We get used to the concept of symmetry from childhood. We know that a butterfly is symmetrical: its right and left wings are the same; a symmetrical wheel whose sectors are identical; symmetrical patterns of ornaments, stars of snowflakes.
A truly vast literature is devoted to the problem of symmetry. From textbooks and scientific monographs to works that pay attention not so much to drawings and formulas, but to artistic images.
The very term “symmetry” in Greek means “proportionality,” which ancient philosophers understood as a special case of harmony - the coordination of parts within the whole. Since ancient times, many peoples have had the idea of symmetry in the broad sense - as the equivalent of balance and harmony.
Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society. We meet her everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception. Truly symmetrical objects surround us literally on all sides; we are dealing with symmetry wherever any order is observed. It turns out that symmetry is balance, orderliness, beauty, perfection. It is diverse, omnipresent. She creates beauty and harmony. Symmetry literally permeates the entire world around us, which is why the topic I have chosen will always be relevant.
Symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability not only of the object itself, but also of any of its properties in relation to transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc. In this regard, different types of symmetry are distinguished. Let's look at all types in more detail.
AXIAL SYMMETRY.
Symmetry about a straight line is called axial symmetry (mirror reflection about a straight line).
If point A lies on the l axis, then it is symmetrical to itself, i.e. A coincides with A1.
In particular, if, when transforming symmetry with respect to the l axis, the figure F transforms into itself, then it is called symmetric with respect to the l axis, and the l axis is called its symmetry axis.
CENTRAL SYMMETRY.
A figure is called centrally symmetric if there is a point relative to which each point of the figure is symmetrical to some point of the same figure. Namely: movement that changes directions to opposite ones is central symmetry.
Point O is called the center of symmetry and is motionless. This transformation has no other fixed points. Examples of figures that have a center of symmetry are a parallelogram, a circle, etc.
The familiar concepts of rotation and parallel translation are used in the definition of so-called translational symmetry. Let's look at translation symmetry in more detail.
1. TURN
A transformation in which each point A of a figure (body) is rotated by the same angle α around a given center O is called rotation or rotation of the plane. Point O is called the center of rotation, and angle α is called the angle of rotation. Point O is a fixed point of this transformation.
The rotational symmetry of the circular cylinder is interesting. It has an infinite number of 2nd order rotary axes and one infinitely high order rotary axis.
2. PARALLEL TRANSFER
A transformation in which each point of a figure (body) moves in the same direction by the same distance is called parallel translation.
To specify a parallel translation transformation, it is enough to specify the vector a.
3. SLIDING SYMMETRY
Sliding symmetry is a transformation in which axial symmetry and parallel translation are performed sequentially. Sliding symmetry is an isometry of the Euclidean plane. Gliding symmetry is a composition of symmetry with respect to some line l and translation to a vector parallel to l (this vector may also be zero).
Gliding symmetry can be represented as a composition of 3 axial symmetries (Chales' theorem).
MIRROR SYMMETRY
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 the real hand.
Immanuel Kant.
If a transformation of symmetry relative to a plane transforms a figure (body) into itself, then the figure is called symmetrical relative to the plane, and this plane is called the plane of symmetry of this figure. This symmetry is called mirror symmetry. As the name itself suggests, mirror symmetry connects an object and its reflection in a plane mirror. Two symmetrical bodies cannot be “nested into each other”, since in comparison with the object itself, its mirror-mirror double turns out to be turned out along the direction perpendicular to the plane of the mirror.
Symmetrical figures, for all their similarities, differ significantly from each other. The double observed in the mirror is not an exact copy of the object itself. The mirror does not simply copy the object, but swaps (represents) the front and rear parts of the object in relation to the mirror. For example, if your mole is on your right cheek, then your looking-glass double’s is on your left. Hold a book up to the mirror and you will see that the letters seem to be turned inside out. Everything in the mirror is rearranged from right to left.
Bodies are called mirror-equal bodies if, with proper displacement, they can form two halves of a mirror-symmetrical body.
2. 2 Symmetry in nature
A figure has symmetry if there is a movement (non-identical transformation) that transforms it into itself. For example, a figure has rotational symmetry if it is translated into itself by some rotation. But in nature, with the help of mathematics, beauty is not created, as in technology and art, but is only recorded and expressed. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.
The structure of any living form is based on the principle of symmetry. From direct observation we can deduce the laws of geometry and feel their incomparable perfection. This order, which is a natural necessity, since nothing in nature serves purely decorative purposes, helps us find the general harmony on which the entire universe is based.
We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification.
The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, and particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. We are talking not only about physical laws, but also others, for example, biological ones.
Speaking about the role of symmetry in the process of scientific knowledge, we should especially highlight the use of the method of analogies. According to the French mathematician D. Polya, “there are, perhaps, no discoveries either in elementary or higher mathematics, or, perhaps, in any other field that could be made without analogies.” Most of these analogies are based on common roots, general patterns that manifest themselves in the same way at different levels of the hierarchy.
So, in the modern understanding, symmetry is a general scientific philosophical category that characterizes the structure of the organization of systems. The most important property of symmetry is the preservation (invariance) of certain features (geometric, physical, biological, etc.) in relation to well-defined transformations. The mathematical apparatus for studying symmetry today is the theory of groups and the theory of invariants.
Symmetry in the plant world
The specific structure of plants is determined by the characteristics of the habitat to which they adapt. Any tree has a base and a top, a “top” and a “bottom” that perform different functions. The significance of the difference between the upper and lower parts, as well as the direction of gravity, determine the vertical orientation of the rotary axis of the “wood cone” and the planes of symmetry. A tree, with the help of its root system, absorbs moisture and nutrients from the soil, that is, from below, and the remaining vital functions are performed by the crown, that is, above. At the same time, directions in a plane perpendicular to the vertical are virtually indistinguishable for a tree; in all these directions, air, light, and moisture enter the tree equally.
The tree has a vertical rotary axis (cone axis) and vertical planes of symmetry.
When we want to draw a leaf of a plant or a butterfly, we have to take into account their axial symmetry. The midrib for the leaf serves as an axis of symmetry. Leaves, branches, flowers, and fruits have pronounced symmetry. The leaves are characterized by mirror symmetry. The same symmetry is also found in flowers, but in them mirror symmetry often appears in combination with rotational symmetry. There are also frequent cases of figurative symmetry (acacia branches, rowan trees).
In the diverse world of colors, there are rotary axes of different orders. However, the most common is 5th order rotational symmetry. This symmetry is found in many wildflowers (bell, forget-me-not, geranium, carnation, St. John's wort, cinquefoil), in the flowers of fruit trees (cherry, apple, pear, tangerine, etc.), in the flowers of fruit and berry plants (strawberries, raspberries, viburnum , bird cherry, rowan, rose hip, hawthorn), etc.
Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of instrument of the struggle for existence, “insurance against petrification, crystallization, the first step of which would be their capture by the grid.” Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are represented very widely in it.
In his book “This Right, Left World,” M. Gardner writes: “On Earth, life originated in spherically symmetrical forms, and then began to develop along two main lines: the world of plants with cone symmetry was formed, and the world of animals with bilateral symmetry.”
In nature, there are bodies that have helical symmetry, that is, alignment with their original position after rotation by an angle around an axis, with an additional shift along the same axis.
If is a rational number, then the rotary axis also turns out to be the translation axis.
The leaves on the stem are not arranged in a straight line, but surround the branch in a spiral. The sum of all previous steps of the spiral, starting from the top, is equal to the value of the subsequent step A+B=C, B+C=D, etc.
Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Arranging in a spiral along the stem, the leaves seem to spread out in all directions and do not block each other from the light, which is extremely necessary for plant life. This interesting botanical phenomenon is called phyllotaxis (literally “leaf arrangement”).
Another manifestation of phyllotaxis is the structure of the inflorescence of a sunflower or the scales of a fir cone, in which the scales are arranged in the form of spirals and helical lines. This arrangement is especially clear in the pineapple, which has more or less hexagonal cells that form rows running in different directions.
Symmetry in the animal world
The significance of the form of symmetry for an animal is easy to understand if it is connected with the way of life and environmental conditions. Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line.
Rotational symmetry of the 5th order is also found in the animal world. This is a symmetry in which an object aligns with itself when rotated around a rotary axis 5 times. Examples include the starfish and sea urchin shell. The entire skin of starfish is as if encrusted with small plates of calcium carbonate; needles extend from some of the plates, some of which are movable. An ordinary starfish has 5 planes of symmetry and 1 axis of rotation of the 5th order (this is the highest symmetry among animals). Her ancestors appear to have had lower symmetry. This is evidenced, in particular, by the structure of the star larvae: they, like most living beings, including humans, have only one plane of symmetry. Starfish do not have a horizontal plane of symmetry: they have a “top” and a “bottom.” Sea urchins are like living pincushions; their spherical body bears long and movable needles. In these animals, the calcareous plates of the skin merged and formed a spherical carapace. There is a mouth in the center of the lower surface. The ambulacral legs (water-vascular system) are collected in 5 stripes on the surface of the shell.
However, unlike the plant world, rotational symmetry is rarely observed in the animal world.
Insects, fish, eggs, and animals are characterized by a difference between the directions “forward” and “backward” that is incompatible with rotational symmetry.
The direction of movement is a fundamentally selected direction, with respect to which there is no symmetry in any insect, any bird or fish, any animal. In this direction the animal rushes for food, in the same direction it escapes from its pursuers.
In addition to the direction of movement, the symmetry of living beings is determined by another direction - the direction of gravity. Both directions are significant; they define the plane of symmetry of an animal being.
Bilateral (mirror) symmetry is the characteristic symmetry of all representatives of the animal world. This symmetry is clearly visible in the butterfly. The symmetry of the left and right wings appears here with almost mathematical rigor.
We can say that every animal (as well as insects, fish, birds) consists of two enantiomorphs - the right and left halves. Enantiomorphs are also paired parts, one of which falls into the right and the other into the left half of the animal’s body. Thus, enantiomorphs are the right and left ear, right and left eye, right and left horn, etc.
Simplification of living conditions can lead to a violation of bilateral symmetry, and animals from being bilaterally symmetrical become radially symmetrical. This applies to echinoderms (starfish, sea urchins, crinoids). All marine animals have radial symmetry, in which parts of the body radiate away from a central axis, like the spokes of a wheel. The degree of activity of animals correlates with their type of symmetry. Radially symmetrical echinoderms are usually poorly mobile, move slowly, or are attached to the seabed. The body of a starfish consists of a central disk and 5-20 or more rays radiating from it. In mathematical language, this symmetry is called rotational symmetry.
Let us finally note the mirror symmetry of the human body (we are talking about the appearance and structure of the skeleton). This symmetry has always been and is the main source of our aesthetic admiration for the well-proportioned human body. Let’s not figure out for now whether an absolutely symmetrical person actually exists. Everyone, of course, will have a mole, a strand of hair or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least for most people. Yet these are only minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same.
Everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. It is the issues of symmetry and mirror reflection that are given attention here.
Many artists paid close attention to the symmetry and proportions of the human body, at least as long as they were guided by the desire to follow nature as closely as possible in their works.
In modern schools of painting, the vertical size of the head is most often taken as a single measure. With a certain assumption, we can assume that the length of the body is eight times the size of the head. The size of the head is proportional not only to the length of the body, but also to the size of other parts of the body. All people are built on this principle, which is why we are, in general, similar to each other. However, our proportions are only approximately consistent, and therefore people are only similar, but not the same. In any case, we are all symmetrical! In addition, some artists especially emphasize this symmetry in their works.
Our own mirror symmetry is very convenient for us, it allows us to move straight and turn right and left with equal ease. Mirror symmetry is equally convenient for birds, fish and other actively moving creatures.
Bilateral symmetry means that one side of an animal's body is a mirror image of the other side. This type of organization is characteristic of most invertebrates, especially annelids and arthropods - crustaceans, arachnids, insects, butterflies; for vertebrates - fish, birds, mammals. Bilateral symmetry first appears in flatworms, in which the anterior and posterior ends of the body differ from each other.
Let's consider another type of symmetry that is found in the animal world. This is helical or spiral symmetry. Helical symmetry is symmetry with respect to the combination of two transformations - rotation and translation along the axis of rotation, i.e. there is movement along the axis of the screw and around the axis of the screw.
Examples of natural propellers are: tusk of a narwhal (a small cetacean that lives in the northern seas) - left propeller; snail shell – right screw; The horns of the Pamir ram are enantiomorphs (one horn is twisted in a left-handed spiral, and the other in a right-handed spiral). Spiral symmetry is not ideal, for example, the shell of mollusks narrows or widens at the end. Although external helical symmetry is rare in multicellular animals, many important molecules from which living organisms are built - proteins, deoxyribonucleic acids - DNA have a helical structure.
Symmetry in inanimate nature
Crystal symmetry is the property of crystals to align with themselves in various positions by rotation, reflection, parallel translation, or part or combination of these operations. The symmetry of the external shape (cut) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical properties of the crystal.
Let's take a closer look at the multifaceted shapes of crystals. First of all, it is clear that crystals of different substances differ from each other in their shapes. Rock salt is always cubes; rock crystal - always hexagonal prisms, sometimes with heads in the form of trihedral or hexagonal pyramids; diamond - most often regular octahedrons (octahedrons); ice is hexagonal prisms, very similar to rock crystal, and snowflakes are always six-pointed stars. What catches your eye when you look at crystals? First of all, their symmetry.
Many people think that crystals are beautiful, rare stones. They come in different colors, are usually transparent and, best of all, have a beautiful, regular shape. Most often, crystals are polyhedra, their sides (faces) are perfectly flat, and their edges are strictly straight. They delight the eye with the wonderful play of light in their edges and the amazing correctness of their structure.
However, crystals are not museum rarities at all. Crystals surround us everywhere. The solids from which we build houses and machines, the substances that we use in everyday life - almost all of them belong to crystals. Why don't we see this? The fact is that in nature one rarely comes across bodies in the form of separate single crystals (or, as they say, single crystals). Most often, the substance is found in the form of tightly adhered crystalline grains of a very small size - less than a thousandth of a millimeter. This structure can only be seen through a microscope.
Bodies consisting of crystalline grains are called finely crystalline, or polycrystalline (“poly” - in Greek “many”).
Of course, finely crystalline bodies should also be classified as crystals. Then it turns out that almost all the solid bodies around us are crystals. Sand and granite, copper and iron, paints - all these are crystals.
There are exceptions; glass and plastics do not consist of crystals. Such solids are called amorphous.
Studying crystals means studying almost all the bodies around us. It's clear how important this is.
Single crystals are immediately recognizable by their regular shape. Flat faces and straight edges are a characteristic property of the crystal; the correctness of the form is undoubtedly related to the correctness of the internal structure of the crystal. If a crystal is especially elongated in a certain direction, it means that the structure of the crystal in that direction is somehow special.
There is a center of symmetry in a cube of rock salt, in the octahedron of a diamond, and in the star of a snowflake. But in a quartz crystal there is no center of symmetry.
The most accurate symmetry is achieved in the world of crystals, but even here it is not ideal: cracks and scratches invisible to the eye always make equal faces slightly different from each other.
All crystals are symmetrical. This means that in each crystalline polyhedron one can find planes of symmetry, axes of symmetry, a center of symmetry or other symmetry elements so that identical parts of the polyhedron are aligned with each other.
All elements of symmetry repeat the same parts of the figure, all give it symmetrical beauty and completeness, but the center of symmetry is the most interesting. Not only the shape, but also many physical properties of the crystal can depend on whether a crystal has a center of symmetry or not.
Honeycombs are a real design masterpiece. They consist of a number of hexagonal cells. This is the densest packaging, allowing the most advantageous placement of the larva in the cell and, with the maximum possible volume, the most economical use of the building material - wax.
III Conclusion
Symmetry permeates literally everything around, capturing seemingly completely unexpected areas and objects. It, manifesting itself in the most diverse objects of the material world, undoubtedly reflects its most general, most fundamental properties. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music.
We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification. Therefore, the study of the symmetry of various natural objects and the comparison of its results is a convenient and reliable tool for understanding the basic laws of the existence of matter.
The laws of nature that govern the inexhaustible picture of phenomena in their diversity, in turn, are subject to the principles of symmetry. There are many types of symmetry, both in the plant and animal world, but with all the diversity of living organisms, the principle of symmetry always operates, and this fact once again emphasizes the harmony of our world. Symmetry underlies things and phenomena, expressing something common, characteristic of different objects, while asymmetry is associated with the individual embodiment of this common thing in a specific object.
So, on the plane we have four types of movements that transform figure F into an equal figure F1:
1) parallel transfer;
2) axial symmetry (reflection from a straight line);
3) rotation around a point (Partial case - central symmetry);
4) “sliding” reflection.
In space, mirror symmetry is added to the above types of symmetry.
I believe that the goal set in the abstract has been achieved. When writing my essay, the greatest difficulty for me was drawing my own conclusions. I think that my work will help schoolchildren expand their understanding of symmetry. I hope that my essay will be included in the methodological fund of the mathematics classroom.
Slide 2
Definition of symmetry; Central symmetry; Axial symmetry; Symmetry relative to the plane; Rotation symmetry; Mirror symmetry; Symmetry of similarity; Plant symmetry; Animal symmetry; Symmetry in architecture; Is man a symmetrical creature? Symmetry of words and numbers;
Slide 3
Definition of symmetry
SYMMETRY - proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane. (Ozhegov's Explanatory Dictionary) So, a geometric object is considered symmetrical if something can be done with it, after which it will remain unchanged.
Slide 4
Central symmetry
A figure is said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure. Point O is called the center of symmetry of the figure.
Slide 5
Examples of figures that have central symmetry are the circle and parallelogram. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals. Any straight line also has central symmetry (any point on the straight line is its center of symmetry). The graph of an odd function is symmetrical about the origin. An example of a figure that does not have a center of symmetry is an arbitrary triangle.
Slide 6
Axial symmetry
A figure is called symmetrical with respect to line a if for each point of the figure a point symmetrical with respect to line a also belongs to this figure. The straight line is called the axis of symmetry of the figure.
Slide 7
An undeveloped angle has one axis of symmetry - the straight line on which the angle's bisector is located. An isosceles triangle also has one axis of symmetry, and an equilateral triangle has three axes of symmetry. A rectangle and a rhombus, which are not squares, each have two axes of symmetry, and a square has four axes of symmetry. The circle has an infinite number of them. The graph of an even function when constructed is symmetrical about the ordinate axis. There are figures that do not have a single axis of symmetry. Such figures include a parallelogram, different from a rectangle, and a scalene triangle.
Slide 8
Symmetry relative to the plane
Points A and A1 are called symmetrical relative to plane a (plane of symmetry) if plane a passes through the middle of segment AA1 and is perpendicular to this segment. Each point of the plane is considered symmetrical to itself. Two figures are called symmetrical relative to the plane (or mirror-symmetrical relative) if they consist of pairwise symmetrical points. This means that for each point of one figure, a point symmetrical (relatively) to it lies in another figure.
Slide 9
Rotation symmetry
A body (or figure) has rotational symmetry if, when rotated through an angle of 360º/n, where n is an integer, near some straight line AB (symmetry axis), it is completely aligned with its original position. Radial symmetry is a form of symmetry that is preserved when an object is rotated around a specific point or line. Often this point coincides with the center of gravity of the object, that is, the point at which an infinite number of axes of symmetry intersect. Such objects can be a circle, a sphere, a cylinder or a cone.
Slide 10
Mirror symmetry
Mirror symmetry connects any object and its reflection in a flat mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror symmetrical figure (or body). Symmetrically mirrored figures, for all their similarities, differ significantly from each other. Two mirror-symmetrical flat figures can always be superimposed on each other. However, to do this it is necessary to remove one of them (or both) from their common plane.
Slide 11
Symmetry of similarity
Similarity symmetries are peculiar analogues of previous symmetries, with the only difference being that they are associated with a simultaneous decrease or increase in similar parts of the figure and the distances between them. The simplest example of such symmetry are nesting dolls. Sometimes figures can have different types of symmetry. For example, some letters have rotational and mirror symmetry: Ж, Н, М, О, А.
Slide 12
There are many other types of symmetries that are abstract in nature. For example: Commutative symmetry, which consists in the fact that if identical particles are swapped, then no changes occur; Gauge symmetries involve changes in scale. In inanimate nature, symmetry primarily arises in such a natural phenomenon as crystals, of which almost all solids are composed. It is this that determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake.
Slide 13
We encounter symmetry everywhere: in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music. The laws of nature are also subject to the principles of symmetry.
Slide 14
Plant symmetry
Many flowers have an interesting property: they can be rotated so that each petal takes the position of its neighbor, and the flower aligns with itself. Such a flower has an axis of symmetry. Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Arranging in a spiral along the stem, the leaves seem to spread out in all directions and do not block each other from the light, which is extremely necessary for plant life. Plant organs, for example, the stems of many cacti, also have bilateral symmetry. In botany, radially symmetrically constructed flowers are often found.
Slide 15
Animal symmetry
Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line. The main types of symmetry are radial (ray) - it is possessed by echinoderms, coelenterates, jellyfish, etc.; or bilateral (two-sided) - we can say that every animal (be it an insect, fish or bird) consists of two halves - right and left. Spherical symmetry occurs in radiolarians and sunfishes. Any plane drawn through the center divides the animal into equal halves.
Slide 16
Symmetry in architecture
The symmetry of a structure is associated with the organization of its functions. The projection of the plane of symmetry - the axis of the building - usually determines the location of the main entrance and the beginning of the main traffic flows. Each part in a symmetrical system exists as a double of its obligatory pair located on the other side of the axis, and thanks to this it can only be considered as a part of the whole. Mirror symmetry is the most common in architecture. The buildings of Ancient Egypt and the temples of ancient Greece, amphitheatres, baths, basilicas and triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous structures of modern architecture are subordinate to it.
Slide 17
To better reflect symmetry, accents are placed on buildings - especially significant elements (domes, spiers, tents, main entrances and staircases, balconies and bay windows). To design the decoration of architecture, an ornament is used - a rhythmically repeating pattern based on the symmetrical composition of its elements and expressed by line, color or relief. Historically, several types of ornaments have developed based on two sources - natural forms and geometric figures. But an architect is first and foremost an artist. And therefore, even the most “classical” styles more often used dissymmetry - a nuanced deviation from pure symmetry or asymmetry - a deliberately asymmetrical construction.
Slide 18
Is man a symmetrical creature?
No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same. But the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. The asymmetry of the face of the Venus de Milo statue is expressed by the displacement of the nose to the right of the midline, in a higher position of the left auricle and left orbit, and a shorter distance from the midline of the left orbit than the right. Proponents of symmetry believed that Venus's face would be much more beautiful if it were symmetrical.
Slide 19
Numerous measurements of facial parameters in men and women have shown that the right half, compared to the left, has more pronounced transverse dimensions, which gives the face rougher features inherent in the male sex. The left half of the face has more pronounced longitudinal dimensions, which gives it smooth lines and femininity. This fact explains the predominant desire of females to pose in front of artists with the left side of their faces, and males - with the right.
Slide 20
Symmetry of words and numbers
A palindrome (from the gr. Palindromos - running back) is an object in which the symmetry of its components is specified from beginning to end and from end to beginning. For example, a phrase or text. The direct text of a palindrome, read in accordance with the normal reading direction in a given script (usually from left to right), is called forward, the reverse is called rakoho or reverse (right to left). Some numbers also have symmetry. The path led to the left, to the port Lesha found a bug on a shelf Argentina beckons a black man 101 2002 6996
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"Mathematical symmetry" - Types of symmetry. Symmetry in mathematics. HAS A LOT IN COMMON WITH AXIAL SYMMETRY IN MATHEMATICS. In poetry, rhyme represents progressive symmetry. Symmetry in chemistry and physics. Physical symmetry. In x and m and i. Bilateral symmetry. The role of symmetry in the world. Spiral symmetry. Symmetry in chemistry.
“Ornament” - Types of ornament. Geometric. a) Inside the strip. 1 2 3. Creating an ornament using axial symmetry and parallel translation. 2011. Transformations used to create an ornament: Planar. c) On both sides of the strip. Turn.
“Movement in Geometry” - Movement in Geometry. What sciences does motion apply to? The concept of movement Axial symmetry Central symmetry. What shape does a segment, angle, etc. transform into when moving? Give examples of movement. What is movement? How is movement used in various areas of human activity? Mathematics is beautiful and harmonious!
“Symmetry in Nature” - We study in the school scientific society because we love to learn something new and unknown. In the 19th century, in Europe, isolated works appeared on the symmetry of plants. Symmetry in nature and in life. One of the main properties of geometric shapes is symmetry. The work was carried out by: Zhavoronkova Tanya Nikolaeva Lera Supervisor: Artemenko Svetlana Yuryevna.
“Symmetry around us” - Rotations (rotational). Central relative to a point. Rotations. Symmetry on a plane. Axial symmetry is relatively straight. Around us. Symmetry in space. Horizontal. Symmetry reigns supreme. Mirror. Two types of symmetry. All types of axial symmetry. The Greek word symmetry means “proportion”, “harmony”.
"Point of symmetry" - Examples of the above types of symmetry. Such figures include a parallelogram, different from a rectangle, and a scalene triangle. We encounter symmetry in nature, everyday life, architecture and technology. Symmetry in architecture. Symmetry in nature. Symmetry of plane figures. A rectangle and a rhombus, which are not squares, have two axes of symmetry.
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