« Symmetry" - word Greek origin. It means proportionality, presence of a certain order, patterns in the arrangement of parts.
Since ancient times, people have used symmetry in drawings, ornaments, and household items.
Symmetry is widespread in nature. It can be observed in the form of leaves and flowers of plants, in the arrangement various organs animals, shaped crystalline bodies, in a fluttering butterfly, a mysterious snowflake, a mosaic in a temple, a starfish.
Symmetry is widely used in practice, in construction and technology. This is strict symmetry in the form of ancient buildings, harmonious ancient Greek vases, the Kremlin building, cars, airplanes and much more. (slide 4) Examples of using symmetry are parquet and borders. (see hyperlink about the use of symmetry in borders and parquets) Let's look at a few examples where you can see symmetry in various subjects, using a slideshow (enable icon).
Definition: – is symmetry about a point.
Definition: Points A and B are symmetrical about some point O if point O is the midpoint of segment AB.
Definition: Point O is called the center of symmetry of the figure, and the figure is called centrally symmetrical.
Property: Figures that are symmetrical about a certain point are equal.
Examples:
Algorithm for constructing a centrally symmetrical figure
1. Construct a triangle A 1B 1 C 1, symmetrical to a triangle ABC, relative to the center (point) O. To do this, we connect points A, B, C with center O and continue these segments;
2. Measure the segments AO, BO, CO and lay off on the other side of point O, segments equal to them (AO=A 1 O 1, BO=B 1 O 1, CO=C 1 O 1);
3. Connect the resulting points with segments A 1 B 1; A 1 C 1; B1 C 1.
We got ∆A 1 B 1 C 1 symmetrical ∆ABC.
– this is symmetry about the drawn axis (straight line).
Definition: Points A and B are symmetrical about a certain line a if these points lie on a line perpendicular to this one and at the same distance.
Definition: An axis of symmetry is a straight line when bent along which the “halves” coincide, and a figure is called symmetrical about a certain axis.
Property: Two symmetrical figures are equal.
Examples:
Algorithm for constructing a figure symmetrical with respect to some straight line
Let's construct a triangle A1B1C1, symmetrical to triangle ABC with respect to straight line a.
For this:
1. Draw from the vertices triangle ABC straight lines perpendicular to straight line a and continue them further.
2. Measure the distances from the vertices of the triangle to the resulting points on the straight line and plot the same distances on the other side of the straight line.
3. Connect the resulting points with segments A1B1, B1C1, B1C1.
We obtained ∆A1B1C1 symmetrical ∆ABC.
Goals:
- educational:
- give an idea of symmetry;
- introduce the main types of symmetry on the plane and in space;
- develop strong construction skills symmetrical figures;
- expand ideas about famous figures, introducing properties associated with symmetry;
- show the possibilities of using symmetry when solving various tasks;
- consolidate acquired knowledge;
- general education:
- teach yourself how to prepare yourself for work;
- teach how to control yourself and your desk neighbor;
- teach to evaluate yourself and your desk neighbor;
- developing:
- intensify independent activity;
- develop cognitive activity;
- learn to summarize and systematize the information received;
- educational:
- develop a “shoulder sense” in students;
- cultivate communication skills;
- instill a culture of communication.
DURING THE CLASSES
In front of each person are scissors and a sheet of paper.
Exercise 1(3 min).
- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.
Question: What function does this line serve?
Suggested answer: This line divides the figure in half.
Question: How are all the points of the figure located on the two resulting halves?
Suggested answer: All points of the halves are on equal distance from the fold line and at the same level.
– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.
Task 2 (2 minutes).
– Cut out a snowflake, find the axis of symmetry, characterize it.
Task 3 (5 minutes).
– Draw a circle in your notebook.
Question: Determine how the axis of symmetry goes?
Suggested answer: Differently.
Question: So how many axes of symmetry does a circle have?
Suggested answer: A lot of.
– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)
Question: What other figures have more than one axis of symmetry?
Suggested answer: Square, rectangle, isosceles and equilateral triangles.
– Let’s consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?
I distribute halves of plasticine figures to students.
Task 4 (3 min).
– Using the information received, complete the missing part of the figure.
Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.
A line (closed, open, with self-intersection, without self-intersection) is laid out from a lace of the same color on the desktop.
Task 5 (group work 5 minutes).
– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.
The correctness of the work performed is determined by the students themselves.
Elements of drawings are presented to students
Task 6 (2 minutes).
– Find the symmetrical parts of these drawings.
To consolidate the material covered, I suggest next tasks provided for 15 minutes:
Name them all equal elements triangle KOR and COM. What type of triangles are these?
2. Draw several isosceles triangles in your notebook with common ground equal to 6 cm.
3. Draw a segment AB. Construct a line segment AB perpendicular and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the straight line AB.
– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?
Suggested answer: wings of butterflies, beetles, tree leaves...
– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.
That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.
Homework:
1. Come up with your own ornament, draw it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, note where elements of symmetry are present.
(means “proportionality”) - the property of geometric objects to be combined with themselves under certain transformations. By “symmetry” we mean any regularity in internal structure bodies or figures.
Central symmetry— symmetry about a point.
relative to the point O, if for each point of a figure a point symmetrical to it relative to point O also belongs to this figure. Point O is called the center of symmetry of the figure.
IN one-dimensional space (on a straight line) central symmetry is mirror symmetry.
On a plane (in 2-dimensional space) symmetry with center A is a rotation of 180 degrees with center A. Central symmetry on a plane, like rotation, preserves orientation.
Central symmetry in three-dimensional space is also called spherical symmetry. It can be represented as a composition of reflection relative to a plane passing through the center of symmetry, with a rotation of 180° relative to a straight line passing through the center of symmetry and perpendicular to the above-mentioned plane of reflection.
IN 4-dimensional space, central symmetry can be represented as a composition of two 180° rotations around two mutually perpendicular planes, passing through the center of symmetry.
Axial symmetry- symmetry relative to a straight line.
The figure is called symmetrical relatively straight a, if for each point of a figure a point symmetrical to it relative to the line a also belongs to this figure. Straight line a is called the axis of symmetry of the figure.
Axial symmetry has two definitions:
- Reflective symmetry.
In mathematics, axial symmetry is a type of motion (mirror reflection) in which the set of fixed points is a straight line, called the axis of symmetry. For example, flat figure A rectangle in space is asymmetrical and has 3 axes of symmetry, unless it is a square.
- Rotational symmetry.
IN natural sciences By axial symmetry we mean rotational symmetry, relative to rotations around a straight line. In this case, bodies are called axisymmetric if they transform into themselves at any rotation around this straight line. In this case, the rectangle will not be an axisymmetric body, but the cone will be.
Images on a plane of many objects in the world around us have an axis of symmetry or a center of symmetry. Many tree leaves and flower petals are symmetrical about the average stem.
We often encounter symmetry in art, architecture, technology, and everyday life. The facades of many buildings have axial symmetry. In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center. Many parts of mechanisms, such as gears, are symmetrical.
Axial symmetry. With axial symmetry, each point of the figure goes to a point that is symmetrical to it relative to a fixed straight line.
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“Point of symmetry” - Central symmetry. A a A1. Axial and central symmetry. Point C is called the center of symmetry. Symmetry in everyday life. A circular cone has axial symmetry; the axis of symmetry is the axis of the cone. Figures that have more than two axes of symmetry. A parallelogram has only central symmetry.
“Mathematical symmetry” - What is symmetry? Physical symmetry. Symmetry in biology. History of symmetry. However, complex molecules, as a rule, there is no symmetry. Palindromes. Symmetry. In x and m and i. HAS A LOT IN COMMON WITH PROGRESSAL SYMMETRY IN MATHEMATICS. But actually, how would we live without symmetry? Axial symmetry.
“Ornament” - b) On the strip. Parallel translation Central symmetry Axial symmetry Rotation. Linear (location options): Creating a pattern using central symmetry and parallel transfer. Planar. One of the varieties of ornament is a mesh ornament. Transformations used to create an ornament:
"Symmetry in nature" - One of the main properties geometric shapes is symmetry. The topic was not chosen by chance, because in next year We have to start studying a new subject - geometry. The phenomenon of symmetry in living nature was noticed back in Ancient Greece. We study at school scientific society because we love to learn something new and unknown.
“Movement in Geometry” - Mathematics is beautiful and harmonious! Give examples of movement. Movement in geometry. What is movement? What sciences does motion apply to? How movement is used in various fields human activity? A group of theorists. The concept of movement Axial symmetry Central symmetry. Can we see movement in nature?
“Symmetry in art” - Levitan. RAPHAEL. II.1. Proportion in architecture. Rhythm is one of the main elements of expressiveness of a melody. R. Descartes. Ship Grove. A.V. Voloshinov. Velazquez "Surrender of Breda" Externally, harmony can manifest itself in melody, rhythm, symmetry, proportionality. II.4.Proportion in literature.
There are a total of 32 presentations in the topic
Axial symmetry and the concept of perfection
Axial symmetry is inherent in all forms in nature and is one of fundamental principles beauty. Since ancient times, man has tried
to comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians of Ancient Greece. And the word “symmetry” itself was invented by them. It denotes proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. Indeed, those phenomena and forms that are proportional and complete “please the eye.” We call them correct.
Axial symmetry as a concept
Symmetry in the world of living beings is manifested in the regular arrangement of identical parts of the body relative to the center or axis. More often in
Axial symmetry occurs in nature. It not only determines general structure organism, but also the possibilities of its subsequent development. Geometric shapes and the proportions of living beings are formed by “axial symmetry”. Its definition is formulated in the following way: this is the property of objects to be combined when various transformations. The ancients believed that the principle of symmetry in the most in full has a sphere. They considered this form harmonious and perfect.
Axial symmetry in living nature
If you look at any Living being, the symmetry of the body’s structure immediately catches the eye. Human: two arms, two legs, two eyes, two ears and so on. Each animal species has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to general laws world order, because nothing in the Universe has a purely aesthetic, decorative purpose. Availability various forms also due to natural necessity.
Axial symmetry in inanimate nature
In the world, we are surrounded everywhere by such phenomena and objects as: typhoon, rainbow, drop, leaves, flowers, etc. Their mirror, radial, central, axial symmetry is obvious. It is largely due to the phenomenon of gravity. Often the concept of symmetry refers to the regularity of changes in certain phenomena: day and night, winter, spring, summer and autumn, and so on. In practice, this property exists wherever order is observed. And the laws of nature themselves - biological, chemical, genetic, astronomical - are subject to the principles of symmetry common to us all, since they have an enviable systematicity. Thus, balance, identity as a principle has a universal scope. Axial symmetry in nature is one of the “cornerstone” laws on which the universe as a whole is based.