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.
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.
Picture 35 from the presentation “Ornament” for geometry lessons on the topic “Symmetry”Dimensions: 360 x 260 pixels, format: jpg. To download a picture for free geometry lesson, right-click on the image and click “Save Image As...”. To display pictures in the lesson, you can also download the entire presentation “Ornament.ppt” with all the pictures in a zip archive for free. The archive size is 3324 KB.
Download presentationSymmetry
“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 of 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
MBOU "Tyukhtetskaya Secondary" comprehensive school No. 1"
Scientific association of students “We want to learn actively”
physico-mathematical and technical direction
Arvinti Tatyana,
Lozhkina Maria,
MBOU "TSOSH No. 1"
5 "A" class
MBOU "TSOSH No. 1"
mathematic teacher
Introduction………………………………………………………………………………...3
I. 1. Symmetry. Types of symmetry..…………………………………………......4
I. 2. Symmetry around us………………………………………………………....6
I. 3. Axial and centrally symmetrical ornaments ….…………………………… 7
II. Symmetry in needlework
II. 1. Symmetry in knitting………………………………………………………...10
II. 2. Symmetry in origami…..……………………………………………………11
II. 3. Symmetry in beading…………………………………………………………….12
II. 4. Symmetry in embroidery………………………………………………………13
II. 5. Symmetry in crafts made from matches………………………………………………………...14
II. 6. Symmetry in Macrame weaving……………………………………………………….15
Conclusion……………………………………………………………………………….16
Bibliography………………………………………………………..17
Introduction
One of the fundamental concepts of science, which, along with the concept of “harmony”, relates to almost all structures of nature, science and art, is “symmetry”.
The outstanding mathematician Hermann Weyl highly appreciated the role of symmetry in modern science:
“Symmetry, no matter how broadly or narrowly we understand the word, is an idea with the help of which man has tried to explain and create order, beauty and perfection.”
We all admire the beauty of geometric shapes and their combination, looking at pillows, knitted napkins, and embroidered clothes.
Many centuries different peoples wonderful views were created decoratively - applied arts. Many people believe that mathematics is not interesting and consists only of formulas, problems, solutions and equations. We want to show with our work that mathematics is a diverse science, and the main objective– to show that mathematics is a very amazing and unusual subject for study, closely connected with human life.
This work examines handicraft items for their symmetry.
The types of needlework we are considering are closely related to mathematics, since the works use various geometric figures that are subject to mathematical transformations. In this regard, the following were studied mathematical concepts like symmetry, types of symmetry.
Purpose of the study: studying information about symmetry, searching for symmetrical handicraft items.
Research objectives:
· Theoretical: study the concepts of symmetry and its types.
· Practical: find symmetrical crafts, determine the type of symmetry.
Symmetry. Types of symmetry
Symmetry(means "proportionality") - the property of geometric objects to combine with themselves under certain transformations. By symmetry we mean any regularity in internal structure bodies or figures.
Symmetry about a point is central symmetry, and symmetry about a line is axial symmetry.
Symmetry about a point (central symmetry) assumes that there is something on both sides of the point at equal distances, for example other points or locus points (straight lines, curved lines, geometric shapes). If you connect a straight line symmetrical points(dots geometric figure) through a point of symmetry, then the symmetric points will lie at the ends of the line, and the point of symmetry will be its middle. If you fix the symmetry point and rotate the straight line, then the symmetrical points will describe curves, each point of which will also be symmetrical to the point of the other curved line.
A rotation around a given point O is a movement in which each ray emanating from this point rotates through the same angle in the same direction.
Symmetry relative to a straight line (axis of symmetry) assumes that along a perpendicular drawn through each point of the axis of symmetry, two symmetrical points are located at the same distance from it. The same geometric figures can be located relative to the axis of symmetry (straight line) as relative to the point of symmetry. An example would be a sheet of notebook that is folded in half if a straight line is drawn along the fold line (axis of symmetry). Each point on one half of the sheet will have a symmetrical point on the second half of the sheet if they are located at the same distance from the fold line and perpendicular to the axis. The axis of symmetry serves as a perpendicular to the midpoints of the horizontal lines bounding the sheet. Symmetrical points are located at the same distance from the axial line - perpendicular to the straight lines connecting these points. Consequently, all points of the perpendicular (axis of symmetry) drawn through the middle of the segment are equidistant from its ends; or any point perpendicular (axis of symmetry) to the middle of a segment and equidistant from the ends of this segment.
Koll" href="/text/category/koll/" rel="bookmark">Hermitage collections special attention used gold jewelry of the ancient Scythians. Extraordinarily thin artwork golden wreaths, tiaras, wood and decorated with precious red-violet garnets.
One of the most obvious uses of the laws of symmetry in life is in architectural structures. This is what we see most often. In architecture, axes of symmetry are used as means of expressing architectural design.
Another example of a person using symmetry in his practice is technology. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind that has a center of symmetry is the wheel; the propeller and other technical means also have a center of symmetry.
Axial and centrally symmetrical ornaments
Compositions built on the principle of a carpet ornament can have symmetrical construction. The drawing in them is organized according to the principle of symmetry relative to one or two axes of symmetry. Carpet patterns often contain a combination of several types of symmetry - axial and central.
Figure 1 shows a diagram for marking the plane for a carpet ornament, the composition of which will be built along the axes of symmetry. On the plane along the perimeter, the location and size of the border are determined. The central field will be occupied by the main ornament.
Options for various compositional solutions of the plane are shown in Figure 1 b-d. In Figure 1 b, the composition is built in the central part of the field. Its outline may vary depending on the shape of the field itself. If the plane has the shape of an elongated rectangle, the composition is given the outline of an elongated rhombus or oval. Square shape the fields would be better supported by a composition outlined by a circle or an equilateral rhombus.
Figure 1. Axial symmetry.
Figure 1c shows the composition diagram discussed in the previous example, which is supplemented with small corner elements. In Figure 1d, the composition diagram is built along the horizontal axis. It includes a central element with two side ones. The considered schemes can serve as the basis for composing compositions that have two axes of symmetry.
Such compositions are perceived equally by viewers from all sides; they, as a rule, do not have a pronounced top and bottom.
Carpet ornaments can contain in their central part compositions that have one axis of symmetry (Figure 1e). Such compositions have a pronounced orientation; they have a top and a bottom.
The central part can not only be made in the form of an abstract ornament, but also have a theme.
All examples of the development of ornaments and compositions based on them discussed above were related to rectangular planes. Rectangular shape surfaces are a common, but not the only type of surface.
Boxes, trays, plates can have surfaces in the shape of a circle or an oval. One of the options for their decor can be centrally symmetrical ornaments. The basis for creating such an ornament is the center of symmetry, through which can pass infinite set symmetry axes (Figure 2a).
Let's look at an example of developing an ornament, bounded by a circle and having central symmetry (Figure 2). The structure of the ornament is radial. Its main elements are located along the radius lines of the circle. The border of the ornament is decorated with a border.
Figure 2. Centrally symmetrical ornaments.
II. Symmetry in needlework
II. 1. Symmetry in knitting
We found knitted crafts with central symmetry:
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Homothety and similarity.Homothety is a transformation in which each point M (plane or space) is assigned to a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than ABOUT. Fixed point ABOUT called the center of homothety. Attitude OM": OM considered positive if M" and M lie on one side of ABOUT, negative - by different sides. Number X called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety turns into a symmetry transformation about a point ABOUT. With homothety, a straight line goes into a straight line, the parallelism of straight lines and planes is preserved, angles (linear and dihedral) are preserved, each figure goes into it similar (Fig. 5.17).
The converse is also true. A homothety can be defined as an affine transformation in which the lines connecting corresponding points, pass through one point - the homothety center. Homothety is used to enlarge images (projection lamp, cinema).
Central and mirror symmetries.Symmetry (in in a broad sense) - a property of a geometric figure F, characterizing a certain correctness of its shape, its invariability under the action of movements and reflections. A figure Φ has symmetry (symmetrical) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Φ with itself is the group of this figure. So, flat figure(Fig. 5.18) with a dot M, transforming-
looking into yourself in the mirror reflection, symmetrical about the straight axis AB. Here the symmetry group consists of two elements - a point M converted to M".
If the figure Φ on the plane is such that rotations relative to any point ABOUT to an angle of 360°/n, where n > 2 is an integer, translate it into itself, then the figure Ф has nth-order symmetry with respect to the point ABOUT - center of symmetry. An example of such figures is regular polygons, for example, star-shaped (Fig. 5.19), which has eighth-order symmetry relative to its center. The symmetry group here is the so-called nth order cyclic group. The circle has symmetry of infinite order (since it is compatible with itself by rotating through any angle).
The simplest types of spatial symmetry are central symmetry (inversion). In this case, relative to the point ABOUT the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, i.e. point ABOUT - the middle of the segment connecting the symmetrical points F. So, for a cube (Fig. 5.20) the point ABOUT is the center of symmetry. Points M and M" cube