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Post by Admin 2012-07-13, 15:49


Nature by Numbers

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Post by Admin 2013-01-10, 19:31


Fibonacci number

In nature

Fibonacci sequences appear in biological settings,[7] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[8] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.[9] In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g., relating to the breeding of rabbits, the seeds on a sunflower, the spirals of shells, and the curve of waves.[50] The Fibonacci numbers are also found in the family tree of honeybees.[51]

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Post by Admin 2013-01-10, 19:32



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Post by Admin 2013-01-10, 19:32


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Post by Admin 2013-01-10, 19:35




06. Can we use any of your images or animations? I think it would be good publicity for you… etc.
If you are a teacher and want to show some of my animations to your students, of course you can. And if you have a personal, non commercial blog and you want to embed some of my videos hosted on Vimeo and YouTube, you can also. In none of these two cases is necessary for me to ask permission, but I would thank if you add a brief note clarifying the authorship of the work.
For any other situation where there is some kind of commercial activity (business sites, advertisements, TV, editorial use, etc) please contact me to negotiate an appropriate license for the use of my images.
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Post by Admin 2013-01-10, 19:38


Nature by Numbers

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Post by Admin 2013-06-20, 18:37

Nature by Numbers  Title_about

This section is meant to be a complement to the animation, in order to better understand the theoretical basis that you can find behind the sequences. It was also, more or less, the appearance of the screenplay in the days that I was planning this project.

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.

Nature by Numbers  Fibon_01

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.

Nature by Numbers  Fibon_02

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.

Nature by Numbers  Espiral

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.

Nature by Numbers  Naut_vs_fib

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 ;-)

Nature by Numbers  Nautilus

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:

Nature by Numbers  Aureo_01

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).

Nature by Numbers  Aureo_02

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 theFibonacci Series by the previous, the resulttends to Phi. The higher the value, the greater the approximation (consider that Phi, like any irrational number, has infinite decimals).

Nature by Numbers  Fibon_03

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:

Nature by Numbers  Aureo_03

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:

Nature by Numbers  Pipas_01

— 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.

Nature by Numbers  Pipas_02

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 andGolden 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: theVoronoi 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 aVoronoi Tiling:

Nature by Numbers  Voronoi_01

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 thebisector 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:

Nature by Numbers  Voronoi_03

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:

Nature by Numbers  Voronoi_02a        Nature by Numbers  Voronoi_02b

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.

Nature by Numbers  Ala_det

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.

Nature by Numbers  Ala

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Post by Admin 2013-06-20, 19:31

Simple logic tell us...

... if we have this PROJECT in all NATURE, and even in ourselves (our own body), then we must see MASTER ARCHITECT of (all) universe.

If so, then we must ask ourself...


*(or don't beleve) 


Rom 1:19  because the thing which may be known of God is clearly revealed within them, for God revealed it to them.
Rom 1:20  For the unseen things of Him from the creation of the world are clearly seen, being realized by the things that are made, even His eternal power and Godhead, for them to be without excuse.
Rom 1:21  Because, knowing God, they did not glorify Him as God, neither were thankful. But they became vain in their imaginations, and their foolish heart was darkened.
Rom 1:22  Professing to be wise, they became fools
Rom 1:23  and changed the glory of the incorruptible God into an image made like corruptible man, and birds, and four-footed animals, and creeping things.
Rom 1:24  Therefore God also gave them up to uncleanness through the lusts of their hearts, to dishonor their own bodies between themselves.
Rom 1:25  For they changed the truth of God into a lie, and they worshiped and served the created thing more than the Creator, who is blessed forever. Amen.
Rom 1:26  For this cause, God gave them up to dishonorable affections. For even their women changed the natural use into that which is against nature.
Rom 1:27  And likewise also the men, leaving the natural use of the woman, burned in their lust toward one another; males with males working out shamefulness, and receiving in themselves the recompense which was fitting for their error.
Rom 1:28  And even as they did not think fit to have God in their knowledge, God gave them over to a reprobate mind, to do the things not right,
Rom 1:29  being filled with all unrighteousness, fornication, wickedness, covetousness, maliciousness; being full of envy, murder, quarrels, deceit, evil habits, becoming whisperers,
Rom 1:30  backbiters, haters of God, insolent, proud, braggarts, inventors of evil things, disobedient to parents,
Rom 1:31  undiscerning, perfidious, without natural affection, unforgiving, unmerciful;
Rom 1:32  who, knowing the righteous order of God, that those practicing such things are worthy of death, not only do them, but have pleasure in those practicing them.

Psa 53:1  The fool has said in his heart, There is no God. They acted corruptly, and have worked out abominable wickedness; there is not one doing good.
Psa 53:2  God looked down from Heaven on the sons of mankind to see if any was discerning, who was seeking God.
Psa 53:3  Every one has turned away; they have altogether become filthy; not one is doing good, no, not even one.


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