Symmetry in space. The concept of a regular polyhedron

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Central symmetry is a mapping of space on itself, in which any point M goes to a point M1 symmetrical to it with respect to a given center O.

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Figures with Central Symmetry

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Art. subway Sokol

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Art. Metro Rimskaya

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Pavilion of Culture, VVC

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.O

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Axial symmetry

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Axial symmetry with the a-axis is such a mapping of space onto itself, in which any point M goes to a point M1 symmetric to it with respect to the a-axis. Axial symmetry is movement. a Axial symmetry M M1

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Х y Z О M(x;y;z) M1(x1;y1;z1) Let us prove that axial symmetry is a movement. To do this, we introduce a rectangular coordinate system Oxyz so that the Oz axis coincides with the symmetry axis, and establish a connection between the coordinates of two points M(x;y;z) and M1(x1;y1 ;z1) symmetrical about the Oz axis. If the point M does not lie on the Oz axis, then the Oz axis: 1) passes through the midpoint of the segment MM1 and 2) is perpendicular to it. From the first condition, using the formulas for the coordinates of the middle of the segment, we obtain (x+x1)/2=0 and (y+y1)/2=0, whence x1=-x and y1=-z. The second condition means that the applicates of points M and M1 are equal: z1=z. Proof

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Proof

Let us now consider any two points A(x1;y1;z1) and B(x2;y2;z2) and prove that the distance between the points A1 and B1 symmetric to them is equal to AB. Points A1 and B1 have coordinates A1(-x1;-y1;-z1) and B1(-x1;-y1;-z1) Using the formula for the distance between two points, we find: AB=\/(x2-x1)²+(y2 -y1)²+(z2-z1), A1B1=\/(-x2+x1)²+(-y2+y1)²+(-z2+z1). It is clear from these relations that AB=A1B1, which was to be proved.

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Application

Axial symmetry is very common. It can be seen both in nature: the leaves of plants or flowers, the body of animal insects and even humans, and in the creation of man himself: buildings, cars, equipment and much more.

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Application of axial symmetry in life

Architectural buildings

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Snowflakes and human body

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eiffel tower owl

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What could be more like my hand or my ear than their own reflection in the mirror? And yet the hand that I see in the mirror cannot be put in the place of a real hand. Emmanuel Kant.Mirror symmetry

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The display of a three-dimensional figure, in which each of its points corresponds to a point symmetrical to it with respect to a given plane, is called a reflection of a three-dimensional figure in this plane (or mirror symmetry).

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Theorem 1. Reflection in a plane preserves distances and, therefore, is a motion. Theorem 2. A motion in which all points of a certain plane are stationary is a reflection in this plane or an identical mapping. Mirror symmetry is specified by specifying one pair of corresponding points that do not lie in the plane of symmetry: the plane of symmetry passes through the middle of the segment connecting these points, perpendicular to it.

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We prove that mirror symmetry is a movement. To do this, we introduce a rectangular coordinate system Оxyz so that the Оxy plane coincides with the symmetry plane, and establish a connection between the coordinates of two points М(x; y; z) and М1(x1; y1; z1), symmetrical relative to the Oxy plane.

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If the point M does not lie in the Oxy plane, then this plane: 1) passes through the midpoint of the segment MM1 and 2) is perpendicular to it. From the first condition, according to the formula for the coordinates of the middle of the segment, we obtain (z+z1)/2=0, whence z1=-z. The second condition means that the segment MM1 is parallel to the Oz axis, and. therefore, x1=x, y1=y. M lies in the Oxy plane. Consider now two points A (x1; y1; z1) and B (x2; y2; z2) and prove that the distance between the points symmetric to them is A1 (x1; y1; -z1) and B (x2; y2; -z2). According to the distance formula between two points, we find: AB \u003d square root of (x2-x1) 2 + (y2-y1) 2 + (z2-z1) 2, A1B1 \u003d square root of (x2-x1) 2 + (y2-y1 )2+(-z2-z1)2. From these relations it is clear what was required to be proved.

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Symmetry with respect to the plane (mirror symmetry) of space is movement, which means that it has all the properties of movements: it translates a straight line into a straight line, a plane into a plane. In addition, this is a space transformation that coincides with its inverse: the composition of two symmetries with respect to the same plane is the identical transformation. With symmetry about a plane, all points of this plane, and only they, remain in place (fixed transformation points). The lines lying in the plane of symmetry and perpendicular to it pass into themselves. The planes perpendicular to the plane of symmetry also transform into themselves. Symmetry with respect to the plane is a movement of the second kind (changes the orientation of the tetrahedron).

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The ball is symmetrical about any axis passing through its center.

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A right circular cylinder is symmetrical with respect to any plane passing through its axis.

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A regular n-gonal pyramid for even n is symmetrical with respect to any plane passing through its height and the longest diagonal of the base.

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It is usually believed that the double observed in the mirror is an exact copy of the object itself. In reality, this is not entirely true. The mirror does not just copy the object, but swaps (rearranges) the parts of the object that are front and back with respect to the mirror. In comparison with the object itself, its mirror twin turns out to be “inverted” along the direction perpendicular to the mirror plane. This effect is clearly visible in one figure and virtually invisible in another.

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Let's assume that one half of the object is a mirror double in relation to its other half. Such an object is called mirror-symmetric. It transforms into itself when reflected in the corresponding mirror plane. This plane is called the plane of symmetry.

Symmetry of space

Tell me what is the symmetry of space?

You need to start with definitions to get to the bottom of it. Many of your physical laws are far from reality, but simply an attempt to describe multidimensional processes with three-dimensional thinking. Symmetry is the construction of a certain order of movement and focusing of energy. The universe is great and diverse, the types of forms of creation are infinitely diverse. Therefore, symmetry in your understanding and symmetry within the framework of the entire universe are different things. This is the same as comparing the decimal number system that you have adopted with, say, the binary or septenary number system. Understand? These are different approaches to organizing structuring. You have countless cubes. You can stack them however you like: in lots of piles of twos or fives or sevens. In two big piles. In five big piles and so on. Further, in each pile, you also define a certain system for distributing cubes. This is the process of structuring space. Since the Divine light is infinite, the number of structuring cubes is also infinite, therefore the variations of the addition of these divine cubes are also infinite, and therefore the variations of the symmetry of space are infinite.

Your concepts of symmetry come from their duality, from systems of single reflection, these are the symmetry properties of the dual world in which you reside. In your world, any form has a symmetrical mirror reflection, any concept and direction of movement has a reflected double.

Reflected double? What do you mean.

It's like the other side of the coin. The same medal, but a look from the other opposite side. Outside view and inside view. The reflected double is a look from within. Any phenomenon and any action can be viewed differently from different points of perception.

Wait, let's get it right. In nature, symmetry is widespread precisely binary symmetry. Snowflakes, plant leaves, crystal lattices, flowers, fruits and much more. Even in the structure of atoms there is symmetry. Why?

Let's go back to the perception filter. You are the source of Divine light, enclosed in a form-lamp. The border form of your lamp is thin, but strong. And it can be organized in different ways. Now it has two holes in it. Therefore, if your light comes out of you, then it always comes out in a binary version. When your light comes out of your holes-sensors of space, then outside of you it also comes across binary rays emanating from other forms reflecting you, is reflected from these rays, refracted and returns to you again through your two holes. This is a very simplified model, it is a binary perception model. Dual reflection model. As your awareness expands, new openings-perceptions open in you, and everything becomes more complicated, as it were, the multivariance increases, the symmetry of space becomes more complicated.

When you talk about the symmetry of, say, a leaf of a tree, you see this symmetry in a plane version. But imagine the symmetry of a plant leaf in 3D, when the reflection mirrors are placed in such a way that three identical parts are created. It is difficult for you, because in your world everything has a couple. Then try to imagine a quaternary system of symmetry, when two leaves intersect in the longitudinal trunk. Or four sheets of paper, as in a book, are bound by a common binding. Now imagine that the book has an infinite number of pages and the interweaving of these pages is also countless.

I feel like your 3D thinking and imagination are at a loss, this is normal. It is difficult to readjust right away, but you must believe that your perception system, which is actually hidden in you and others very deeply, allows you to create and perceive any multidimensionality. Therefore, I will give you examples of spatial models and complicate them so that you gradually get used to multidimensional perception not only mentally, but also in your imagination, although in reality it is one and the same.

So we take a point in space and an infinite number of rays emanating from it. As you understand, this is the description of you in the universe. For if the number of rays emanating from a point is infinite, then it describes all possible rays of space around you. But there are also an infinite number of such points. The points from which the rays emanate are the forms of God. As you can see, the symmetry of space was originally laid down for you and in the space around you. For each ray emanating from the point of reflection will find a reflected pair. But there will be not two such rays, but many pairs. Further, these rays come across, say, a mirror and are reflected from it. If we imagine a ray as a straight line, then its reflection gives a refraction, a bend in the other direction of this straight line. And accordingly, the dual pair of this beam will also be reflected from this mirror and give a symmetrical bend, as it were, in the other direction. This is how fractality is born, that is, the symmetry of reflections or reflected symmetry. And now imagine that the point from which the rays emanate is one, and the mirrors are countless, then there will be countless fractal reflections. Now imagine that it is not the mirrors that someone put up that reflect. And simply the rays emanating from you as points of perception are reflected from the myriad rays of countless other forms of perception, from which countless rays also come. This is the multidimensional symmetry of space.

But in your concept, symmetry is an identical equality of halves. But if you look at a plant leaf or a fruit, then the symmetry is still distorted there. That is, the reflections do not match completely up to a micron and beyond. So in your perception, the symmetry of space is also partially violated. When both beams that touch and reflect from each other have the same strength and direction, then the created reflection symmetry is more accurate, when this is not the case, then the reflection of one beam differs from the reflection of the other beam. But this is if we talk about the space as a whole. But then your reflected beam returns to you, and therefore it is for you, as for everyone, that the power of direction and the power of reflection are equal, since this is your power.

Then tell me, in nature we observe certain symmetrical figures: spheres, triangles, rectangles. These figures are present in everything. Why? Moreover, there are experiments with sound. When sand is poured onto the surface of the speaker, under the influence of sound vibrations, it takes certain geometric shapes.

There are many questions here. But you again try to think linearly. Let's take a snowflake whose symmetry you can see. She is beautiful and never repeats. Why? Because the microscopic particles of snow are structured in a certain order, each time representing a different reflection of energy on the parameters of cold, on the parameters of the environment in which they are reflected. But if you imagine a snowball, then there are a huge number of snowflakes in it, a huge number of non-repeating symmetries. And if you could see this new pattern, you would find a certain symmetry in it too. That is, everything is structured in interaction with each other.

Vibrations of sound are just the reflected energy. Its fluctuations in the reflective spectrum. In principle, everything is the reflected energy and its fluctuations in the reflecting spectrum. You can simply perceive some of these vibrations with your eyes, some with your ears, some with your sense of smell, and so on. And some are not yet able to perceive.

Now let's go further. When you observe the world around you, you see in it the symmetry of reflections in the form of certain figures and symbols. But if you look deep into you, then there is also an infinity of symmetry and reflections. You just haven't learned to look deep into yourself yet. You have created devices in the form of microscopes and magnifying structures, but with the power of your thought you yourself can penetrate into all your components up to the first particles, and if you do this, you will find amazing fractality and symmetry deep within yourself. You have been turned outside of yourself all the time. But inside you is the same endless world, what you call the microcosm, it is not known to you at all.

So now, in our example, countless rays emanate from a point, not only outside the point, but also inside the point, in the opposite direction. And these rays of perception are also reflected, structured, fractalized.

There are many experiences with water, when the sounds of certain vibrations, say kind words or classical music structures snowflakes in a very beautiful patterns. There are many examples of the harmonizing effect on a person of music, certain colors and smells, paintings in the form of symmetrical mandalas, and so on. What it is? What is happening?

Reflection. For example, a mandala is an energy image of certain interconnections of the rays of perception, built symmetrically. For you, it's just a picture. But imagine it as an energy picture. When you meditate on it, your directed energy is reflected from the energy of the mandala and, as it were, copies it, makes a mold from it, and is reflected symmetrically to it. Understand? And it comes back to you, structures your energy in a certain way and is again reflected outside. If you sit in a mandala meditation for a long time, you kind of become attuned. If you turn off all other sources of perception and completely focus on the mandala, then gradually your internal structuring becomes similar to the structure of the mandala, it reflects symmetrically from it, and a mandala is also born inside you, somewhat similar to the reflected one, but still possessing your features and characteristics. characteristics. The same happens with music, and with smells, and with flowers, and so on. You simply perceive more deeply the symmetry of the other form and structure your form accordingly.

Why exactly the sounds of nature or certain music or certain signs harmonize a person? If everything is only a type of reflection and its diversity, why can't we equally endure, let's say, a cacophony of sounds or, for example, the smells of decomposition? If there are no bad and good perceptions, why are we sufficiently equally attuned to certain perceptions?

Sustainability. Why is everything around you symmetrical? Because symmetrical configurations are stable. It's like a chair with one leg, three or four. What you call harmony is the most stable viable configurations of space. Unstable configurations break up. If the paper is consistently and symmetrically folded and folded many times, then you can roll it to a point, to a small ball, while there will be symmetry inside it, and many faces of a sheet of paper will have a huge number of contacts, clutches with each other. And if a sheet of paper is simply crumpled, then there will be much less contact between the points of the paper and the adhesion is correspondingly less, and the volume of the crumpled sheet is greater. This design is less stable. If, say, you sit on a folded sheet of paper, then it almost does not deform, and more importantly, the relationships do not deform. And if you sit on a crumpled sheet of paper, it will deform and many connections-contacts will be broken. Therefore, symmetry is a consistent compaction.

Does this mean that there is some original, unmanifested chaos, which, under a certain creative influence, takes on symmetrical forms?

You are all mixed up. Non-manifestation is the absence of movement. The movement itself is either chaos or symmetry, that is, when particles move randomly, this is already a manifestation. When the rays are reflected asymmetrically, this is also manifestation. just eat different types manifestations, and chaotic movement is no worse than symmetrical movement, it is just different. In the universe there are different kinds construction of space, including what you call chaos.

But you say that symmetrical configurations are more stable. Then why chaotic configurations?

This is various forms creations of space, its organization and structuring. Sometimes chaotic movement gives new directions for structuring. Just as you cannot reject the energy of destruction, since it is also used in creation, so you should not reject the chaotic structuring that is also used in creation. The symmetrical structuring of space is more stable, but also more rigid, less mobile. It's like a pre-created zone for choosing the movement of energy, you know? If you take your freedom of choice - this is just randomness. If we take any hierarchy, this is rigid symmetry and fractality.

It turns out that a chaotic structuring was introduced into the symmetry of space?

Or vice versa, symmetry was introduced into the chaotic structuring.

If everything that I see around me is just an agreement between people how to see it, then why do I see space precisely symmetrically, and not chaotically? If everything around is energy, then why do all people see the symmetry of a flower in a certain way? Why not chaos?

Because the reflected rays of the flower as the form of God are symmetrical. And you perceive exactly the direction of these rays. Look with light vision. When you look at a luminous object, when you close your eyes, light configurations appear on the inner screen, this is light vision. If you imagine the world around you in the form of energy, you will see fluctuations and movement of light lines and points of other figures. When you look at objects that seem formless to you and give them a shape in your imagination, as in the case of clouds, this means that either the object does not have rigid structuring bonds, that is, elements of chaos predominate, or you are simply not able to perceive such structuring. It's like with a snowball, inside of which there are billions of amazing snowy symmetry, but the snowball itself is not very symmetrical.

I'm asking about the observer effect. If the motion of, say, elementary particles depends on the observer, does this mean that the observed symmetry of the space of nature also depends on us, on the observers of this symmetry, and not on the space itself?

Certainly. Remember the example with your rays reflecting. The reflection of your beam is up to you. That is, from the properties of the beam itself. Passing the Divine light through your prism of perception, you give it certain characteristics of perception, a certain degree of reflection. Therefore, the effect of the observer consists precisely in the fact that you and only you are reflected in your own way from other rays of perception. But at some point or in some space of a certain extent, your rays are combined, this is the reflection of the outside world, this is your general picture of the world, this is the symmetry of space you see.

So, if we start to reflect chaotically, then the picture of the world will change?

You're putting the accents a little wrong. You are reflecting all the time. It's just that some of you and of God's forms reflect more symmetrically, and some more chaotically. Therefore, those who reflect more chaotically come into contact, intersect in their perception with those who also reflect more chaotically. This is the law of similarity, like does not just attract like. Like intersects only with like. You cannot intersect with someone who is directed, relatively speaking, in the other direction. Like non-intersecting roads in your world, they exist and lead in certain directions. But your road is in a different area and goes in a different direction. But if your road encircles the whole Earth, then sooner or later it will intersect with all other roads.

Therefore, if you see symmetry in the surrounding space, it is simply an intersection of your perception with those who are also reflected more symmetrically.

So somewhere there are worlds and spaces where everything is asymmetrical?

Certainly. Again, in your world the concept of chaos has a negative connotation. And imagine if you lived in a universe that is predominantly built on the chaotic movement of energy. Then any symmetry for you would seem to be something alien and, in terms of duality, negative and dark.

That is, the fact that we are directed towards the light, goodness is only a consequence of the fact that our universe is more built on the symmetry of space?

Yes. You got it right. However, your concept of light is the opposite of the concept of darkness. But everything, both light in your understanding and darkness in your understanding, is the reflected light of God, the reflected energy of God. Therefore, light in your understanding is a symmetrical reflection of the energy of God. And darkness is a chaotic reflection of the energy of God. And in fact your universe is an attempt to balance both. Give symmetry to chaos, and introduce chaotic components into symmetry. To get something in between. Since the symmetrical configuration is more stable, and the chaotic configuration is more multivariable.

It seems to me that harmony still wins, that is, symmetry. If you look at nature, it is clearly visible.

The development of any form and any system has stages of direction. Symmetry replaces chaos. Chaos replaces symmetry. Now you are at the stage of symmetrical infusion of configurations, like the process of crystallization of let's say salt, your space is crystallizing into certain harmonious structures and new forms of connection, new configurations, new crystals are created. But then, in order to test the stability of these forms, there will come a period of chaotic movement, like the impact of wind and rain on geological rocks and mountains. And then the mountains undergo changes. The mountain is symmetry or not? This is a combination of both. When symmetrical shape under the influence of chaotic processes, it changes its configuration, and this configuration is neither good nor bad. It's just a new combination of symmetry and chaos.

How can a person use the symmetry of space other than to harmonize himself?

Oh this is very interest Ask and you have a lot to understand on this subject. He can use this symmetry in everything. For example, he can configure himself symmetrically to an external object and thus repeat, copy it. That is to become similar with this object.

Did I understand correctly: if a person copies, say, the configuration of a plant, then he will become this plant?

It will almost become, because it will always be something different from the original. It will only be a copy. But you got it right. Those magicians who could transform into plants and animals did exactly that, copied the energy configuration of another object.

But that is not all. You, knowing the configuration and symmetry of space, can get from one point of space to any other. Now you are doing this chaotically by chance in your dreams and over very small, as it were, distances. But it is like a network of roads, a coordinate grid of the space of the universe. Knowing the coordinates, you kind of know the picture of the configuration, the picture of the symmetry of space, and having reproduced it with your consciousness, rebuilding your configuration in this way, you get, combined with this space, as if you get into a puzzle. If you, by your configuration, cannot fit into the picture like a puzzle, then you cannot perceive the boundaries of contact with other puzzles of the picture, understand? And much more in the symmetry of space you have to master. But it's too early for that.