Statement: graphically determine a circle c equivalent (like air) to the given irregular hexagon ABCDEF . - transform the irregular hexagon in an irregular pentagon of equivalent area. "The Pentagon as the hexagon now transformed into a triangle of area equal (divide the pentagon into triangles for keeping it moving its vertices heights). - We mean proportional between half the height of the triangle base and getting the side of the square equivalent. - The side from the split into eleven parts, took seven of them and half of those seven. - In the proportional average of 7 / 2 and the side of the square radius of the circle we obtain the equivalent.

That group willing to accept grief can comment here and create the exercise to be solved, otherwise it will choose a random group.

The Platonic solids, also known as Platonic bodies, bodies cosmic Pythagorean solids, solid perfect, Plato's polyhedra or, more accurately, convex regular polyhedra, are characterized as geometric polyhedra whose faces are equal regular polygons and whose vertices bind the same number of faces. DEFINITION

will take that as a reference for defining the Platonic solids (which obviously are named after Plato who first studied them)

All those who read this blog is sure to have known any of them, but may not call it that. The list of the Platonic solids is small, since no other solid meets those conditions. are 5 Platonic solids:

The tetrahedron with 4 equilateral triangular faces and an angle of 70, 53 º

The cube or hexahedron, with 6 square faces and an angle of 90 °

The octahedron, with 8 equilateral triangular faces and an angle of 109.47 º

The icosahedron, with 20 equilateral triangular faces and an angle of 138.19 º

The dodecahedron, with 12 regular pentagonal faces and an angle of 116.56 º

Once known, study their properties

- Regular:

All faces of a Platonic solid are equal regular polygons.

In all the vertices of a Platonic solid attend the same number of faces and edges.

All edges of a Platonic solid are the same length.

All dihedral angles formed by the faces of a Platonic solid with each other are equal.

All its vertices are convex to the icosahedron.

- Symmetry:

The Platonic solids are highly symmetrical:

All of them are central symmetry about a point in space equidistant center of symmetry of its faces, vertices and edges.

They also have axial symmetry on a number of lines of symmetry passing through the center of symmetry above.

They also have mirror symmetry about a series of planes of symmetry (or principal planes), which divided into two equal parts.

geometric

Following the above, can be traced all Platonic solid three particular areas, all focused in the center of symmetry of the polyhedron:

A sphere inscribed tangent to all sides in the middle.

A second area tangent to all edges in the center.

A circumscribed sphere, passing through all vertices of the polyhedron.

Projecting the centers of the edges a platonic polyhedron on its circumscribed sphere from the center of symmetry of the polyhedron is obtained a regular spherical network, consisting of equal-circle arcs, which are regular spherical polygons.

-conjugation:

If you draw a polyhedron using as vertices the centers of the faces of a Platonic solid Platonic solid gains another, called conjugate of the first, with many vertices and faces was the original sound, and same number of edges. The conjugate of a polyhedron is a dodecahedron, icosahedron, and vice versa, that of a cube is an octahedron, and tetrahedron conjugate polyhedron is another tetrahedron.

The Platonic solids are present in nature (such as the basic structure of HIV, which is an icosahedron) and man has used for such things as well you can guess.

For example, the dice.

The Omnipoliedro

A omnipoliedro is a composite made with the frames of the five Platonic solids, so that each one of them is enrolled in another. place to begin the octahedron (white), enrolled in the tetrahedron (red), for which we make its vertices coincide with the center of the edges of the tetrahedron. Then we could put the cube (yellow), matching the four corners of the tetrahedron with as many of that.

now surround the dodecahedron (green), for which it would seek the agreement of the eight corners of the cube. Finally, we have the icosahedron (blue). The edges of this and the dodecahedron, are cut at the midpoints. The centers of the faces of a dodecahedron and the icosahedron determined.

It seems that Escher liked omnipoliedro always have a hand built with wires. Escher more than a curiosity (if you like Escher, a fellow have a great article about the artist here )

Information on omnipoliedro is taken from this blog

By the way, here a video of how some students of 2 º ESO build a omniedro:

Plato's beliefs

The history of the Platonic solids can be traced back to Plato and Pythagoras. They thought that the Platonic solids had magical properties and was what the universe was formed. The 4 elements (which in ancient Greece were all supposed to things, and now the chemical elements) were attributed to each figure: the tetrahedron symbolized fire, earth cube, octahedron icosahedron air and water. The dodecahedron was the most special of all, since it was thought that the cosmos and the key to unlock the secrets of the world. Here is an explanatory text from Wikipedia and an entertaining video on beliefs about the Platonic solids. (Incidentally, some of our comrades prepared an article on the Pythagorean Theorem, if you're interested in him )

They even came to attribute magical and mythological properties, Timaeus of Locri, in Plato's dialogue says "The fire is made up of tetrahedrons, the air of octahedra, the water of icosahedra, the land of cubes, and as it is still possible fifth way, God has used it, the pentagonal dodecahedron, to serve as a limit on world ". The ancient Greeks studied the Platonic solids thoroughly.

Science and art are two activities that we usually think never intersect, due largely to a trend today day break in most aspects of life in "Science" or "letters".

But there is no science without lyrics, and vice versa. It po r Thus, if we look closely at works of art, we see that are composed of infinite geometric elements, both simple shapes such as methods and systems of representation.

Defining the conical perspective scientific mind, we say q ue is a graphical method of representation through which we can translate in a plane h orizontal the projection of a three-dimensional body with straight lines that intersect at one point, thus obtaining a rough picture of what would the human eye at a distance from the object.

In the picture above you can see a cube of edge x drawn in perspective frontal cone. The side of the cube that is facing the viewer has d ibujado in size (although it could also be applied on any scale, if necessary.) To prepare the drawing is necessary to know certain information, such as the position of the points D (which allows us to find the depth) and F (vanishing point). Here we have a single vanishing point, but when it purports to represent an image with oblique perspective, it is necessary a second vanishing point F ', for directions. The horizontal gray lines represent the ground line and horizon line, whose position depends largely on the outcome or etting.

To relate this method to the actual drawing, attached the following and you sq UEMA , which shows ta nt elements that are part of the design co mo e LEMENTS external.

Now however, let's talk about the same from an artistic point of view.

Back in the fifteenth century a new artistic movement emerged in Europe came close, as far as art is concerned, to classical antiquity, the Renaissance.

was then that the concept d and the perspective gained strength in the paint. A century later (1500-1520) Italian artists such as Raphael, da Vinci and Michelangelo reached the pinnacle as far as artistic and intellectual innovation is concerned.

The main feature of Renaissance painting is precisely what interests us: the use of perspective. But as the saying goes, a picture is worth a thousand words, so here goes:

At first glance, it appears that these two images there is no link, but if we combine ...

is now obvious that the first image was nothing more than a simple outline or sketch of what would become the School of Athens , Rafael

.

We can see horizontal and vertical lines (some of which are highlighted in red and green, respectively) did not deviate and are maintained parallel and perpendicular to each other. But if you look at the depth contours (highlighted in black) we can see as lines that are parallel in reality get closer to converging on a single point, which in this case is approximately the center of the work.

There are many other paintings in which you can see the use conical perspective, that cover centuries, from The ideal city of Piero della Francesca (1470) ...

... until today works as Gran Via, Antonio López García.

Passing by artists like Van Gogh, who skillfully used perspective in his famous Van Gogh's Room (1888), since, although at first it was thought that the apparent deformation of objects around the corner right of the room was due to his mental imbalance, years later, discovered the plans of the room, shown below, justifying the above deformation.

what looks like a fern, the coast and a snowflake?

All three are elements of nature. And three, with its complicated forms and repetitions seem fractals. And if you're wondering what the hell is a fractal, you will be surprised to know these fascinating structures that seem taken from a book that one of mathematics.

Fractals are geometric shapes, like triangles and rectangles, but special properties that distinguish themselves from them. First, they are very complex, any size. They have self-similarity , ie which can be divided into parts that are small copies of the total. Unlike other geometrical dimension is a fraction.

Fractals often look like objects of nature. Many objects natural, such as ferns, snowflakes, the coasts of the country rocks, have shapes similar to fractals. An interesting thing about fractals is that their study is new and are currently being investigated.

Here you have a rather interesting video on fractals.

has traditionally been defined as part of geometry that studies the properties of incidence figures, completely abstracting the concept of measurement. In considering such a complete quadrilateral, the Projective Geometry learning no properties in size between sides and angles, but all that is derived from having a figure formed by four straight non-concurrent three to three that identifies six breakpoints .

This cuatrilátero has six vertices.

These properties that at first glance may seem irrelevant, may become powerful tools without which we could not unwrapped olvernos in geometry that requires the design engineering.

The problem of the painter

Consider for a moment the problem that an artist must paint faces when a real object on the canvas: it should play almost identically that perceived reality.

In an effort to produce more realistic paintings, artists of the Renaissance was deeply interested in discovering the laws governing constructio n of the projection of real objects onto a plane .

Masaccio, 1435. First paint

perspective.

Leonardo Da Vinci.

The Last Supper.

So how can we represent a three dimensional object on a plane? What is preserved by projection if they do neither the length nor the angles? What is the relationship between two sections of the same figure? How far away are the stars?

All these aspects are included in Projective Geometry.

We can find the definition of symmetry in many ways:

The symmetry is a characteristic of geometrical shapes, system, equations, and other material objects or abstract entities, related with invariance certain transformations, movements or exchanges. "If we are to the definition of the SAR, third meaning: BODY { g_RealURL "Geom. Correspon dence exact regular arrangement of parts or points of a body or figure relation to a center, an axis or plane.

Symmetry surrounds us in all aspects and m ay very different, and we have an abstract conception of symmetry in our minds. Consider the types of symmetry mples:

* Central

* Axial

Certain shapes ic present several lines of symmetry or any line that passes through a certain point will axis of symmetry. Can you imagine that?

Lets look at some examples. Referring to our fellow LoshombresdeThales and entry of the golden ratio, we note that the Vitruvian Man remains the golden proportions but also provides a human body symmetry:

But we can go back further to find clearly symmetrical elements:

Indeed the Greeks were the first to study in essence, along with arithmetic. Pythagoras himself used simple concepts of symmetry plane to demonstrate his famous theorem. An interesting link: Proof of Theorem: Pythagoras on the beach

If we want a practical application of the symmetry, we can always play pool:

All this based on pure reflection, in which the angle of the ball is equal to the output. The theory of reflection and diffraction has even belong in the military aviation industry with Lockheed F-117 Nighthawk and "short-wave method in theoretical physics of diffraction."

Finally I leave you with this creator of symmetries very useful and even fun with which you can do to siemtrías radio: Symmetry Artist

Pictures Optical Illusions and impossible constructions "?

"The optical illusions are effects that play with the sense of sight characterized by visually perceived images that are false or misleading. False if there is not really what the brain sees or wrong if the brain misinterprets the visual information. " This is one of the possible definitions of optical illusions, which, most often surprise us with images that seem very peculiar sight. Thus we could ask the question Do we deceive our senses?. But as the saying goes a picture is worth a thousand words so we put some examples:

(This image shows the inability to determine the highest point of the ladder.)

Is not it a bit strange the construction of this triangle ?

MORE CASES TO RECREATE THE VIEW (comment that you observe in the different images and the peculiarities tengais)