Symmetry in space. Regular polyhedron concept

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Central symmetry is a mapping of space onto itself, in which any point M goes into a point M1 symmetric to it relative to a given center O.

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Shapes with central symmetry

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Art. metro Sokol

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Art. metro Rimskaya

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Pavilion Culture, VVC

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.ABOUT

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Axial symmetry

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

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X y Z О M (x; y; z) M1 (x1; y1; z1) Let us prove that axial symmetry is motion. To do this, we introduce a rectangular coordinate system Oxyz so that the Oz axis coincides with the symmetry axis, and we establish a connection between the coordinates of two points M (x; y; z) and M1 (x1; y1; z1) symmetric about the Oz axis. If point M does not lie on the Oz axis, then the Oz axis: 1) passes through the middle of the segment MM1 and 2) is perpendicular to it. From the first condition, using the formulas for the coordinates of the midpoint of the segment, we obtain (x + x1) / 2 \u003d 0 and (y + y1) / 2 \u003d 0, whence x1 \u003d -x and y1 \u003d -z. The second condition means that the applicates of points M and M1 are equal: z1 \u003d z. Evidence

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Evidence

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 distance formula between two points, we find: AB \u003d \\ / (x2-x1) ² + (y2 -y1) ² + (z2-z1), A1B1 \u003d \\ / (- x2 + x1) ² + (- y2 + y1) ² + (- z2 + z1). It is clear from these relations that AB \u003d A1B1, as required.

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Application

Axial symmetry is very common. It can be seen both in nature: plant leaves 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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Applying axial symmetry to life

Architectural buildings

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Snowflakes and the 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? Yet the hand that I see in the mirror cannot be replaced by a real hand. Emmanuel Kant, mirror symmetry

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The display of a volumetric figure, in which each point of it corresponds to a point symmetrical to it relative to a given plane, is called a reflection of a volumetric figure in this plane (or mirror symmetry).

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Theorem 1. Reflection in a plane preserves distances and, therefore, is motion. Theorem 2. A motion in which all points of a certain plane are immovable is a reflection in this plane or an identity 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 midpoint of the line segment connecting these points, perpendicular to it.

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Let us prove that mirror symmetry is motion To do this, we introduce a rectangular coordinate system Оxyz so that the Оxy plane coincides with the symmetry plane, and we establish a connection between the coordinates of two points M (x; y; z) and M1 (x1; y1; z1), symmetric relative to the Oxy plane.

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If point M does not lie in the plane Oxy, then this plane: 1) passes through the middle of the segment MM1 and 2) is perpendicular to it. From the first condition, using the formula for the coordinates of the midpoint of the segment, we obtain (z + z1) / 2 \u003d 0, whence z1 \u003d -z. The second condition means that the segment ММ1 is parallel to the Оz axis, and. therefore, x1 \u003d x, y1 \u003d 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 A1 (x1; y1; -z1) and B (x2; y2; -z2) symmetric to them. Using the formula for the distance 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. It is clear from these relations what was required to be proved.

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Symmetry about a plane (mirror symmetry) of space is motion, and therefore has all the properties of motion: it transforms a straight line into a straight line, a plane into a plane. In addition, this is a transformation of space that coincides with its inverse: the composition of two symmetries relative to the same plane is an identical transformation. With symmetry about a plane, all points of this plane, and only they, remain in place (fixed transformation points). The straight lines lying in the plane of symmetry and perpendicular to it pass into themselves. The planes perpendicular to the plane of symmetry also pass into themselves. Symmetry about the plane is a motion 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 straight circular cylinder is symmetrical about any plane passing through its axis.

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A regular n-gonal pyramid for even n is symmetric with respect to any plane passing through its height and the largest 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 front and rear parts of the object with respect to the mirror. In comparison with the object itself, its mirror counterpart turns out to be "inverted" along the direction perpendicular to the plane of the mirror. This effect is clearly visible in one drawing and practically invisible in the other.

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Suppose one half of an object is a mirror image of the other half. Such an object is called mirror-symmetrical, and 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 space symmetry?

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 by three-dimensional thinking. Symmetry is the construction of a certain order of movement and focusing of energy. The universe is large 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 two different things. This is the same as comparing the decimal number system, which is accepted by you, with, say, the binary, or septenary number system. Do you understand? These are different approaches to organizing structuring. You have countless cubes. You can add them as you want: in many piles of two or five or seven cubes each. In two large heaps. Five big piles and so on. Further, in each pile you also define a certain distribution system of 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 in the addition of these divine cubes are also infinite, and therefore the variations in the symmetry of space are infinite.

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

A reflected double? What do you mean.

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

Wait, let's go in order. In nature, binary symmetry is widespread. Snowflakes, plant leaves, crystal lattices, flowers, fruits and 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 lamp-form. The border of the shape of your lamp is thin but strong. And it can be organized in different ways. Now there are relatively speaking two holes in it. Therefore, if your light goes outside you, then it always comes out in binary form. When your light comes out of your holes-sensors of space, then outside of you it also stumbles upon binary rays emanating from other forms reflecting you, reflects from these rays, refracts 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 perception holes open in you and everything seems to become more complicated, 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 a three-dimensional version, when the reflection mirrors are set so that three identical parts are created. It's difficult for you, because everything in your world has a pair. Then try to imagine a quaternary symmetry system when two leaves intersect in the longitudinal trunk. Or four sheets of paper, like in a book, are united 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 is at a loss, this is normal. It is difficult to readjust right away, but you must believe that your system of perception, 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 fact they are one and the same.

So we take a point in space and an infinite number of rays emanating from it. As you understood, 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 countless 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 you 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 ray 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 there is only one point from which the rays emanate, and there are countless mirrors, then there will be countless fractal reflections. Now imagine that it is not the mirrors that have been installed by someone. And simply the rays emanating from you as points of perception are reflected from the myriads of rays of an innumerable number of other forms of perception, from which an innumerable number of rays also emanate. This is the multidimensional symmetry of space.

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

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

There are many questions here. But you're trying to think linearly again. Let's take a snowflake whose symmetry you can see. She is beautiful and never repeats. Why? Because microscopic snow particles 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 it contains a huge number of snowflakes, a huge number of non-repeating symmetries. And if you could look at this new pattern, you would find a certain symmetry in it. That is, everything is structured in interaction with each other.

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

Now let's move on. 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. It's just that you have not yet learned to look deeply into yourself. You have created devices in the form of microscopes and magnifying structures, but by the power of your thoughts you yourself can penetrate into all your components, down to the first particles, and if you do this, you will discover amazing fractality and symmetry deep within yourself. You have been turned outside yourself all the time. But inside you is the same infinite world, what you call a microcosm, you have not known it at all.

So now, in our example, an infinite number of 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, pictures in the form of symmetrical mandalas, and so on. What it is? What happens then?

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

Why exactly 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 tolerate, say, the cacaphony of sounds or, for example, the smell of decomposition? If there are no good and bad perceptions, why are we sufficiently attuned to certain perceptions?

Stability. Why is so much symmetrical around you? Because symmetrical configurations are stable. It's like a chair with one leg, three legs and four legs. What you call harmony are the most stable viable configurations of space. Unstable configurations disintegrate. If the paper is sequentially and symmetrically bent and folded many times, then you can fold it to a point, to a small ball, while there will be symmetry inside it, and many edges of a sheet of paper will have a huge number of contacts, adhesions to 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 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, then it deforms and many connections-contacts are broken. Therefore, symmetry is a series compaction.

So there is some primordial unmanifest chaos, which, under a certain creative influence, takes symmetrical forms?

Everything is mixed with you. Non-manifestation is the absence of movement. The movement itself is either chaos or symmetry, that is, when particles move chaotically, this is already manifestation. When the rays are reflected asymmetrically, this is also a manifestation. Just eat different types manifestation, and chaotic movement is no worse than symmetrical movement, it is just different. The universe contains different kinds building space, including what you call chaos.

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

it various forms creation 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. 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 precisely chaos. If we take any hierarchy, this is rigid symmetry and fractality.

It turns out that chaotic structuring has been introduced into the symmetry of space?

Or vice versa, symmetry has been introduced into the chaotic structurization.

If everything that I see around me is just an agreement between people how to see it, then why do I see space exactly symmetrically and not chaotically? If everything 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 forms of God are symmetrical. And it is the direction of these rays that you perceive. Look with light vision. When you look at a luminous object, then 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 vibrations and movement of light lines and points of other figures. When you look at what you think are formless objects and give them a shape in your imagination, as in the case of clouds, it means that either the object does not have rigid structural connections, that is, elements of chaos prevail, or you are simply not able to perceive such structuring. It's like with a snowball, inside which there are billions of snowy amazing symmetry, but the snowball itself is not very symmetrical.

I am 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 space itself?

Of course. Remember the example of your reflecting rays. The reflection of your ray depends on you. That is, from the properties of the ray 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 is precisely 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 length, your rays are combined, this is the reflection of the external world, that is your general picture of the world, this is the symmetry of space you see.

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

You place accents a little differently. You reflect all the time. It's just that some of you and the forms of God reflect more symmetrically, and some more chaotic. Therefore, those who reflect more chaotically, touch, intersect with their perception with those who also reflect more chaotically. This is the law of similarity, like not only attracts like. Like meets only 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 covers 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 the intersection of your perception with those who are also reflected more symmetrically.

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

Of course. 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 would seem to you to be something alien and negative and dark in assessments of duality.

That is, the fact that we are directed to the light, is this good 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 God's energy. And in fact, your universe is an attempt to balance both. Give chaos symmetry, and add chaotic components to symmetry. To get somewhere in between. Since a symmetric configuration is more stable, and a chaotic configuration is more multivariable.

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

The development of any form and any system has stages of direction. Symmetry gives way to chaos. Chaos replaces symmetry. Now you are at the stage of symmetric infusion of configurations, like the crystallization process of, say, salt, your space crystallizes into certain harmonious structures and new forms of connection, new configurations, new crystals are created. But further, in order to check the stability of these forms, a period of chaotic movement will come, like the effect of wind and rain on geological rocks and mountains. And then the mountains undergo changes. Is a mountain symmetrical 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 bad nor good. It's just a new combination of symmetry and chaos.

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

Oh this is very interest Ask and you still have a lot to understand on this topic. 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 to 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, since it will all early be somewhat 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 just that, copied the energy configuration of another object.

But that's not all. Knowing the configuration and symmetry of space, you can get from one point in space to any other. Now you do it chaotically by chance in your dreams and at very small, as it were, distances. But this is like a network of roads, a coordinate grid of the space of the universe. Knowing the coordinates, you seem to know the picture of the configuration, the picture of the symmetry of space, and by reproducing it with your consciousness, thus rearranging your configuration, you find yourself, combine with this space, as if you find yourself in a puzzle. If by your configuration you cannot fit into the picture like a puzzle, then you cannot perceive the boundaries of contact with other puzzles of the picture, do you understand? And there is much more to master in the symmetry of space. But this is too early.