous traveling to the center.
- Turn other 137.5º
- Add a third ocher seed and make the previous traveling to the center, to stay side by side with the first one.
- Turn other 137.5º…

…and so on, seed after seed, we will obtain gradually a kind of distributions like the ones you have in the following figures.

This leads to the characteristic structure in which all seeds are arranged into a sunflower, which is as compact as possible. We have always said: nature is wise :-)**ANOTHER CURIOSITY:** Do you remember we had commented that there had a deep connection between the *Fibonacci Sequence* and*Golden Ratio*? Well, next we have another meeting point between both concepts. Look at the following images of a sunflower:

By observing closely the seeds configuration you will see how appears a kind of **spiral patterns**. In the top left picture we have highlighted three of the spirals typologies that could be found on almost any sunflower.

Well, if you look at one of the typologies, for example the one in green, and you go to the illustration above right you can check that there is a certain number of spirals like this, specifically **55 spirals**. Coincidentally a number that is within the *Fibonacci Sequence* ;-)

And we have more examples in the two upper panels, cyan and orange, they are also arranged following values that are within the sequence: **34** and **21 spirals**.

In principle, all the sunflowers in the world show a number of spirals that are within the *Fibonacci Sequence*. You could go out to the countryside and look for a plantation to be sure :-)

You can also use this image of a real sunflower or go to this websitewhere this is explained, along with another curiosities.

By the way, I recommend the rest of the *Ron Knott* site, a mathematician at the *University of Surrey* in England. His web is full of invaluable and educational information, all very well explained and with large doses of curious and funney elements*.*

Finally we reached the third segment of the animation in which we work with a concept that is a little less known than the others: the Voronoi Tessellations, also called Dirichlet Tessellation.

I discovered this issue thanks to Hector Garcia's personal site, which I visit almost daily (and despite being a blog dedicated to Japanese culture and everything that is related to that country, also delights us from time to time with other interesting topics, like this one about Delaunay and Voronoi).

These geometric formations are based on a distribution pattern that is easily recognizable in many natural structures, like the wings of some insects or these small capillary ramifications in some plant's leaves.

It is also widely used to optimize the distribution systems based on areas of influence, at the time to decide, for example, where to install phone antennas, or where to build the different delegations for a pizza chain.

Let me show you a very intuitive way to understand how it forms a Voronoi Tiling:

Imagine we have two points: one red and another blue (top left). Start by drawing a segment joining these dots and then a second orthogonal line who is right in the middle. We have just found the**bisector** of the segment joining these two points.

Above right we added a third green point, generating two new bisectors that intersect with the first.

If we continue adding points to generate succesive bisectors, with their intersections, will lead to a series of polygons *—Voronoi Tiles—*around a set of "control points". Thus, **the perimeter of each one of these tiles is equidistant to neighboring points** and defines their area of influence.

All these segments that interconnect the points form a triangular structure called *Delaunay Triangulation*. In the illustration below you can see the process as we continue adding points:

We can find interactive sites on the internet (like **this**) to draw points, move them, and check how the structure becomes updated in real time.

In fact, if we have a series of random dots scattered in the plane, the best way of finding the correct *Voronoi Telesación* for this set is using the Delaunay triangulation. And in fact, this is precisely the idea shown on the animation: first the *Delaunay Triangulation* and then, subsequently, the *Voronoi Tessellation*.

But to draw a correct *Delaunay Triangulation* is necessary to meet the so-called ** “Delaunay Condition”. **This means that:

*a network of triangles could be considered Delaunay Triangulation if all circumcircles of all triangles of the network are “empty”.*

Notice that actually, given a certain number of points in the plane there is no single way to draw triangles, there are many. But only one possible triangulation meets this condition. It is very simple: we draw a triangle using 3 points only if the circumcircle created using these 3 points is "empty" (not enclosing any other dot).

You see that in the graph below, extracted from Wikipedia:

We could rotate 90 degrees each side of the triangle using the the midpoint after defining the *Delaunay Triangulation* (top left), to construct the *Voronoi Tiling* (top right). This is exactly what the animation shows just before that the camera pulls back to show us the structure of our dragonfly wing.

We could also use the centers of each circle, marked in red, as they describe the vertices of *Voronoi Tilings*.

Of course, I am pretty sure of one thing: if we take a real dragonfly, and we analyze their wings with the help of a magnifying glass or microscope (example), we find exceptions and deviations. But it is clear the similarity of both structures.