The inner chambers of the palace of Sultan Ibrahim ibn Said consist of one hundred identical square rooms, arranged in the form of a 10x10 square. If two rooms have a common wall, then it must have exactly one door. And if the wall is external, then it must have exactly one window. How many windows and doors are there in the chambers of Ibrahim ibn Said? A flying carpet only flies when it is rectangular in shape. Ivan Tsarevich had a magic carpet measuring 9x12. One day the Serpent Gorynych crept up and cut a small rug measuring 1x8 from this carpet. Ivan Tsarevich was very upset and wanted to cut off another piece of 1x4 to make an 8x12 rectangle. But Vasilisa the Wise suggested doing things differently. She cut the carpet into three pieces, from which she used magic threads to sew a square 10x10 flying carpet. Can you guess how Vasilisa the Wise remade the damaged carpet? A point was marked in the circle. Cut the circle a) into three; b) into two parts so that a new circle can be made from them, with the marked point in the center.
a) Draw a straight line containing one of the sides of the square. If a point lies on the other side of this line than the square, then it definitely does not lie in the square. Otherwise, we draw a straight line containing the second side of the square and carry out similar reasoning. By drawing all four lines containing the sides of the square, we find out whether the point lies inside the square.
b) First draw two straight lines containing the diagonals of the square. They will break the piece of paper into four parts. Based on the answers to the first two questions, we will find out which of these parts the point lies in. Now let's draw a straight line containing the side of the square that lies in the same part of the plane as the point. This straight line will divide the indicated part of the plane into two more parts, one of which will be inside the square, and the other outside. By answering the third question, we find out whether the point lies inside the square. Please note: now we know not only whether the point lies inside the square, but also in which of the four parts of the square (into which the diagonals divide it) it lies.
This solution says nothing about cases where the point happens to be on one or more of the lines we draw. Look into these cases yourself.
Sometimes there is a need to print out a pre-lined sheet for various occasions. For example, to pass the time playing tic-tac-toe, you need a sheet of paper in a box, what to do if you don’t have it at hand. You can draw it yourself using a ruler, but it’s easier to print it on a printer. All you need is a ready-made template. On this page you can download and print a checkered sheet, a lined sheet or a sheet of music.
Print and download checkered sheet
A checkered sheet may turn out to be useful for children to solve an example in mathematics, and sometimes even for adults for various board games, for example in sea battle, tic-tac-toe or dots. Making your own checkered sheet in Word is not at all difficult; create a table measuring 37 by 56 cells. The result will be an even square, like a checkered notebook.
You can print or download the A4 squared sheet in PDF format. If you need to change a check, for example, its size or color, for example, to print a sheet not with black, but with a gray or light gray check, then below is a link to a sheet of checks in Word format.
Print and download lined sheet
You can download or print a lined sheet in A4 format. The sheet is lined in a large line with margins like a notebook. You can use a lined sheet for penmanship. On our website you can find an online copybook generator for children.
You can print or download the lined sheet on A4 paper using a PDF file. If you need to change the distance between the rulers or remove margins, then below is a link to a sheet with a ruler in Word format.
Music sheet print and download
To record notes in music schools, special notebooks are used. The staff consists of five lines on which notes are written. You can print out a lined sheet of music in A4 format. The sheet music is presented in two versions: blank - only lines and with a treble clef already printed. In order to print a sheet of music in A4 format, you can use the PDF files below. You can download the sheet music by saving the PDF file to your computer.
Children learn the art of origami from school, and in most cases they become familiar with paper, glue and scissors when they are still young. kindergarten, where together with the teachers they make the first primitive crafts. By cutting out figures from paper, first flat, then three-dimensional, the child is preparing to study geometry, and it is very important to instill in him a love for Aristotelian science right away, so that interest does not disappear later, and everything works out. A volumetric square made of paper is a figure with which one usually begins to get acquainted with stereometry. Having learned simple techniques for making this polygon, you will be able to build amazing structures based on an ordinary cube.
Basics of geometry
Anyone without advanced mathematical knowledge can make origami out of paper. A square - at first glance, a very simple figure, and constructing it will not be difficult. However, you need to start the process prepared, having an understanding of paper assembly and not being too upset at the first failures. Before you start getting interested in how to make a square out of paper, you need to familiarize yourself with the basics of geometry. Do not forget that a cube is a rectangle in which all sides are equal and the angles are right. This condition is mandatory, but errors of a couple of millimeters are acceptable.
Assembly materials
The production of a square occurs in several stages. You will need a sheet of paper, glue or tape, a ruler, a pencil and scissors. If you are afraid to draw a diagram yourself, you can print the finished diagram on a printer. To make the cube bright, each side can be covered with colored paper, shiny tape, or painted with multi-colored paints or crayons. To prevent the sides of the square from getting dirty over time and to maintain their original appearance, it is recommended to cover it with a transparent film.
Paper square: manufacturing diagram
Children and adults will love making geometric figures - the process is so exciting that very soon you will have a whole collection of various cubes. Before starting work, you need to stock up on the required amount of paper, which will be enough for all sides, so that later you do not have to add missing elements to the existing frame. The development is performed on whatman paper or cardboard. The material should be moderately dense so that the figure does not wrinkle during assembly, but a sheet of paper that is too thick will also not work. A special condition is the presence of additional edges on the sides of some sides, with the help of which the cube will be glued together. A volumetric square consists of eight identical sides. In order not to make a mistake with the choice of whatman paper, perform all the calculations before drawing the drawing on a sheet of paper.
Cube assembly
Once the development is ready, it must be cut with scissors. The connection can be made with any glue or To ensure that the edges of the square are even, each fold line must be bent several times before gluing. Now the pencil drawing can be erased so that the cube has an aesthetic appearance. Depending on the purpose for which the craft is being made, it can be decorated different ways: cover with shiny glossy film or colored paper, paint or stick bright stickers.
Make it a fun process, and soon your home will have a collection of cubes of different colors and sizes. Once you know how to make a square out of paper, it will be easy for you to create miniature crafts that can be turned into, for example, dice. If it grows in the house Small child, you can cover several figures with colored tape to get a unique set of doll furniture.
A little imagination...
If you decide to make an unusual origami out of paper, the square can be turned into For this, it should be made from the material white. The optimal length of each face is 1.5-2 cm. Before you start making a miniature cube, you need to practice and start learning the art of origami from a figure bigger size. As soon as the technique is fully studied, you will be able to create masterpieces based on simple Knowing how to make a square out of paper, you can easily move on to the next stage of learning - modeling original three-dimensional crafts.
Paper people
In Russia and the West, a tendency has long been noticed to produce children's kits for creating people and animals, which are based on a three-dimensional square of paper. The diagram is no different from the development of a simple cube, only for some parts the proportions are changed. So, by putting together squares of different sizes, you can design interesting crafts. Once your child understands this simple technique, you will be able to create masterpieces with your own hands and not spend money on expensive handicraft kits. It is better if the drawing is applied to the paper immediately, and not after assembling the entire figure.
To preserve the aesthetics of the product, the edges must be glued together with hard glue, not tape. is clearly visible on paper, and the craft will not look its best. Having figured out how to make a square out of paper, you can easily make unique crafts and collect a whole collection, because origami is a fascinating art that is accessible to people of all ages.
Which paper is best for crafts? How to make a lot faster? What is the optimal module size? This section will reveal the secrets modular origami
and will help you take your first creative steps.
Paper selection
Office paper. For modular origami, office paper is most suitable because it is not too smooth and quite dense, the modules stick well to each other and there is no slipping when connected. Office paper of different color shades is colored on both sides and does not turn white on the folds.
Blocks for notes (stickers). For the Trefoil module and the Superball kusudama modules, square stickers are used. Triangular modules are usually folded from rectangular pieces of paper. The aspect ratio of the rectangular leaf is 2:3 (for example 4 x 6 cm). However, you can use square stickers. cut in half. In this case, the sides will have a ratio of 2:1. The module itself will be thicker, which is good for some crafts. Sometimes you can find rectangular-shaped stickers that do not need to be cut.
Origami paper. In Japanese, special origami paper is called “kami”. The sheets are already cut square shape, but sheet size may vary. This set of paper contains several colors of paper, one side of the sheet is usually white, and the other is colored. You can find paper with ornaments or colored on both sides.
Colored paper for children's creativity. Before use, such paper must be checked for strength on folds - it can tear easily. For modular origami, you need a large number of paper of the same color, and these sets usually contain several sheets of the same color. Sometimes it is difficult to find black or brown paper, so you will have to use just such a set.
Magazine paper. Paper from modern glossy magazines is perfect for modular origami. It is advisable to choose paper that is not very thin. By sorting the modules by color, you can achieve interesting pictorial effects in the assembled product.
Paper size for modules
You can fold triangular modules from pieces of paper of different sizes - small and large. The size of the future product depends on this. It is convenient to use sheets of paper measuring 1/16 or 1/32 of a standard A4 sheet for triangular modules.
If the sides of an A4 sheet (long and short) are divided into 4 equal parts, and the sheet is cut along the marked lines, the rectangles will come out measuring approximately 53 x 74 mm.
If the long side of an A4 sheet is divided into 8 equal parts, and the short side into 4, and the sheet is cut along the marked lines, then the rectangles will come out measuring approximately 37 x 53 mm.
You can choose a different size. Choose one module size for your craft and stop there. And if you make all the crafts from modules of the same size, then all the products will be compatible with each other and the Christmas tree, folded for the New Year, can turn into a palm tree, and then into a peacock or something else.
Tools and additional materials
Stationery knife and ruler. First, you will have to prepare and cut the paper into squares or rectangles before starting to fold the modules. Cut several layers of paper at once. It will be difficult to cut with scissors, so use stationery knife and cut along the ruler. The ruler must be solid. Place a board under the paper. You can purchase a special paper cutter if possible.
Glue. Adhesive used for joining small parts- eyelet, decorations, etc. Try to make all products without glue, this way any craft can be disassembled and created something new. But if you use the craft as a toy for children or a gift, then of course you need to take care of its durability.
When assembling, you can use thick PVA glue or a glue stick. And when working with the Trefoil module, glue is required. Flower petals are connected with glue; it is also needed when attaching leaves and making stems.
Adhesive mass. Great invention! The mass resembles a cross between white plasticine and double-sided tape. You can tear off a piece, roll it into a ball and secure any part inside or outside the craft. When necessary, the part can be removed. The adhesive mass leaves no traces.
Paints. If you make the entire craft from white office paper, you can cover it with any colored paint from a spray can.
Sewing threads with a needle. Kusudama “Superball” modules are connected to each other using a needle and thread.
Additional details. Ready-made eyes for toys will come in handy when making crafts. You can draw the eyes yourself on thick paper or print ready-made images of the eyes of the desired size and type on a printer. Beads, ribbons, beads, sequins, etc. are suitable for decorating crafts.
One of the ideals of origami requires folding the original square without preliminary drawings or measurements. With this folding, a problem arises if it is necessary to fold a sheet of paper, for example, in three. How to do this without using a pencil? This article discusses the issue of dividing a square sheet of paper into two, three, four, ... ten equal parts.
Dividing a sheet of paper into two parts is not difficult, since it is accomplished simply by folding the basic book shape. Let's move on to the more complex task of dividing a square sheet into three parts.
This task is no longer so simple. To solve it we need Hag's theorem. Fold the corner of the square to the middle of the opposite side. In this case, the point of intersection of the other side opposite this angle and the side adjacent to it divides the side in the ratio of one to two. Thus, using only folds we have found a third of the side of the square.
The next task is dividing the side of a square into four equal parts. To do this, it is enough to divide them in half, and then, each of the halves in half again. This is exactly what happens when we fold the basic shape of a door.
As you might guess, dividing a square into five parts by folding is a much more difficult task. Her solution is shown in the figure. Try to prove for yourself that in this way we really divide the square into five parts.
In order to divide the side of a square into six parts, we just need to divide it into three parts, as shown earlier. And then, divide each part in half.
It can be noted that dividing a square into a number of parts that is a prime number causes particular difficulties. Let's start dividing the side into seven equal parts. To do this, first divide the square into five equal parts, and then do the action shown in the picture.
Dividing a square into eight equal parts is quite simple. To do this, it is enough to divide it into four equal parts, and then divide each of them in half.
To divide into nine equal parts, two methods can be proposed. The first is to first divide into three equal parts, and then repeat the division by three for the small square. However, this method is bad in that when used in practice it will be difficult to maintain sufficient accuracy, since the errors made at different stages add up.
The other method is more original and is based on the development of Haga's theorem proposed by Koji and Mitsuya Fushimi. Note that this method was used when dividing a square into seven and nine parts. Maybe it can also be used to divide it into nine parts?