Illusions of size perception. straight line AB
A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the scope of the task, only its location is important
The point is indicated by a number or a capital (capital) Latin letter. Several dots - with different numbers or different letters so that they can be distinguished
point A, point B, point C
A B Cpoint 1, point 2, point 3
1 2 3You can draw three dots “A” on a piece of paper and invite the child to draw a line through the two dots “A”. But how to understand through which ones? A A A
A line is a set of points. Only the length is measured. It has no width or thickness
Indicated by lowercase (small) in Latin letters
line a, line b, line c
a b cThe line may be
- closed if its beginning and end are at the same point,
- open if its beginning and end are not connected
closed lines
open lines
You left the apartment, bought bread at the store and returned back to the apartment. What line did you get? That's right, closed. You are back to your starting point. You left the apartment, bought bread at the store, went into the entrance and started talking with your neighbor. What line did you get? Open. You haven't returned to your starting point. You left the apartment and bought bread at the store. What line did you get? Open. You haven't returned to your starting point.- self-intersecting
- without self-intersections
self-intersecting lines
lines without self-intersections
- direct
- broken
- crooked
straight lines
broken lines
curved lines
A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions
Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions
Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line
straight line a
astraight line AB
B ADirect may be
- intersecting if they have a common point. Two lines can intersect only at one point.
- perpendicular if they intersect at right angles (90°).
- Parallel, if they do not intersect, do not have a common point.
parallel lines
intersecting lines
perpendicular lines
A ray is a part of a straight line that has a beginning but no end; it can be continued indefinitely in only one direction
The ray of light in the picture has its starting point as the sun.
Sun
A point divides a straight line into two parts - two rays A A
The beam is designated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters, where the first is the point from which the ray begins, and the second is the point lying on the ray
ray a
abeam AB
B AThe rays coincide if
- located on the same straight line
- start at one point
- directed in one direction
rays AB and AC coincide
rays CB and CA coincide
C B AA segment is a part of a line that is limited by two points, that is, it has both a beginning and an end, which means its length can be measured. The length of a segment is the distance between its starting and ending points
Through one point you can draw any number of lines, including straight lines
Through two points - an unlimited number of curves, but only one straight line
curved lines passing through two points
B Astraight line AB
B AA piece was “cut off” from the straight line and a segment remained. From the example above you can see that its length is the shortest distance between two points. ✂ B A ✂
A segment is denoted by two capital (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends
segment AB
B AProblem: where is the line, ray, segment, curve?
A broken line is a line consisting of consecutively connected segments not at an angle of 180°
A long segment was “broken” into several short ones
The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.
The vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.
A broken line is designated by listing all its vertices.
broken line ABCDE
vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E
broken link AB, broken link BC, broken link CD, broken link DE
link AB and link BC are adjacent
link BC and link CD are adjacent
link CD and link DE are adjacent
A B C D E 64 62 127 52The length of a broken line is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305
Task: which broken line is longer, A which has more vertices? The first line has all the links of the same length, namely 13 cm. The second line has all links of the same length, namely 49 cm. The third line has all the links of the same length, namely 41 cm.
A polygon is a closed polyline
The sides of the polygon (the expressions will help you remember: “go in all four directions”, “run towards the house”, “which side of the table will you sit on?”) are the links of a broken line. Adjacent sides polygon is adjacent links broken.
The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the endpoints of one side of the polygon.
A polygon is denoted by listing all its vertices.
closed polyline without self-intersection, ABCDEF
polygon ABCDEF
polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F
vertex A and vertex B are adjacent
vertex B and vertex C are adjacent
vertex C and vertex D are adjacent
vertex D and vertex E are adjacent
vertex E and vertex F are adjacent
vertex F and vertex A are adjacent
polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF
side AB and side BC are adjacent
side BC and side CD are adjacent
CD side and DE side are adjacent
side DE and side EF are adjacent
side EF and side FA are adjacent
A B C D E F 120 60 58 122 98 141The perimeter of a polygon is the length of the broken line: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599
A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, etc.
When solving problems, you can also use a paper prototype of a geoplan - an ordinary student notebook with a square grid pricked with an awl or a thin nail on all its sheets.
Segments
1. Construct two segments, each 5 dm long, on the geoplan so that they intersect at a point dividing them into four segments of length 1 dm, 2 dm, 3 dm, 4 dm.
2. On the fourth part of the geoplan (5x5 dm), place ten sections of length 1 dm, 1 dm, 1 dm, 2 dm, 2 dm, 3 dm, 3 dm, 4 dm, 4 dm and 5 dm so that no two of they had no common point.
3. Construct three segments with a common end so that the length of the first of them is 2 dm, the second is 3 dm, and the length of the third is greater than the length of the first, but less than the length of the second. Find two solutions.
4. Select a point and build on your geoplan the three smallest pairwise unequal segments with ends at this point.
5. Construct the shortest and longest segments of the geoplan so that their common point divides one of them into two parts of equal length.
6. Construct a segment that is the diagonal of a rectangle with sides of 4 dm and 6 dm. Construct two more segments intersecting the first and dividing it into three parts of equal length.
1. Construct a broken line of five links, each 3 dm long, so that the distance between its ends is 9 dm; was more than 9 dm; was less than 9 dm.
2. From segments of length equal to the length of the diagonal of a rectangle with sides 2 dm and 1 dm, build a broken line consisting of three, five, seven links, so that the distance between its ends is 1 dm.
3. Construct a polyline consisting of six links so that its length is more than 18 dm, but less than 19 dm.
4. Construct a broken line in the form of a letter of the Russian alphabet, consisting of two, three, four links.
5. Construct a broken line in the form of the letter M of the Russian alphabet. Move one of its vertices so that a broken line is formed in the form of another letter of the Russian alphabet.
6. The tourist changed the direction of his movement several times during the day. Before lunch, he walked 4 km to the north, then turned east and moved 2 km, and then walked some distance in the direction of the northeast, more than two km, but less than 3 km, and finally, a km to the east. After lunch, he started moving south and walked 1 km, then turned west and moved 3 km, and then he walked in the southwest direction the same distance that he walked in the northeast direction before lunch. As a result, the tourist ended up at a point 2 km away from the starting point of movement in the direction of the east. Select an appropriate scale and draw a polyline depicting the tourist route.
*In these tasks we're talking about only about an open simple broken line, i.e. about one in which the end of the last link does not coincide with the beginning of the first and non-adjacent links do not intersect.
Angles
1. Construct angles of 45, 90, 135, 180 degrees in such a way that they all have a common vertex and each smaller angle is contained inside a larger one.
2. Construct adjacent angles so that the value of one of them is greater than 135 degrees.
3. Draw on the geoplan several words consisting of letters of the Russian alphabet, in the writing of which only right angles are found.
4. Construct an acute angle whose value is 45 degrees. Select a point inside it and construct another angle so that the sides of both angles are respectively perpendicular.
5. Construct two angles, the sides of which are parallel in pairs, so that the intersection of these sides forms a rectangle with an area of 6 dm 2.
6. Construct two angles, the sides of which are perpendicular in pairs, so that when these sides intersect, a segment is formed with a length of 2 dm.
Triangles
1. Construct a triangle in which the length of the first side is more than 2 inches, but less than 3 inches, the length of the second side is more than 3 inches, but less than 4 inches, the length of the third side is more than 4 inches, but less than 5 inches.
Quadrilaterals
1. Construct a quadrilateral, all sides of which have a length equal to the diagonal of a rectangle measuring 3x1 in. Find several solutions.
2. Construct a quadrilateral, all sides of which have different lengths from 4 to 5 dm.
3. Construct a square with a side of 6 dm. Construct all the different squares whose vertices lie on the sides of the original square.
4. Construct a rectangle with an area of 12 dm 2 in four different ways.
5. Construct six squares with areas of 4 dm 2, 16 dm 2, 64 dm 2, so that each smaller square is contained inside each larger one.
6. Construct two rectangles having: a) equal perimeters and equal areas; b) equal areas and different perimeters.
2.3 Geometry on checkered paper
It is advisable to start teaching schoolchildren from the fifth grade.
Teaching should be conducted in a relaxed, almost improvisational style. This apparent ease actually requires a lot of serious preparation from the teacher.
It is better to conduct classes in a non-standard form.
It is necessary to use as much as possible in lessons. visual material: various cards, pictures, sets of figures, illustrations for solving problems, diagrams.
When analyzing a topic, you should try to achieve understanding, not memorization.
Lesson #1
Goal: develop combinatorial skills (consider various ways constructing a cutting line for figures, rules that allow you not to lose solutions when constructing this line), develop ideas about symmetry.
We solve problems 1-4 in class, problem 5 - at home.
1. A square contains 16 cells. Divide the square into two equal parts so that the cut line goes along the sides of the cells. (Methods of cutting a square into two parts will be considered different if the parts of the square obtained using one cutting method are not equal to the parts obtained using another method). How many cuts does the problem have?
Note. Finding multiple solutions to this problem is not that difficult. The figure shows some of them, and solutions b) and c) are the same, so the figures obtained in them can be combined by overlapping (if you rotate square c) by 90 degrees).
But finding all the solutions and not losing a single solution is already more difficult. Note that the broken line dividing the square into two equal parts is symmetrical relative to the center of the square. This observation allows you to draw a polyline step by step from both ends. For example, if the beginning of a polyline is at point A, then its end will be at point B. Make sure that for this problem, the beginning and end of the polyline can be drawn in two ways.
When constructing a polyline, in order not to lose any solution, you can adhere to this rule. If the next link of a broken line can be drawn in two ways, then you first need to prepare a second similar drawing and perform this step in one drawing in the first way, and in the other in the second way. You need to do the same when there are not two, but three ways. The specified procedure helps to find all solutions.
2. A 3x4 rectangle contains 12 cells. Find five ways to cut a rectangle into two equal parts so that the cut line goes along the sides of the cells (cutting methods are considered different if the parts obtained by one cutting method are not equal to the parts obtained by another method).
3. A 3x5 rectangle contains 15 cells and the central cell has been removed. Find five ways to cut the remaining figure into two equal parts so that the cut line goes along the sides of the cells.
4. A 6x6 square is divided into 36 identical squares. Find five ways to cut a square into two equal parts so that the cut line goes along the sides of the square.
5. Problem 4 has more than 200 solutions. Find at least 5 of them.
Lesson #2
Goal: continue to develop ideas about symmetry (axial, central).
1. Cut the shapes shown in the figure into two equal parts along the grid lines, with each part containing a circle.
2. The figures shown in the figure must be cut along the grid lines into four equal parts so that each part has a circle. How to do this?
3. Cut the figure shown in the figure along the grid lines into four equal parts and fold them into a square so that the circles and stars are located symmetrically relative to all axes of symmetry of the square.
4. Cut this square along the sides of the cells so that all parts are the same size and shape and so that each contains one circle and an asterisk.
5. Cut the 6x6 checkered paper square shown in the picture into four equal pieces so that each piece contains three shaded squares.
Illusions of size perception
Are the top and bottom of the numbers the same?
Now let's turn them upside down. So how?
Which segment is longer: AB or BC?
Sander's parallelogram, discovered by him in 1926. Segments AB and BC are equal.
Which segment is larger: AB or BC?
AB and BC are equal. The effect is mainly due to the fact that the figure on top is generally larger. Therefore, its separate segment seems larger.
Which line is larger: A or B?
Baldwin illusion. Lines A and B are absolutely equal.
Which of the red lines are longer?
Which circle is bigger? The one surrounded by small circles or large ones?
The Ebbin Gause illusion, discovered in 1902. Both central circles are the same size.
Which line is longer: AC or AB?
Both lines are the same size.
Which ice cream is bigger?
Both are the same. The effect is based on the following. In life, figures that are far away from us seem much smaller than their actual size. Our consciousness adapts to this feature of perception and automatically, as it were, adds size to distant figures in order to correctly evaluate them. In a flat drawing, all the figures are at the same distance from us. But the drawing itself depicts a tunnel going into the distance, prompting our consciousness that the second ice cream is in the distance (perspective). Consciousness is deceived and “adds” its size.
Which of the inner squares is larger: black or white?
The phenomenon of irradiation. The phenomenon is that light objects on dark background they seem larger than their actual size, as they seem to capture part of the dark background. When we look at a light surface against a dark background, due to the imperfection of the lens of the eye, the boundaries of this surface supposedly move apart and it seems to us larger than its true geometric dimensions. In the picture due to the brightness of the colors white square seems much larger relative to the black square on a white background.
Which circle is larger?
The left circle appears larger than the right, but it is not. The circles are the same size.
Which man is taller?
All little people are the same. The same effect of violating the law of perspective is at work here as in the example with ice cream.
Who is the longest person? And the shortest?
Here the illusion of perspective (we automatically add size to figures in the distance) is enhanced by the effect of comparison ( tall man stands next to the low one). In fact, the person in the background and the “dwarf” in the foreground are one and the same person.
Which of the horizontal segments is longer?
Müller Layer illusion, 1889. Both segments have the same length. The property of the whole figure is transferred to its individual part, and since the upper figure as a whole is longer, its straight segment also seems larger.
Which figure is bigger?
Jastrow's illusion (1891). Both figures are exactly the same.
Which horizontal line is longer?
Train track illusion. The top horizontal line appears longer. This line continues to be perceived as longer, no matter in what position we view the drawing. In fact, both lines are the same.
Which parallelepiped is larger?
All bars are the same. And here we return to the fact that the law of perspective is violated, as has already been shown in the examples above.
Which pillar is taller?
And one more variation on the theme of violating the law of perspective. All columns are the same size.
Which circle is the smallest?
The “bottom of the bucket” and the circle in the center of the lid are the same size.
Which line is longer?
Vertical-horizontal illusion. The lines are the same, but vertical line perceived as longer. If you look at the drawing with one eye, you will see how the effect changes.
Which girl is slimmer?
The effect is well known to any woman. In fact, both girls are the same size. But the longitudinal stripes on the dress visually reduce the figure (picture on the left), while the transverse stripes visually increase the volume (picture on the right).
Which of the figure parameters is greater: length or width?
The figure is the same in length and width, but the accordion shape and white wedges, as if inserted into the figure, visually elongate the object.
Are the top and bottom of the numbers the same?
Now let's turn them upside down. So how?
Which segment is longer: AB or BC?
Sander's parallelogram, discovered by him in 1926. Segments AB and BC are equal.
———————————————————————————————————
Which segment is larger: AB or BC?
AB and BC are equal. The effect is mainly due to the fact that the figure on top is generally larger. Therefore, its separate segment seems larger.
———————————————————————————————————
Which line is larger: A or B?
Baldwin illusion. Lines A and B are absolutely equal.
———————————————————————————————————
Which of the red lines are longer?
Picture tube illusion. The red lines in the figure are the same length.
———————————————————————————————————
Which circle is bigger? The one surrounded by small circles or large ones?
The Ebbin Gause illusion, discovered in 1902. Both central circles are the same size.
———————————————————————————————————
Which line is longer: AC or AB?
Both lines are the same size.
_____________________________________________________________________
Which ice cream is bigger?
Both are the same. The effect is based on the following. In life, figures that are far away from us seem much smaller than their real size. Our consciousness adapts to this feature of perception and automatically, as it were, adds size to distant figures in order to correctly evaluate them. In a flat drawing, all the figures are at the same distance from us. But the drawing itself depicts a tunnel going into the distance, prompting our consciousness that the second ice cream is in the distance (perspective). Consciousness is deceived and “adds” its size.
———————————————————————————————————
Which of the inner squares is larger: black or white?
The phenomenon of irradiation.
The phenomenon is that light objects against a dark background appear larger than their actual size, as they seem to capture part of the dark background. When we look at a light surface against a dark background, due to the imperfection of the lens of the eye, the boundaries of this surface supposedly move apart and it seems to us larger than its true geometric dimensions. In the picture, due to the brightness of the colors, the white square appears much larger relative to the black square on a white background.
———————————————————————————————————
Which circle is larger?
The left circle appears larger than the right, but it is not. The circles are the same size.
———————————————————————————————————
Which man is taller?
All little people are the same. The same effect of violating the law of perspective is at work here as in the example with ice cream.
———————————————————————————————————
Who is the longest person? And the shortest?
Here the illusion of perspective (we automatically add size to figures in the distance) is enhanced by the effect of comparison (a tall person standing next to a short one). In fact, the person in the background and the “dwarf” in the foreground are one and the same person.
———————————————————————————————————
Which of the horizontal segments is longer?
Müller Layer illusion, 1889. Both segments have the same length. The property of the whole figure is transferred to its individual part, and since the upper figure as a whole is longer, its straight segment also seems larger.
———————————————————————————————————
Which figure is bigger?
Jastrow's illusion (1891). Both figures are exactly the same.
———————————————————————————————————
Which horizontal line is longer?
Train track illusion. The top horizontal line appears longer. This line continues to be perceived as longer, no matter in what position we view the drawing. In fact, both lines are the same.
———————————————————————————————————
Which parallelepiped is larger?
All bars are the same. And here we return to the fact that the law of perspective is violated, as has already been shown in the examples above.
———————————————————————————————————
Which pillar is taller?
And one more variation on the theme of violating the law of perspective. All columns are the same size.
———————————————————————————————————
Which circle is the smallest?
The “bottom of the bucket” and the circle in the center of the lid are the same size.
———————————————————————————————————
Which line is longer?
Vertical-horizontal illusion. The lines are the same, but the vertical line is perceived as longer. If you look at the drawing with one eye, you will see how the effect changes.
———————————————————————————————————
Which girl is slimmer?
The effect is well known to any woman. In fact, both girls are the same size. But the longitudinal stripes on the dress visually reduce the figure (picture on the left), while the transverse stripes visually increase the volume (picture on the right).
———————————————————————————————————
Which of the figure parameters is greater: length or width?
The figure is the same in length and width, but the accordion shape and white wedges, as if inserted into the figure, visually elongate the object.
Someone knocked. I opened my bedroom door. This tall guy was standing there. I had never seen him before in my life. He seemed somehow shy; he said, “I had to come here to tell you something.” I asked his name and what he needed.
“Well,” he said, “I was sent here by the Masons to tell you about the circle and the square.”
This really amazed me. I seemed numb and just looked at him for a moment, trying to understand how this was happening. Then I decided that I didn't really care that much about how it happened, only that it actually happened. I grabbed his hand and said, “Come in here,” and pushed him into the room and locked the door behind him. I said, “I want to know everything you have to tell me.” And then he drew this drawing (Fig. 7-22). First he drew a square, then he described a circle around this square in a special way - in front of me was an image that I saw glowing in the room! I thought: this will be great. He divided the square into four sections, then drew diagonals from the corners through the middle to the opposite corners. She then drew diagonals across the four smaller squares. Then he drew lines from I to E and from E to J. After that, he drew lines from I to H and from H to J (E and H are points on the circle line where the vertical center line intersects it).
I had no problem up to this point, but then he drew a line from A to nowhere (G) and back to B, from D to nowhere (F) and back to C. I said, “Wait a minute, this is not as specified.” to me conditions. This doesn’t fit – there’s nothing here.” He said, "That's okay, because this line (A-G) is parallel to this line (I-H), and this line (D-F) is parallel to this line (J-E)."
"Okay," I said, "This is a new condition. I didn't have it before. I mean, there's nothing there... Parallel lines? “Well, okay, I’ll listen.”
Then he started telling me a lot of things. He said the first clue is that the circumference of a circle and the perimeter of a square are equal, which I told you before. This circle and square represent the same picture that appears from the air when looking at Great Pyramid, when there is a ship on top of it.
Proportion Φ (phi ratio)
He started telling me about the proportion Φ of 1.618 (here rounded to the third digit decimal). The proportion Φ is a very simple ratio. If you had a rod and were going to put a sign somewhere on it, then the Proportion Φ would determine only two places; in his illustration this is shown by points A and B (Fig. 7-23).
There are only two places - depending on which end you are coming from. The lower figure shows a ratio in which, dividing segment D by segment C and segment E by segment D, the two answers will be the same - 1.618.... So you divide the long line into a short line and this gives you a ratio of 1.618. If you divide the entire length of segment E by the next segment, which is shorter than segment D, you will get the same proportion. This is a magical place. Although I studied mathematics in college, when we passed this place, the information about the proportion Φ somehow went over my head. I haven't figured it out. I had to go back and study it all again.
This guy also gave an example of Leonardo's drawing of a circle inside a square and gave me more information that I will tell you about later. I asked him a lot of questions and about half the time he didn't know the answer. He would simply say, “This is how it happens,” or “I don’t know; We don’t know that.” Although I cannot say for sure, I suspect that the Masons have lost large number your information. I think they once had absolutely full knowledge, very similar to the knowledge of the Egyptians, but both of these teachings fell into decay.
Before leaving, he made a sketch under his diagram (see Fig. 7-22). There was a picture of a square and someone's right eye - I can't say it's the Mountain, because I don't know who it is. Then he left. Since then I have never seen him. I don't even remember his name.
Using the Key to Metatron's Cube
This Masonic gentleman did not directly answer the question of how a circle and a square fit into Metatron's Cube. In fact, I don't think he ever even saw Metatron's Cube. But he said something that touched something in me and I realized what it was. Immediately after he left, I already knew the answer. As you know, Metatron's Cube is not actually a flat object, but a three-dimensional one. The three-dimensional Metatron's Cube looks like this (Fig. 7-24). It's a cube within a cube, in three dimensions. Then, by turning it at a certain angle (Fig. 7-25), you can get its square aspect.
Having done this, you get Fig. 7-26. At this point the external aspect can be dropped; all you need is just the initial eight squares. Around these eight cells there is already a sphere, zona pellucida. The cells are arranged in the shape of a cube. So, having described them both in a circle and in straight lines, you get a circle and a square, which the Angels showed me. I was happy!
