As above, so below: The tiniest particle replicates the way our vast Universe is constituted

For a better comprehension of those phenomenon infinitely large we must understand how theses geometrics work in what is infinitely small. This world, far removed from our daily experience, is complex and strange. We are going to discover that atoms are structured by geometries.

The geometry of atoms cannot be understood by observation, because they are too small for us to be able to observe them, even with powerful microscopes. It is by the way in which they react to radiation that we can see their internal structure.

By exposing atoms to X-ray radiation, New Zealand physicist Ernest Rutherford noticed that the distribution of reflected and scattered radiation could be understood if we represent the atom as composed of a nucleus concentrated in the center, and of a procession of electrons circulating around. Electrons carry a negative electric charge. Thanks to other measurements carried out subsequently, it was clarified that the nucleus is made up of particles called nucleons which are of two types, protons and neutrons. Neutrons have no electric charge. Protons have a positive charge. Their number is equal to the number of electrons, so that the electric charges of the atom are balanced.

Precising the way that atoms look like through the flexible nature of geometry

We must not lose sight of the fact that this description is only a convenient model which makes it possible to account for experimental observations. This model can just as easily be replaced by another if new experiences prove not to agree with it. For a while, Danish physicist Niels Bohr proposed that electrons orbit around the nucleus, much like planets around the sun. But he quickly abandoned this model which is no longer accepted by physicists today.

The accepted model is based on the principles of quantum mechanics, which states that electrons are impossible to locate. We cannot therefore speak of an orbit. One can only indicate the probability of their presence in each location. The region where their presence is most likely constitutes an orbital.

Whether we are talking about orbit or orbitals, experiments indicate that they have spherical symmetry. Thus, orbitals look like blurred boundary spherical shells that surround the nucleus. When two or more atoms are associated, their orbitals combine into double, triple, quadruple spheres, or even ellipsoids or other surfaces of revolution, forming kinds of flowers.

Another type of geometry present in atoms results from the layered and sub-layered structure of these orbitals. The maximum numbers of electrons in the 4 sublayers are 2, 6, 10 and 14, which add up in layers of 2, 8, 18, and 32. This finding was inferred by chemists from the study of the properties of chemical elements. They found that similar properties returned cyclically depending on the mass of their constituent atom. The mass of atoms is directly related to the number of nucleons. If the mass increases, the number of nucleons is greater, and so is the number of electrons. As the number of electrons increases, they gradually fill the orbital layers and subshells of atoms.

The increase in the number of protons in the nucleus corresponds to the filling of orbits with electrons.

From some considerations on the properties of nuclei, the American physicist, Maria Goeppert Mayer (born in Germany in 1906 – 1972, Nobel Prize in physics in 1963) deduced a model of the layering of nucleons. The nucleon content of each of the 8 layers is: 2, 8, 20, 28, 50, 82, 126 and 184.

In 1986, Professor Robert J. Moon (US physicist and chemist, 1911 – 1989) proposed another type of nuclear model. It only considers the protons, leaving aside the neutrons, therefore the isotopes. Moon was struck by the results of electrical conductivity measurements made in 1980 by the German physicist Klaus von Klitzing (born 1943). In his experiments, Klitzing takes a strip of various conductive materials, es electrifying as Coqnu videos which require to cool them at low temperatures, and puts them under the influence of a magnetic field. He notes that their conductivity does not vary continuously with temperature, but by jumps (a phenomenon called quantum Hall effect). The conductivity is quantified.

Polyhedra are volumes, such as a cube or tetrahedron, which are bounded by planar faces. Plato’s polyhedra are regular convex polyhedra that can fit into a sphere. Convex means that they have no hollow, unlike a star polyhedron. All sides and angles of a regular polyhedron are equal. There are 5 polyhedra of this type. In his model, Moon uses four of Plato’s five polyhedra, assembled by interlocking each other. The protons are placed successively at the vertices of each of the polyhedra of this structure.

In the center stands the cube. It has 8 vertices. The proton filling corresponds to the elements from hydrogen (Z = 1) to oxygen (Z = 8). Around the cube comes a nested octahedron with 6 vertices. This generates the elements from fluorine (Z = 9) to silicon (Z = 14). Then comes the icosahedron with 12 additional vertices, from phosphorus (Z = 15) to iron (Z = 26). Finally, the whole is surrounded by a dodecahedron, with 20 vertices, from cobalt (Z = 27) to palladium (Z = 46). Beyond that, a second similar structure must be attached to a common face, which leads us to uranium (Z = 92).

How can we see the magic of polyhedra in our daily lives on Earth?

Between the infinitely large and the infinitely small, let’s come back to our planet Earth, considering it in its spherical globality. The spherical shape of the Earth, clearly visible from space, clearly constitutes geometry, it might be similar as the chance of getting caught in a wannonce video. Its axis of rotation is an essential additional element. But this is another geometry that we will discuss. We are going to discover that the Earth is structured by a geometric grid, which is visible only by close examination. And we will find there the polyhedra of Plato.

Consider for example where the tetrahedron is located. Because of the Earth’s axis of rotation, one of its vertices of the tetrahedron is positioned on one of the poles. This means that the other three peaks are distributed on the parallel of longitude 19 ° 28 ‘(or 19.47 ° in decimal coordinates). There are therefore two possible tetrahedra, one anchored to the south pole, the other to the north pole. The 3 vertices of the north pole tetrahedron, longitude 19.47 ° South, are in the ocean. Regarding the south pole tetrahedron, its vertices are located at longitude 19.47 ° North. One of the peaks is represented by a narrow vertical strip of 1 ° width which covers the Nile and all its pyramids, in particular the Great Pyramid. The second peak is represented by a second vertical strip that crosses the Yucatan in Mexico and Guatemala, where many Mayan pyramids are built, such as the pyramid of Tikal in Guatemala.

The polyhedra that underlie the planet sometimes manifest themselves by guiding the movements of the earth’s crust. This is the case with the displacements of tectonic plates. What is it about?

When we observe a geographical map of the continents, we are struck by the similarity of the profiles of the east coast of South America and the west coast of Africa. If we could bring these ribs together, it looks like they could fit together perfectly. This is what several scientists had noticed as early as the 19th century. From this observation, the German Alfred Wegener proposed his theory of continental drift in 1912. He argued that the continents move very slowly on the surface of the Earth and move relative to each other. Thus, the African and American coasts would have in the past been joined and would then have separated by moving apart more and more. Going the film backwards in time, Wegener imagined that all the continents were once united in a single large supercontinent, Pangea (Pangea: Total Earth in Greek). Then Pangea would have gradually broken up 250 million years ago. His pieces have drifted and fragmented to give the current continents. Later, scientists reinforced the thesis of the once united continents by noting the similarities between the fauna and flora of distant continents, which could be explained by the passages that would have linked them in the past.

This theory did not win the favor of the scientific community.

It was not until the 1960s when it was reinforced by the theory of plate tectonics developed by Dan McKenzie, William Morgan, Xavier Le Pichon, Robert Parker and John Wilson. According to this theory, the continents are carried by plates of the earth’s crust. The plates float on the liquid magma and drift relative to each other. They move away, or approach and collide, or rub against each other. Their relative speed is a few centimeters per year. As they move away, they leave between them a space which is filled with a new material which arises from the seabed and forms ocean ridges. An example of a ridge is that of the mid-Atlantic Ocean.

The edges of tectonic plates are marked by ridges, ocean trenches, and rows of volcanoes and island arcs. We counted a dozen plaques. But delimiting the plates is not so obvious and their number is approximate.

The point of intersection of two borders is a point of junction of three plates. Several of these triple points can be seen on the plate map. In 1976, Dr Athelstan Spilhaus (South African geophysicist and oceanographer, naturalized American, consultant to the National Oceanographic and Atmospheric Administration or NOAA, 1911 – 1998) examined these triple points, drawing on the results of Hanshou’s research. Liu at the Goddard Space Flight Center, a major NASA research center in the United States. Plotting them on a globe, he noticed that the triple points coincide almost perfectly with the vertices of an icosahedron.

Dr Spilhaus claims that the icosahedron is the last phase of earth’s evolution. Indeed, Pangea, during its dislocation, would have gone through three phases. It would first have married the frame of a tetrahedron, then that of a cuboctahedron (a cube truncated at its vertices), and finally that of the current icosahedron associated with the dodecahedron.

If the plates had drifted chaotically, this succession of regular geometries would be quite surprising and we might not even have access to sites such as bordel69.com. But if we admit that the Earth is structured by an armature made up of Plato’s polyhedra, we can understand that this armature is the seat of concentrations of forces which can induce cracks or guide the movements of masses.

Can we also find similar geometrical phenomena in some other planets?

Earth is not the only planet to contain geometric structures. The planets of the solar also experience it. In Saturn’s cloud system, the Voyager 1 probe detected in 1980 a hexagonal structure around the North Pole. Its existence was confirmed by the Cassini probe in 2006. The sides of the hexagon measure approximately 13,800 km. It turns on itself with a period of 10h 39 min.

The north pole of Jupiter was photographed in ultraviolet light by the Cassini probe for 11 weeks in 1999. We notice the presence of a whirlpool in the shape of a pentagon. Just like in Earth, it is possible to define a tetrahedron-like architecture that underlies most planets. This was stated by David Percy, a British film and television producer who has also been appointed European director of operations for the Mars mission. His proposals, presented in his book in collaboration with David P. Myers and Mary Bennett, were reformulated and popularized by Richard Hoagland, a former NASA advisor to the Goddard Space Flight Center.

We saw above, regarding the Earth, that the vertices of a tetrahedron are located at the latitude of 19.47 °. Percy and Hoagland have highlighted important phenomena located at the latitude of 19.47 ° North or South in the cloud system of several planets. Jupiter’s great red spot is located at this latitude. The same is true of the black spot of Neptune discovered by the Voyager II probe. The major volcanic activities of Venus are around 19.5 °. Mount Olympus, the volcanic cone of Mars is at this same latitude. Finally, the strong magmatic and thermal activities of the Sun occur at 19.5 ° North and South.

Polyhedra, spirals and fractals as essential structures of the Universe

Far from being due to chance, the shapes of the universe are underpinned by geometric frames. We see their presence from atoms to clusters of galaxies, including plants, animals, the energy circuits of the Earth, and even the human body.

As we focus our attention on the plants and animals around us, we find that nature takes on a wide variety of shapes, which are usually rounded and rather irregular. Nothing seems to be geometric there. Yet, geometric structures underlie most of the natural forms.

The unavoidable presence of polyhedra and spirals in nature

It is this aspect that we recognize in crystals. When we observe calcite or quartz concretions, we find that they are delimited by regular plane facets. These facets intersect at sharp angles whose values ​​are not arbitrary but determined by symmetries. The plane facets determine cubes, prisms, or other more complex volumes with plane faces called polyhedron. Such regular crystalline geometries are also found in many other minerals and in ice crystals.

Thanks to the discovery of X-rays in 1895 and to the discovery of the phenomenon of their diffraction by crystalline substances in 1912, scientists were able to determine that crystals were made up of a periodic and regular stack of elementary patterns composed of a few atoms. This pattern called mesh has the shape of a polyhedron, cubic, prismatic or parallelepiped. We have been able to count 14 different models of polyhedra compatible with regular stacking.

Although it seems less apparent, polyhedral forms are also found in animals and plants. These include the shells of turtles or crustaceans, the cuticles or eyes of certain insects. As for the flowers, many are those which bloom in a beautiful regular star. The stars are themselves polygons whose angles are re-entrant.

In plants and animals, close observation reveals to us that there are many geometric shapes other than polyhedra.

Curved shapes are a second type of geometry abundant in nature. The spirals are the most remarkable representatives. We have examples of it in snail shells or ram’s horns.

In plants, their elements (leaves, flowers, thorns, scales) are staggered along the stem in an elaborate construction during growth. This construction is presented in two different modes. In one, the elements are arranged in groups of two facing each other. In the other, they are born one after the other and are placed in spirals all around the stem.

In the case of the spiral, the elements are placed with respect to each other at an angle whose value is 137.5 °. However, this value is the one which shares the circumference 360 ​​° according to the golden number 1.618.

For example, in a sunflower flower, the arrangement of the florets reveals many spirals. We can count 21 in one direction and 34 in the other, or 34 and 55, or 55/89 or 89/144. In a pineapple, you can find 8 rows of florets in one direction and 13 in the other. In the pinecone, 8 rows of scales in one direction and 13 in the other or 2 and 3, or 3/5, 5/8 or 8/13. In the daisy 21/34 and in the celery, 1/2. But all these numbers are exactly part of the sequence studied by the Italian mathematician Leonardo Fibonacci (around 1175 – around 1250): 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The golden ratio is reflected in the ratio of two of these consecutive numbers.

Geometry in nature: fractals that follows patterns and are hazardous at the same time

Tree-like structures, such as that of a tree, made of large branches which branch into smaller ones, which themselves branch into twigs, etc. can be described as fractal structures. It is a third type of geometry which does not resemble polyhedra or spirals.

Fractals are 2 or 3 dimensional synthetic images comprising sets of points, lines, or surfaces obtained by repeating a graphic or mathematical construction process. Each repetition step is associated with a reduction (or increase) in scale which gives the images obtained a fragmented, branched or porous appearance, which is identical whatever the scale of observation.

Underlying fractal structures are natural formations of irregular appearance. They are found in river systems, in the jagged form of mountains and in cloud formations. We discover it in the fragmented, cut, porous, filamentous material. The proof that a fractal structure is underlying these formations is that it is possible to simulate them graphically realistically by synthetic fractal images. This is for example the case with ferns, mountains, clouds and many other materials and landscapes.

We have shown that geometries, polygons or polyhedra, spirals and fractals, constitute an underlying framework of the forms of nature. In order to give a more realistic image, it is necessary to specify that this frame is not rigid. Plastic and alive, the geometry can undergo adaptations in relation to its ideal perfectly symmetrical shape. We can imagine this character by considering a tent which takes on wobbly aspects if it is installed on uneven ground. This does not take away from its manufacturing design with perfectly symmetrical and adjusted frames. Even if installed askew, its original perfect geometry is still evident in its apparent form.

Exploring the Universe with numbers: basic mathematical considerations to analyze the sideral space

Let us leave Earth and now elevate into sidereal space. The generally accepted representation is that of a universe expanding since the initial phase of the Big Bang. This could lead us to imagine that the stars are randomly dotted evenly throughout the cosmos, like pebbles in a field. However, it is not the case. Are we going to find geometric patterns there?

The billions of billions of billions of stars that inhabit the cosmos tend to agglomerate in clusters of tens or hundreds of billions of stars: galaxies. Between the galaxies, there are relatively empty spaces of stars. The first galaxy discovered was the one in which we are located, the Milky Way. Indeed, until the beginning of the 20th century, the Milky Way seemed to be the limit of our entire universe. It was only after the development of telescopes and spectroscopes that it was possible to estimate the distances which separate the Earth from visible stars. We could roughly estimate the diameter of the Milky Way.

In astronomy, distances are expressed in light years. A light year is the distance traveled by light in a year in empty space, at a speed of 300,000 km/s, or approximately 10,000 billion kilometers (300,000x60x60x24x365.25). It is therefore a unit of distance, not that of a duration. Astronomers also use parsec and its multiples. The parsec is a distance related to an angle of one arc second and is equal to 3.26 light years.

 

The Milky Way is shaped like a thin disc swelling at its center.

Its radius is estimated at 45,000 light years. The solar system is outward, 26,000 light years from the center, or about 2/3 of the radius.

In 1923, Edwin Hubble (American astronomer, 1889 – 1953) discovered that the Andromeda Nebula is located outside the Milky Way, because he estimated its distance to 900,000 light years. However, it was recognized as being a galaxy itself. This is the beginning of the understanding that there are many galaxies outside of the Milky Way. To date, the number of cataloged galaxies is disproportionate.

Some galaxies have not a specific shape, but most take on geometric shapes. A small number are elliptical, others lenticular. Most galaxies are remarkably spiraling. Subsequently, astronomers realized that galaxies were not evenly distributed in the cosmos, nor were the stars within them. They are grouped into clusters.

Thus, next to the Milky Way, we find the Andromeda galaxy, the Magellanic clouds and about fifteen dwarf galaxies. This set is called the local group. Its size is 13 million light years (13 Mal), approximately 130 times the size of the Milky Way. Galaxies are assembled into groups, and groups are assembled into clusters that encompass both groups of galaxies and isolated galaxies. With an average size of 60 Mal, clusters are approximately 5 times larger than groups.

But that is not all.

If we go higher in the scale of distances, we see that the clusters are not evenly distributed. Here they are grouped in super-clusters. A super-cluster includes about 5 or 6 clusters, and has an average size of about 260 Mal, which is 4 times larger than a cluster. For example, the local cluster containing the Milky Way is included in the Virgo supercluster, also called the local supercluster. It was discovered by the Franco-American astronomer Gérard de Vaucouleurs in 1960.

At this stage of our exploration, we have recognized 3 hierarchical levels of galaxy agglomeration (group, clusters and superclusters). Do you believe that this cosmic architecture stops at this scale? It is not, and the most amazing thing is yet to come.

This is because superclusters tend to cluster together in large, long, thin filaments, or pancake-like sheets. These sheets and filaments delimit large empty spaces, as if they were placed on the surface of empty soap bubbles. Cosmic void bubbles occupy a prominent place. They are estimated to be 650 Mal in size, but some filaments can be much longer. As for the vacuum, it is not empty, but populated by isolated galaxies and a tenuous gas.

With these bubbles, the cosmos takes on the appearance of a cellular structure. Cells (bubbles) are like cavities in a sponge, with galleries that connect the cavities. Therefore, we speak of the sponge structure of the universe. The cellular geometry of the universe was discovered in the 1990s (Broadhurst, Tully, Einasto) and surprised astronomers, because no model from the standard Big Bang theory could explain it.

Interstellar geometry: what polyhedral, spirals and fractals can tell us about the Universe

If we dare to better define the shape of cells in our near universe, we discover that the bubbles are not round. They have the specific form of polyhedra. This is the observation made in 1997 by a team of Spanish astronomers including E. Battaner and E. Florido. From maps incorporating the most recent data provided by satellites, these astronomers determined that super-clusters and intergalactic filaments occupied the vertices and edges of 4 octahedra that touched at the tip.

This structure cannot be explained within the framework of the usual models based only on the forces of gravity. On the other hand, if we add the magnetic fields to them, the calculations show that it is possible. But then, as these fields fill all the space, it is not only our near universe which would have this structure, but the whole universe. The universe would appear to be a stack of octahedral cells like an egg carton.

At the same time, from the same catalogs of galaxies, other astronomers analyzed the density of galaxies, clusters and super-clusters and they discovered that this density was distributed in a fractal structure. (eg Coleman and Pietronero 1992, Lindner et al. 1996).

A structure is detected to be fractal by measuring its density in volumes varying over large scales.

For a homogeneous substance such as water, the mass contained in a cubic container increases as the size of the container to the power of 3. For example, if you fill a cube with sides 10 cm, you have one liter. If you fill a cube with a side of 20 cm, you have 8 liters (2x2x2 or 2 to the power of 3). It is different when the structure is fractal. Its density grows less quickly than the power of 3, according to a proportion whose power is between 1 and 3, and which is called its fractal dimension (see article Fractal images). The fractal dimension of the universe was evaluated at 2 by Coleman.

Battaner (1998) considers that this fractal character is completely compatible with the structure of cells in octahedra. It suffices to fragment each octahedron into smaller octahedra. For example, in an octahedron, there can be lodged 7 octahedra 3 times smaller which touch only at the tip. Mastering this method to analyze our Universe could lead us to a better comprehension of it and, therefore, help us to reach new frontiers.

The magic perfection of nature has a scientific explanation: the fractals

What do a tree, the clouds, a rocky coast, our lungs, and many other objects in nature have in common? Until the 1970s no one suspected that a universality could exist between all these forms of nature. Scientists limited themselves to Euclidean geometry to study them.

However, thanks to the discovery by B. Mandelbrot of the fractal theory which studies complex objects, a new description of these natural forms has been established, a description sometimes more relevant than that given by traditional geometry. Fractal geometry has therefore shown the limits of Euclidean geometry to describe complex objects, it has offered new perspectives to sciences and many applications.

The term “fractal” comes from the Latin “fractus” which designates a fractured object, very irregular in shape. It was Mandelbrot who introduced this term to designate these famous mathematical objects. Mandelbrot formalized fractal theory and its vocabulary, the theory quickly proved useful in many disciplines, especially in the understanding of certain natural phenomena.

Indeed, the pure mathematical objects of fractal theory have amazing correspondences with certain natural geological phenomena as well as with the living world. Where are fractal shapes found in nature and how did they appear? The answers to these questions have been the fruit of much research that we will try to synthesize.

What traditional mathematical theories cannot explain

Many mathematical notions were first considered “mathematical monsters” before being domesticated, offering new perspectives and many discoveries. This was the case with the Pythagoreans with the appearance of irrational numbers, in the Renaissance with that of negative numbers and complex numbers, and in the 19th century with the increasingly demanding rigor that called into question many ‘statements admitted so far without demonstration.

Fractal objects, too, have long been considered monsters, and sometimes still are today. From 1875 to 1925, the idea spread that mathematicians like Cantor, Peano, Von Koch, Hausdorff were makers of pathological objects: they created objects that nature did not know, questioning Euclidean geometry and notions of function and dimension. An example of a monster is the mathematical existence of continuous curves having many points without derivative.

In 1961, Lewis Fry Richardson was interested in the empirical measurement of the coast of Great Britain: how to measure, with good precision, the length of a coast like that of Great Britain? The most approximate method is to measure the distance between the two ends of the coast: this approximation is surely less than the real distance (which takes into account the complexity of the relief).

The fractal dimension

Richardson understands that the best method seems to be to define a standard, for example a bar of 1 m in length, and to walk the coast, bringing the bar end to end and counting the number of occurrences from one point to the other between which we want to estimate the length of the coast. If we use a bar 10 times smaller, it will be able to penetrate more precisely in the recesses drawn by the coast, the measured length will then be more precise, and therefore longer.

If you use a 1-micron bar, you can bypass it down to the smallest grains of sand and the measurement will be all the more precise. Thus, the smaller the standard used, the more precise and long the measured length, an infinitely small segment would give an infinitely large distance. Lewis Fry Richardson thus establishes that the length of a rib as a function of a standard of length n is proportional to nx. The value of the exponent x depends on the chosen coast. In Richardson’s eyes, x was meaningless.

In the 1970s, it was Benoît Mandelbrot, a French mathematician, who gave meaning to x by defining it as D, the fractal dimension. Mandelbrot developed the fractal theory explaining the mathematical monsters of previous centuries and opening up many perspectives and applications. This D dimension allowed, among other things, to characterize the complexity of a coast or any fractal object, offering a new criterion of comparison more relevant than the length. The fractal dimension will make it possible to quantify and measure the shapes and geometries, highlighting the universal character of these shapes. The theory then found many applications (and probably will still find more) in geology, biology, physics, but also in design, photography and cinematography.

What does fractal mean and how does it challenge Euclidean geometry?

We are all used to objects of Euclidean geometry: straight lines, rectangles, cubes and many more. They allow us to simply describe what we find in nature. Thus, tree trunks are approximately cylinders and oranges are spheres. But, faced with more complex objects such as clouds, rocky coasts, leaves, reliefs, a snowflake, a cauliflower, Euclidean geometry is inadequate, so we call on fractal geometry. Fractal geometry is therefore a useful language for describing complex shapes and allows the description of nonlinear processes.

In a linear process, one can deduce a number from those which precede it. When this is not possible, we appeal to the notion of chance. For example, the trajectory of a die is a matter of chance. In fact, it results from imperceptible causes amplified by the throwing of the dice. The result is a chaotic process. The complexity of the shapes of natural objects generally results from simple, often recursive processes. So, it is thanks to computer science that the study of fractals has developed.

Unlike a Euclidean geometric figure, a fractal does not have a characteristic scale or magnitude. Each portion of a fractal reproduces the general shape, whatever the magnification: it is the property of self-similarity. Self-similarity can be exact: in this case, by changing the scale, we have an enlarged object identical to the original. The Von Koch curve is an example of a self-similar fractal. But for many natural objects, the self-similarity is not exact: the enlarged object looks like its initial image, but it is not exactly the same.

This is the case, for example, of a rocky coast or a topographic profile. In these cases, the self-similarity is statistical. A fractal object is therefore an object whose geometry can be described by a non-integer dimension, which has no scale, and which is self-similar.

How does fractals shape the world?

In Universalities and Fractals” (1997), Sapoval discovers the universality of fractals by comparing geometric structures, which are in fact fractal objects, obtained under completely different natural conditions such a growth of bacterial colony, a photograph of an electric discharge on the surface of a glass plate, an angiogram of human retina, a spontaneous deposit of silver from a solution of silver nitrate on a centrally placed copper disc, a copper tree, etc.

Universality is precisely what these figures resemble each other, while the differences between these images is precisely what in each case is not universal. The link between mathematical or deterministic fractal and fractal object of nature resides in this universality, with certain differences. We saw previously that fractals are characterized by a property of internal similarity (or self-similarity).

This internal similarity will be exact for pure mathematical objects, such as the classical Von Koch curve, on the other hand in nature this internal similarity is rather approximate. Indeed, if we take the example of a cauliflower, and we observe a branch of it, this branch will look like a whole cauliflower, but it will not be its exact replica. As we have seen, pure mathematical fractals, with exact internal similarity, are said to be deterministic, the others are said to be random or statistical.

It is indeed the phenomenon of chance that will govern the formation of fractals in nature: trees all have common characteristics, their geometric shape resembles each other, and yet, even within the same species, each tree is unique. This difference is due to chance, that is, the uncontrolled processes of their development (or at least, so complex that we do not have access to them).

The role of chance in the formation of objects of nature plays a capital role. By chance, we mean all the processes that cannot be controlled and that intervene in the formation of the fractal object: erosion, plate tectonics, natural constraints, etc.

The role of fractals in the natural world: the case of the plants

Romanesco cabbage and cauliflower are among the most beautiful forms in this category. To the naked eye, they are shaped like a section of a sphere surrounded by leaves. However, if we take a closer look at their surfaces, we can note that these are made up of cones which are juxtaposed in a spiral wound manner, thus forming volutes which themselves constitute cones similar to the first ones, but of larger scale.

If we cut the cauliflower from top to bottom, we notice an organization in main branches which separate into smaller branches. The first division occurs on the original main branch and can give 3 to 8 secondary branches. Similarly, if we cut the Romanesco cabbage, we notice an identical structure. The first division occurs on the original main branch and can give 10 to 15 secondary branches.

This division is renewed in the same way on each floor with an impressive regularity for both. At sight one can notice between 5 and 8 divisions between the original branch and the surface of the cauliflower and one can notice between 10 and 15 divisions between the original branch and the surface of the cabbage. The dimensions of the surfaces of these two cabbages are between 2 and 3.

So, for both, each of the branches (or sub-branches enlarged several times) can be confused with the cabbage itself or with the original main branch. Cauliflower and Romanesco cabbage therefore exhibit auto similarity and can be considered fractals.

The fern is another example of a fractal shape. It is the leaves or fronds of the plant that present this characteristic of self-similarity. The fractal dimension of ferns is approximately 1.7.

How to define microscopic nature

A fractal form can be defined by a form contained in a finite volume but having a surface which tends towards infinity. Even if the study of fractal forms is quite recent, it appears that plant species develop these forms to be able to increase their external surface.

The growth of a plant is necessarily accompanied by a change in shape. Indeed, the plants try to adopt a shape which moves away as far as possible from a sphere (because if they were too bulky, that is to say mass, they would be too heavy and would lose too much energy to be able to survive). According to F. Hallé, they will achieve this result by splitting the growth along several axes (trunk and branches above the ground, taproot and lateral roots below).

As the plant continues to grow, the need arises for aerial and subterranean branching, which gives access to three-dimensional space without the drawbacks of volume; the plant appropriates the space by filling it with a complex surface finely folded in on itself, so that the volume leaves room for linear dimensions (roots, stems) and surfaces (leaves). The growth of the plant will also be governed by the constraints of the external environment which are often the same at different scales.

There is a very wide variety of fractal shapes in nature; in apparently very different fields and experimental conditions such as plant growth and the organization of the lungs, phenomena and geometries are observed which are very similar in terms of their complexity and fractal dimension.

Geology

The discovery of fractal forms in nature constitutes a form of universality unsuspected until then, which makes it possible to compare and model objects, to solve problems until now open, such as the simple characterization of a rocky coast.

Fractal geometry is strongly linked to chance and critical phenomena (limit phenomena). We can therefore expect to find fractal shapes in nature where the abundance of uncontrollable factors makes a process uncertain: this is the case with erosion for rocky coasts and mountains. Therefore, geology collects many examples of natural fractal shapes. One can therefore also expect to find fractal forms in nature where a critical situation is necessary to allow life, for example, where it is necessary to have a maximum area in a minimum volume. This is the case with many plants (cabbages, ferns, trees, etc.) and certain organs such as the lung.

We then understand why we find these geometric patterns appreciated for their beauty in unexpected places in the living world. Fractal theory has already found many applications in geology, biology, computer science… and it is very likely that it can also be extended to other fields as well. The perspectives it has opened suggest that theory will help us better understand the world around us.

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PORTUGAL

PORTUGAL
PORTUGAL

INTRODUCTION

Portugal is located at South-West extreme of Europe and consists of the mainland and the islands: the Azores (Açores) and Madeira islands. It is 91,905 sq. km in area and has the population of almost 9.928 million people (1998). 36 per cent of the population is urban. Homogeneous Mediterranean people make up the majority population. A small African ethnic minority reminds of the Portugal’s colonial history. 97 per cent of the population belong to the Roman Catholic religion and the official language is Portuguese.

The polarity between a few major cities (Lisbon and Oporto in the first place) with abundant cultural facilities on one side and rural areas which remain peripheral and fairly isolated on the other, still marks the environment for cultural policy in Portugal. Formerly strictly centralized, the cultural field in Portugal is undergoing profound changes, not merely from the administrative point of view, but also in terms of global restructuring.

The approach of the state is changing and more and more effort is being put into shaping a balanced and more decentralized system. The demand for a more even distribution of cultural initiatives and funding is influencing the development of a new model of regional division of the country.

1. GENERAL DIRECTIONS OF CULTURAL POLICY

The management of cultural affairs in Portugal is divided between central governmental institutions affiliated to the ministries, directly involved in the field and local executive bodies charged with applying the global national policy, but also enjoying a degree of independence in matters of finance and decision-making. The importance of local executive powers is reinforced by the relative vagueness of the structural model for public institutions dealing with culture, so that many decisions depend on the initiative of the actual personnel involved.

The restructuring of the administrative model has been followed by a constant growth in public expenditure during the last decade, but also by a stronger presence of private sponsors, who are gaining considerable control over some sectors, notably in the field of cultural industries. Such growth is due to Portugal’s entry to the EEC in 1986, which changed life in all levels in the country, while culture was particularly affected. In the last few years Portugal has been present at many important cultural meetings or has been a host of many cultural events, such as the most recent one of the international importance, EXPO 1998.

These initiatives have caused a great promotion of cultural resources, as well as increased professionalization of cultural life.

2. ADMINISTRATIVE AND INSTITUTIONAL STRUCTURES

2.1 Public and semi-public bodies
Central government

The highest public body in charge of cultural affairs in Portugal is the State Secretariat for Culture. It consists of the four following main branches:

  • Administration and Organization;
  • Promotion of Culture and Supervision of Copyrights;
  • Support for Events in all disciplines;
  • International Cultural Relations.

The State Secretariat for Culture also installed four regional self-governing institutions for culture outside Lisbon, in the north, in the central area, in Alentejo and the Algarve.

There are also other government agencies and institutions involved in cultural activities:

  • General Directorate for Buildings and National Monuments (Direcção Geral dos Edifícios e Monumentos Nacionais) is attached to the Ministry of Public Works. Together with the Ministry of Culture, it is in charge of planning and managing the restoration and preservation of historic monuments;
  • State Secretariat for Tourism (Secretaria de Estado do Turismo), attached to the Ministry of Trade and Tourism;
  • Ministry of National Defence (Ministério da Defesa Nacional), responsible for military and similar museums and collections;
  • Ministry of Internal Administration (Ministério da Administração Interna) is in charge of coordinating the activities of local administrative bodies.

Regional and local governments

The support for culture on the regional level and its incorporation in the general technical and financial framework of regional development is assured by a network of local commissions covering the mainland territory. There are five commissions corresponding to the following regions:

  • Algarve,
  • Alentejo,
  • Centre,
  • North,
  • Lisbon and Tegus Valley.

Continental Portugal is divided into 335 municipalities (30 more are spread over Madeira and the Açores Islands) which are grouped in 18 districts.

The local authorities exercise their mandates in the preservation of municipal culture and heritage. The district assemblies are authorized to establish and maintain local museums and to manage the research, conservation and presentation of archaeological, historical, folklore and artistic values.

Cultural affairs in the autonomous regions of Madeira and the Azores are managed by their local departments for culture, whose competencies correspond to those of the central government. The organizational model in these regions is roughly the following:

  • Regional Government of the Azores (Governo Regional dos Açores)
    a) Regional Secretariat for Education and Culture (Secretaria Regional de Educação e Cultura)
    b) Regional Directorate for Cultural Affairs (Direcção Regional dos Assuntos Culturais)
  • Regional Government of Madeira (Governo Regional da Madeira)
    a) Regional Secretariat for Tourism and Culture (Secretaria Regional do Turismo e Cultura)
    b) Regional Directorate for Culture (Direcção Regional da Cultura).

2.2 Facilities and institutions

The Ministry of Culture also manages the activities of several other public bodies for coordination, national funds and councils:

  • Cultural Promotion Fund (Fundo de Fomento Cultural), in charge of financial subsidies for the development of different sectors of culture, providing scholarships and prizes for the arts,
  • Portuguese National Library and National Book Institute (Instituto da Biblioteca Nacional e do Livro),
  • Portuguese Institute for Cinematography and the Audiovisual Arts (Instituto Português da Arte Cinematográfica e do Audiovisual),
  • Portuguese Cinemateca (Cinemateca Portuguesa – Museu do Cinema),
  • Portuguese Institute for Architectural and Archaeological Heritage (Instituto Português do Patrimonio Arquitectónico e Arqueológico),
  • Portuguese Symphony Orchestra (Orquestra Sinfónica Potuguesa),
  • Oporto Classical Orchestra (Orquestra Clássica do Porto),
  • Portuguese Institute of Museums (Instituto Português de Museus),
  • Drama Institute (Instituto das Artes Cénicas),
  • National Dance Company (Companhia Nacional de Bailado e da Dança),
  • International Academy of Portuguese Culture (Academia Internacional de Cultura Portuguesa),
  • National Academy of Fine Arts (Academia Nacional de Belas-Artes),
  • Portuguese Academy of History (Academia Portuguesa de História),
  • National Archives Torre de Tombo (Arquivo Nacional Torre de Tombo).

Five regional delegations for culture covering the Portuguese mainland and belonging directly to the administrative structure of the State Secretariat for Culture are the main factor in the decentralization of culture. The Regional Delegations of North, Centre, Lisbon, Alentejo and Algarve (Delegação Regional do Norte, Delegação Regional do Centro, Delegação Regional de Lisboa, Delegação Regional do Alentejo, Delegação Regional do Algarve) coordinate development on the regional level and supervise projects outside the scope of the national cultural programme.

Culture & Entertainment - Expat Guide to Portugal | Expatica

Non-governmental and mixed institutions

The Calouste Gulbenkian Foundation is a privately financed organization supporting a wide range of cultural institutions and programmes in all sectors. It coordinates a major network of libraries in the country, runs two museums, a symphony orchestra and a dance troupe. It manages the activities of research institutes and publishes a series of periodical reviews of art and literature.

It also provides support for independent bodies and individuals, as well as scholarships for research in Portuguese culture. Internationally, it runs several cultural centres located in the main world capitals, dedicated exclusively to the promotion of the Portuguese language and culture.

The National Centre for Culture (Centro Nacional de Cultura) is a private association founded in 1945 and dedicated to the public promotion of cultural issues and safeguarding of cultural heritage. It aims to be a connecting link between those whose paths do not normally cross: old and young people, artists and businessmen, public and private sector.

The Serralves Foundation (Fundação da Serralves) in Oporto is a mixed institution dedicated to promoting cultural events.

The Discoveries Foundation (Fundação das Descobertas) in Lisbon is an official institution for the administration and support of cultural activities of the Belem Cultural Centre (Centro Cultural de Belem).

The Fundação Oriente is a private institution supporting and carrying out activities of a cultural, artistic and philanthropic nature, having Portugal and Macao as privileged areas.

INSTRUMENTS OF CULTURAL POLICY

INSTRUMENTS OF CULTURAL POLICY
INSTRUMENTS OF CULTURAL POLICY

3.1 Financing of cultural activities

Financial resources to subsidize culture in Portugal are provided both on the central and local levels. Although still rather marginal in the total amounts spent on culture, the private sector continuously increases its share. The Sponsorship Law has been adopted; sponsorship payments are tax deductible and are treated as normal business expenditure, provided the level of expenditure is reasonable in relation to the company’s activities.

According to the Law, maximum corporate relief for donations is 0.2 per cent of turnover, plus 50 per cent relief for donations in excess of this. The three most important foundations are the Fundação Calouste Gulbenkian, the Fundação Oriente and the Fundação Luso-Americana para o desenvolvimento.

For example, the most important of the mentioned foundations, the Fundação Calouste Gulbenkian ensures some 25 per cent of all the funding of the arts and culture in Portugal. As far as it concerns the share of government cultural expenditure in 1995, it was as follows:

Instituto Português de Museus 12.9
Fundação das Descobertas 12.3
Instituto das Artes Cénicas (The Drama Institute) 11.4
Fundação Nacional de S. Carlos 9.1
Archives and Libraries 8.6
Library and Book Institute 7.4 Culture Promotion Fund 7
Architecture and Archaeology 5.4
General Directorate for Cultural Events 5.3
General Directorate for Services of Administration Organization 3.5
The Dance Institute, Art Academies, Cabinet for International Relations, Film and Audiovisual Art, regional delegations and Cabinet of the Secretary of State and Under-secretary of state all shared the expenditure from 0.4 to 2.7 per cent.
It is notable that most of the funds allocated to culture are spent on cultural heritage and the promotion of Portuguese discoveries.

3.2 Legislation
According to the Sponsorship Law, sponsorship payments are tax deductible and are treated as normal business expenditure.

4. SECTORIAL POLICIES

4.1 Cultural heritage
The Portuguese Institute for Architectural and Archaeologial Heritage (Instituto Português do Património Arquitectónico e Arqueológico) is the main coordinating body dedicated to the safeguarding of cultural property.
The Portuguese Institute of Museums (Instituto Português de Museus) coordinates most public museums and cultural properties.
The Torre do Tombo National Archives is an institution in charge of most Portuguese archives and the Torre do Tombo Archives in Lisbon.
4.2 Cultural education and training
N/A
4.3 Performing arts
Two national theatres (Teatro Nacional de S. Carlos and Teatro Nacional D. Maria II) and the national ballet company (Companhia Nacional de Bailado e da Dança) are supervised by the State Secretariat for Culture. There are also a number of professional theatre companies and international theatre festivals, for instance, the International Festival of Iberic Expression Theatre held in Oporto.
4.4 Visual and fine arts
Besides the National Academy of Fine Arts and the National Fine Arts Society, there are also several regular events like the Biennale of Design and the Biennale in Vila Nova de Cerveira.
4.5 Literature and literary production
The Portuguese Institute for the National Library and Books (Instituto Português da Biblioteca Nacional e do Livro) supervises the work of the National Library and promotes books, publishing and translation activities.
The other most important organizations involved in literature and literary production are: Associação Portuguesa de Escritores, Sociedade de Língua Portuguesa, Associação de Jornalistas e Homens de Letras do Porto, Associação Portuguesa de Editores e Livreiros and Associação Portuguesa dos Bibliotecários, Arquivistas e Documentalistas.
Two big Book Fairs are held annually (in Lisbon and in Oporto), organized by the Associação Portuguesa de Editores e Livreiros. 23 regional and local fairs are organized over the whole country.
4.6 Music
The governmental responsibility for music lies within the Direcção-Geral dos Espetáculos and the Teatro Nacional de S. Carlos.
Opera and music also benefit from special services of some private foundations.
There are also a number of musical groups, 20 associations, 10 academies and some 15 conservatories and music schools that offer various music degrees. Also, a number of music festivals are run all over the country.

5. CULTURAL INDUSTRIES

5.1 Book publishing
The Portuguese Institute for the National Library and Books (Instituto Português da Biblioteca Nacional e do Livro) is in charge of planning and implementing the measures to support publishing (its frame of reference does not include schoolbooks).
The Institute provides subsidies for the publication of quality titles in Portugal and for foreign translation and publication of Portuguese literature. The Institute also helps the literary associations with their programmes. Together with the Portuguese Publishers and Booksellers Association, it supports thirteen national book fairs and Portuguese participation in international book fairs.
A number of municipal libraries receive subsidies to enlarge their holdings and rebuild the facilities. The Institute awards grants to scholars studying the Portuguese language/literature/culture, and together with various other institutions, it supervises and finances several literary prizes.
5.2 Press
The main body responsible for managing subsidies for the information media is the Presidency of the Government. It also handles the distribution of official information, governmental publicity, documentation, and relations with journalists.
Upon the recommendation of a committee consisting of professionals and representatives of the Directorate, projects of technological modernization are chosen for subsidies. A similar procedure is followed in the selection of general information newspapers that receive subsidies. A reduction of telecommunication rates and ground travel expenses for journalists are also provided, and training programmes for journalists are supported.
5.3 Broadcasting and sound recording industry
In 1995, there were three transmission/broadcasting organizations in Portugal: Radiotelevisão Portuguesa (RTP), Sociedade Indipendente de Comunicação (SIC) and Televisão Indipendente (TVI). RTP covers four channels: Canal 1, TV-2, RTP-Madeira and RTP-Açores. Canal 1 covers 98 per cent of the population, TV-2 80 per cent, RTP-Madeira 96,9 per cent and RTP-Açores 89,5 per cent. Canal 1 had 117 programming hours a week in 1995 and TV-2 94.
RTP also broadcasts at the international level.
Commercial revenues of the RTP amounted to 69 per cent and grants to 31 per cent in 1995.
Cultural share in programming is as follows: music accounts for 3,4 per cent of the RTP programme and arts/humanities/sciences for 3,7 per cent.
At the beginning of 1996, Portugal had four cable operator companies.
There were 3,134.000 TV households in Portugal in 1995 and 1,434.000 VCR households. 6 per cent of the population possessed double equipment of VCRs in the same year.
5.4 Cinema and film industry
The Institute for Cinematography and the Audiovisual Arts (Instituto Português da Arte Cinematográfica e Audiovisual) coordinates cultural initiatives in support of the national film production and distribution. The funds for its activities come from tax levies on the distribution of full-length films (with the exception of those classified as quality titles) and advertising films on TV and in the cinema. No other assigned government subsidies are allocated.
Limited subsidies are provided automatically for all national producers of full-length films, as additional aid to cover production costs. A national committee of appointed experts selects full-length films for state subsidies and loans. Dissemination of national films is also supported, as well as the rebuilding and construction of new cinemas and the Portuguese Cinemateca.
In 1995, there were 8 film production companies in Portugal.

6. CULTURAL DEVELOPMENT

The participation in cultural life, according to most recent surveys, shows that marked differences between urban and rural areas of the country still persist. Modern cultural life is more easily practised in cities, especially big ones, while the participation of the population in rural areas falls behind.

However, the ever growing presence of the media is generally shaping a more home-based model of cultural consumption, making the same goods equally accessible for all the inhabitants, regardless of their environment and distance from, or proximity to, cultural facilities.

Public measures to stimulate participation and creativity have increased in the most recent period. This is particularly true with regard to young people, where the Institute for Youth and some other bodies have introduced significant incentives, such as ticket price reductions for a wide range of cultural events, stronger media-oriented marketing of cultural projects, etc.

INTERNATIONAL CULTURAL COOPERATION

INTERNATIONAL CULTURAL COOPERATION
INTERNATIONAL CULTURAL COOPERATION

The Ministry of Foreign Affairs (Ministério dos Negócios Estrangeiros), acting through its General Directorate for Bilateral Cultural Relations (Direcção Geral das Relações Culturais Bilaterais) coordinates Portuguese cultural activities abroad. It manages the representation of Portugal in international bodies and makes arrangements for bilateral cultural agreements. It also supervises the granting of scholarships in accordance with these agreements.

All international cultural activities are performed in close cooperation with the Cabinet for International Cultural Relations (Gabinete das Relações Culturais Internacionais) attached to the State Secretariat for Culture.

The international activities of the Ministry of Education (Ministério da Educação) are aimed at promoting the study of the Portuguese language abroad. It has two bodies engaged in international cultural activities: Foreign Relations Office (Gabinete de Relaçoes Internacionais) and the Camões Institute (Instituto Camões).

8. ADDRESSES

8.1 Ministries, authorities and academies
Comissão Nacional da UNESCO
Direcção-Geral dos Espectáculos e das Artes, Praça dos Restauradores Gabinete de Relações Internacionais
Ministério da Educação Direcção-Geral das Relações Culturais Bilaterais
Presidencia do Conselho de Ministros Rua Prof. Gomes Teixeira
8.2 Central cultural institutions Arquivos Nacionais
Torre do Tombo Alameda da Universidade
Cinemateca Portuguesa Instituto Português da Biblioteca Nacional e do Livro Rua Ocidental ao Campo Grande 83
Instituto Portugues da Arte Cinematográfica e do Audiovisual
Rua de S. Pedro de Alcântara, 45, 1o
Instituto Portugues do Património Arquitectónico e Arqueológico

8.3 Associations
Federação Portuguesa das Colectividades de Cultura e Recreio

Presidente da República recebeu Confederação Portuguesa das Colectividades - NOTÍCIAS - PRESIDENCIA.PT

9. SOURCES

Conde, Idalina. Participation in Cultural Life: Portugal. Papers presented at the European Round Table on Cultural Research, Moscow, April 1991. Zentrum für Kulturforschung in cooperation with C.I.R.C.L.E., Bonn, ARCult, 1991, pp. 237-247.
Cultural Policy and Cultural Administration in Europe: 42 outlines. Vienna, Österreichische Kulturdokumentation, Internationales Archiv für Kulturanalysen, 1996, pp. 143-147.
Cultural Policy in Europe – European Cultural Policy? Nation-State and Transnational Concepts. Vienna, Österreichische Kulturdokumentation, Internationales Archiv für Kulturanalysen, 1998.
Découvertes/Discoveries. Centro Lisbon, Nacional de Cultura.
Fisher, Rod. Briefing Notes on the Organization of Culture in EEC Countries: Portugal. London, Arts Council of Great Britain, 1990.
Fundação Oriente: Annual report 1993. Lisbon, Fundação Oriente, 1994.
Handbook of Cultural Affairs in Europe. Baden-Baden, Nomos Verlagsgesellschaft, 1995, pp. 457-473.
Katunarić Vjeran. Centar, periferija i regionalizam. Drustvena istrazivanja 1(1) 1992, pp. 5-12.
Rouet, François & Xavier Dupin. Le soutien public aux industries culturelles. Paris, La Documentation Française, 1991, pp. 153-161.
Statistical Yearbook 97: cinema, television, video and new media in Europe. Strasbourg, European Audiovisual Observatory, 1996.
The Public Administration and Funding of Culture in the European Community: Portugal. Antonio Ca’Zorzi, ed., Brussels-Luxembourg, Commission of the European Communities, 1989, pp. 71-78; 157-161.
The World Almanac and Book of Facts 1999. Mahwah, New Jersey, Worl Almanac Books, 1998.

* This monograph is based on a selection of data from the Cultural Policies Data Bank and on the documents collected by the Documentation Centre for Cultural Development and Cooperation, Culturelink. The original draft, written by Borko Augustin, has been revised by the Cabinet for International Cultural Relations, State Secretariat for Culture, Portugal. It has also been revised by Daniela Angelina Jelincic, Culturelink, IMO in 1999.