There can be many answers to the question of what a square is. It all depends on who you addressed this question to. The musician will say that the square is 4, 8, 16, 32 bars or jazz improvisation. A child - that this is a ball game or a children's magazine. The printer will send you to study the pins of the font, and the technician will send you variations of the metal profile.
There are many other meanings of this word, but today we will ask a question to mathematics. So...
We will deal with this figure gradually, from simple to complex, and begin with the history of the square. How did he appear, how did people, scientists from different countries and civilizations perceive him?
History of the study of the square
The ancient world perceives the square mainly as the four cardinal points. In general, despite the many quadrangles, it is the square that has the main number - four. For the Assyrians and Peruvians, the square is the whole world, that is, it represents four main directions, the cardinal points.
Even the Universe was presented as a square, also divided into four parts - this is a vision of the inhabitants of North America. For the Celts, the universe is three squares enclosed in each other, and four (!) Rivers flow from the center. And the Egyptians generally deified this figure!
For the first time, the square was described by means of mathematical formulas by the Greeks. But for them, this polygon had only negative characteristics. Pythagoras did not like even numbers at all, seeing in them weakness and femininity.
Even in religions there is a square. In Islam, the Kaaba - the navel of the Earth - does not have any spherical, but rather a cubic shape.
In India, the main grapheme representing the Earth, or the symbol of the earth, was a crossed square. And again we are talking about the four cardinal points, the four regions of the earth.
In China, a square is peace, harmony and order. Chaos is defeated by building square Vara. And the square inscribed in the circle is the basis of the vision of the world, symbolizing the unity and connection of the Cosmos and the Earth.
Pagan Russia - Svarog Square. This symbol is also called the Star of Svarog, or the Star of Russia. It is quite complex, as it is composed of intersecting and closed lines. Svarog - the Blacksmith God, the most important creator, creator and the sky itself in the representation of the Russians. There is a rhombus in this symbol, which again speaks of the Earth and its four directions. And a star with four rays - 4 cardinal points, 4 faces of Svarog - its omniscience. And the intersection of rays is a hotbed.
Interesting about the square
The most popular phrase that comes to mind about our protagonist is Black Square.
Malevich’s painting is still very popular. After its creation, the author himself was tormented for a long time with the question of what it is and why a simple black square on a white background attracts attention to itself.
But if you look closely, you will notice that the plane of the square is not smooth, and there are many colorful shades in the cracks of black paint. Apparently, at the beginning there was a certain composition, which the author did not like, and he closed it from our eyes with this figure. Black square, like nothing - a black hole, only a magical square shape. And emptiness, as you know, attracts ...
The "magic squares" are also very popular. In essence, this is a table, naturally square, filled with numbers in each column. The sum of these numbers is the same in all rows, columns, and diagonals (individually). If diagonals are excluded from the equality, then the square is semi-magic.
Albrecht Dürer in 1514 created the painting "Melancholy I", on which he depicted a 4x4 magic square. In it, the sum of the numbers of all columns, rows, diagonals, and even inner squares is thirty-four.
Based on these tables, very interesting and popular puzzles appeared - Sudoku.
The Egyptians were the first to draw lines of the relationship of numbers (date of birth) and qualities of character, abilities and talents of a person. Pythagoras took this knowledge, somewhat reworked and put it in a square. The result was the Pythagorean Square.
This is a separate direction in numerology. Four basic numbers are calculated from the person’s birth date by additions, which are placed in the psychomatrix (square). And lay out all the secret information about your energy, health, talent, luck, temperament and other things on the shelves. On average, according to surveys, the reliability is 60% -80%.
What is a square?
A square is called a geometric figure. The shape of the square is a quadrangle that has equal sides and angles. More precisely, this quadrangle is called regular.
The square has its own characteristics. It:
- sides of equal length;
- angles equal to each other - straight (90 degrees).
Due to these signs and features, you can enter a circle into a square and describe it around it. The circumscribed circle will touch all its vertices, and the inscribed circle will be the midpoints of all its sides. Their center will coincide with the center of the square and divide all its diagonals in half. The latter, in turn, are equal to each other and divide the corners of the square into equal parts.
One diagonal divides the square into two isosceles triangles, both into four.
Thus, if the length of the side of the square is t, the radius of the circumscribed circle is R, and the length of the inscribed circle is r, then
- the area of the base of the square, or the area of the square (S) will be S = t 2 = 2R 2 = 4r 2 ;
- the perimeter of the square P should be calculated by the formula P = 4t = 4√2R = 8r;
- the radius length of the circumscribed circle R = (√2 / 2) t;
The base area of the square can still be calculated by knowing its side (a) or the length of its diagonal (c), then the formulas will look accordingly: S = a 2 and S = 1 / 2c 2 .
What is a square, we found out. Let's take a closer look at the details, because the square figure is the most symmetrical quadrangle. It has five axes of symmetry, with one (fourth order) passing through the center and perpendicular to the plane of the square itself, and the other four axis of symmetry of the second order, two of them parallel to the sides, and two more through the diagonals of the square.
Ways to build a square
Based on the definitions, it seems that there is nothing easier than building the right square. This is true, but provided that you have all the measuring tools. And if something is not available?
Let's look at existing ways to help us build this figure.
A measuring ruler and a square are the main tools with which you can most easily build a square.
First mark a point, let's say A, from it we will build the base of the square.
Using a ruler, set aside a distance equal to the length of the side from it to the right, say 30 mm, and set point B.
Now from both points, using a square, draw up perpendiculars of 30 mm each. At the ends of the perpendiculars we put points B and D, which we connect together using a ruler - that's it, the square ABVG with a side of 30 mm is ready!
Using a ruler and a protractor is also quite easy to build a square. Start, as in the previous case, from a point, say H, postpone a horizontal segment from it, for example 50 mm. Put point O.
Now connect the center of the protractor with point H, put a mark at the angle value of 90 0 , through it and point H, build a vertical segment of 50 mm, put point P. at its end. Next, in a similar way, build the third segment from point O through angle 90 0 equal to 50 mm, let it end with point P. Connect the points P and R. You have a square NORP with a side length of 50 mm.
You can build a square using only a pair of compasses and a ruler. If the size of the square is important to you and the length of the side is known, then you will also need a calculator.
So, put the first point E - this will be it from the vertices of the square. Next, specify the place where the opposite vertex F will be located, that is, stand the diagonal of the hedgehog of your figure. If you build a square in size, then having the length of the side, calculate the length of the diagonal using the formula:
d = √2 * a, where a is the length of the side.
After you know the length of the diagonal, build a segment of the hedgehog of this magnitude. From point E, use a compass in the direction of point G to draw a semicircle with a radius of hedgehog. And vice versa, from point - a semicircle towards the point , with radius . Through the intersection points of these semicircles, using a ruler, build a segment of ZI. Hedgehog and ZI intersect at right angles and are the diagonals of the future square. Connecting the points , , and with the help of a ruler, you will get the inscribed square .
There is still the opportunity to build a square using one ruler. What is a square? This is a section of a plane bounded by intersecting segments (lines, rays). Therefore, we can construct a square according to the coordinates of its vertices. First draw the coordinate axes. The sides of the square can lie on them, or the center of intersection of the diagonals will coincide with the origin — this depends on your desire or the conditions of the task. Perhaps your figure will stand at some distance from the axes. In any case, first you mark two points by numerical values (arbitrarily or conditionally), then you will know the length of the side of the square. Now you can calculate the coordinates of the remaining two vertices, remembering that the sides of the square are equal and pairwise parallel to each other. The last step is to connect all the points in series with each other using a ruler.
What are the squares?
A square is a figure clearly defined and strictly limited by its definitions, therefore, the types of squares do not differ in variety.
In non-Euclidean geometry, a square is perceived more widely - it is a quadrangle with equal sides and angles, but the degree of angles is not specified. This means that angles can be 120 degrees ("convex" square) and, for example, 72 degrees ("concave" square).
If you ask what a square is for a geometer or computer scientist, they will answer you that it is a complete or planar graph (columns K 1 to K 4 ). And this is absolutely fair. The graph has vertices and edges. When they get into an ordered pair, a graph is formed. The number of vertices is the order of the graph, the number of edges is its size. Thus, a square is a planar graph with four vertices and six edges, or K 4 : 6.
Side of a square
One of the main conditions for the existence of a square — the presence of equal sides — makes the side very important for various calculations. But at the same time, it gives many ways for the side length of the square to be calculated in the presence of a variety of source data.
So how to find the value of the side of the square?
- If you know only the diagonal length of the square d, then you can calculate the side using the following formula: a = d / √2.
- The diameter of the inscribed circle is equal to the side of the square and, therefore, to two radii, that is: a = D = 2R.
- The radius of the circumscribed circle can also help to calculate what the side of the square is. We can find out the diameter D from the radius R, which, in turn, is equal to the diagonal of the square d, and we already know the formula for the side of the square through the diagonal: a = D / √2 = d / √2 = 2R / √2.
- It follows from the equality of sides that you can find out the side of a square (a) using its perimeter P or area S: a = √ S = P / 4.
- If we know the length of the line that goes out of the corner of the square and intersects the middle of its adjacent side C, then we will also be able to find out what the length of the side of the square is: a = 2C / √5.
Here are how many ways exist to figure out such an important parameter as the length of the side of the square.
Square volume
The phrase itself is absurd. What is a square? This is a flat figure with only two parameters - length and width. And the volume? This is a quantitative characteristic of the space occupied by an object, that is, it can only be calculated for volumetric bodies.
A volumetric body, all faces of which are squares, is a cube. Despite the enormous and fundamental difference, schoolchildren quite often try to calculate the volume of a square. If this succeeds, the Nobel Prize is guaranteed.
And to find out the volume of the cube V, it is enough to multiply all three of its edges - a, b, c: V = a * b * c. And since they are equal by definition, the formula may look different: V = a 3 .
Values, parts and specifications
A square, like any polygon, has vertices - these are the points at which its sides intersect. The vertices of the square lie on the circumference described around it. A diagonal passes through the vertex to the center of the square, which is also the bisector and radius of the circumscribed circle.
Since a square is a flat figure, it is impossible to cut and construct a section of the square. But it can be the result of the intersection of many volumetric bodies by a plane. For example, a cylinder. The axial section of the cylinder is a rectangle or square. Even when the body intersects the plane at an arbitrary angle, a square can come out!
But the square has one more relation to the section, and not to any, but to the Golden section.
We all know that the Golden Ratio is a proportion in which one quantity relates to another in the same way as their sum to a larger value. In generalized percentage terms, this is as follows: the initial value (amount) is divided into 62 and 38 percent.
The golden ratio is very popular. It is used in design, architecture, and anywhere, even in economics. But this is far from the only proportion derived by Pythagoras. There is, for example, the expression "√2". On its basis, dynamic rectangles are constructed, which, in turn, are the founders of the formats of group A (A6, A5, A4, etc.). Why is it about dynamic rectangles? Because their construction begins with a square.
Yes, first you need to build a square. Its side will be equal to the smaller side of the future rectangle. Then it is necessary to draw the diagonal of this square and, using the compass, lay the length of this diagonal on the extension of the side of the square. From the point obtained at the intersection, we build a rectangle, in which we again build the diagonal and set its length on the continuation of the side. If we continue to work on this scheme, we get the very dynamic rectangles.
The ratio of the long side of the first rectangle to the short will be 0.7. This is almost 0.68 in the Golden Ratio.
Square corners
Actually, it’s already difficult to say something fresh about the corners. All properties, they are also signs of a square, we have listed. As for the corners, there are four of them (as in any quadrangle), each corner squared is a straight line, that is, it has a size of ninety degrees. By definition, there is only a rectangular square. If the corners are larger or smaller - this is another figure.
The diagonals of a square divide its angles in half, that is, they are bisectors.
Square equation
If it is necessary to calculate the value of various values of a square (area, perimeter, lengths of sides or diagonals), various equations are used that are derived from the properties of the square, the basic laws and rules of geometry.
1. The equation of the square area
From the equations for calculating the area of the quadrangles, we know that it (the area) is equal to the product of length and width. And since the sides of the square are the same in length, its area will be equal to the length of any side raised to the second power
S = a 2 .
Using the Pythagorean theorem, we can calculate the area of a square, knowing the length of its diagonal.
S = d 2/2.
2. The equation of the perimeter of the square
The perimeter of a square, like all quadrangles, is equal to the sum of the lengths of its sides, and since they are all the same, we can say that the perimeter of a square is equal to the length of the side multiplied by four
P = a + a + a + a = 4a.
Again, the Pythagorean theorem will help us find the perimeter through the diagonal. You need to multiply the value of the length of the diagonal by two roots of two
P = 2√2d
3. The equation of the diagonal of the square
The diagonals of the square are equal, intersect at right angles and divide the intersection point in half.
You can find them based on the above equations of the area and perimeter of the square
d = √2 * a, d = √2S, d = P / 2√2
There are still ways to find out how long the diagonal of a square is. The radius of the inscribed circle is equal to half its diagonal, hence
d = √2D = 2√2R, where D is the diameter and R is the radius of the inscribed circle.
Knowing the radius of the circumscribed circle, it is even easier to calculate the diagonal, because it is the diameter, that is, d = D = 2R.
It is also possible to calculate the length of the diagonal, knowing the length of the line extending from the corner to the center of the side of the square C: d = √8 / 5 * C.
But do not forget that a square is a section of a plane bounded by four intersecting lines.
For lines (and the shapes they form) there are enough equations that do not need an additional description, but the line is infinite. And polygons are limited by the intersection of lines. For them, you can use linear equations, combined into a system that defines straight lines. But it is necessary to specify additional parameters, conditions.
To determine the polygons, it is necessary to compose an equation that describes not a line, but a separate arbitrary segment without the intervention of additional conditions and descriptions.
[x / x i ] * [x i / x] * y i - this is a special equation for polygons.
The square brackets in it indicate the condition for excluding the fractional part of the number, that is, we should leave only the integer. y i is a function that will be executed in the parameter range from x to x i .
Using this equation, we can derive new equations for calculating segments and lines consisting of several segments. It is basic, universal for polygons.
Remember that a square is a part of the plane, therefore its description of the type y = f (x) can be represented, most often, only as a multi-valued function, which, in turn, can be expressed in terms of single-valued, if they are represented parametrically, that is, depending on any parameter t:
x = f (t), y = f (t).
So, if we use together the universal equation and the parametric representation, then we can really derive an equation for expressing polygons:
x = ((A2 + A3) * A5 + A4 * P) * Cos (L)
y = ((A1 + A4) * A5 + A3 * P) * Sin (L),
Where
A1 = [1 / [T / P]] * [T / P]; A2 = [2 / [T / P]] * [[T / P] / 2]; A3 = [3 / [T / P]] * [[T / P] / 3]; A4 = [4 / [T / P]] * [[T / P] / 4]; A5 = TP * [T / P],
where P is the diagonal of the rectangle, L is the angle of inclination to the horizontal of the diagonal P, T is the parameter varying in the range from P to 5P.
If L = 3.14 / 4, then the equation will describe squares of different sizes, depending on the size of the diagonal P.
Square application
In the modern world, technologies make it possible to give various materials a square shape, more precisely a square section.
It is much more profitable, cheaper, more durable and safer. So, now they make square pipes, piles, wire (s) and even square threads.
The main advantages are obvious, they come out of elementary geometry. With the same size, the area of the inscribed circle is less than the area of the square into which it is inscribed; therefore, the throughput of the square pipe or the energy consumption of the square wire will be higher than that of round analogs.
Often consumables of a square section are more aesthetic and convenient to use, install, fasten.
When choosing these materials, it is important to correctly calculate the cross-section of the square so that the wire or pipe can withstand the necessary load. In each individual case, of course, such parameters as current strength or pressure will be necessary, but one can not do without the basic geometric rules of the square. Although the sizes of square sections are no longer so much calculated as they are selected according to the specified parameters from the tables established by GOSTs for different industries.