How to make a polyhedron out of paper. Paper polyhedrons - schemes

Paper crafts are not only various postcards and applications made in the form of flat products. Volume models of figures are very original (photo 1). For example, you can design a polyhedron out of paper. Let's look at some ways to accomplish it using diagrams and photographs.

History of figures

Ancient mathematical science has its roots in the distant past, during the prosperity of ancient Rome and Greece. Then it was customary to connect technical aspects with philosophical ones. Therefore, according to the teachings of Plato (one of the ancient Greek thinkers), each of the polyhedra, consisting of a certain number of identical planes, symbolizes one element. The figures from the triangles - the octahedron, icosahedron and tetrahedron - are associated with air, water and fire, respectively, and can be transformed into each other due to the same type of faces, each of which has three vertices. The earth is symbolized by the hexahedron of squares. And the dodecahedron, thanks to its special pentagonal faces, plays a decorative role and is a prototype of harmony and peace.

It is also known that one of the Greek mathematicians, Euclid, proved in his teaching “Beginnings” the uniqueness of the mentioned Platonic solids and their property of “fitting” into the sphere (photo 2). The polyhedron shown from paper is made by folding twenty isosceles triangles closed between themselves. The diagram clearly demonstrates the pattern for making the figure. Let us consider in more detail all the stages of work on the creation of an icosahedron.

We make a hexahedron

The icosahedron consists of equal-sized isosceles triangles. It can be easily folded using the scan shown in Figure 2. Take a rectangular sheet of paper. Draw on it twenty triangles of equal size and shape, placing them in four rows. Moreover, each face of one will simultaneously be the side of the other. Use the resulting template for the manufacture of the workpiece. It will differ from the base-scan by the presence of allowances for gluing along all external lines. After cutting the paper from the paper, bend it along the lines. Forming a polyhedron from paper, close the extreme rows between each other. In this case, the vertices of the triangles are connected to one point.

Regular Polyhedrons

All figures differ from each other in different number of faces and their shape. In addition, some models can be folded from a single sheet (as described in the example of manufacturing an icosahedron), while others can only be assembled from several modules. Regular polyhedrons are considered classical. They are made of paper, adhering to the main rule of symmetry - the presence in the template of completely identical faces. There are five main types of such figures. The table shows information about their names, number and shape of faces:

Variety of shapes

Based on the five types given, using skill and imagination, craftsmen easily construct many different models from paper. The polyhedron can be completely different from the above five shapes, being formed simultaneously from faces of different shapes, for example from squares and triangles. This is how Archimedean bodies are obtained. And if you skip one or more faces, you get an open figure that can be viewed both outside and inside. For the manufacture of volumetric models, special patterns are used, cut out of a sufficiently dense, well-kept paper. They also make special polyhedrons of paper. Schemes of such products include the presence of additional, protruding modules. Let us examine the ways how to construct a very beautiful figure using the example of a dodecahedron (photo 3).

How to make paper a polyhedron with twelve vertices: the first way

This figure is also called a star-shaped dodecahedron. Each of its vertices at its base is a regular pentagon. Therefore, such polyhedra are made in two ways from paper. Schemes for the manufacture will be slightly different from each other. In the first case, this is a single part (photo 4), as a result of folding of which the finished product is obtained. In addition to the main faces, on the drawing there are connecting parts for gluing, thanks to which the figure closes together. To make a polyhedron in the second way, you need to make several patterns separately. Consider the work process in more detail.

How to make a polyhedron out of paper: the second way

Make two main templates (photo 5):

- The first one. Draw a circle on the sheet and divide it across into two parts. One will be the basis for the pattern, immediately erase the second arc for convenience. Divide the part into five equal parts and limit all radii to transverse segments. The result is five identical isosceles triangles connected together. Draw next to the middle segment exactly the same semicircle, only in mirror image. The resulting part when folding looks like two cones. Make such similar patterns in just six pieces. For gluing them, a second part is used, which will be placed inside.

- The second one. This pattern is a five-pointed star. Complete the same twelve blanks. Forming a polyhedron, each of the stars with the ends bent upward is placed inside the conical parts and glued to the faces.

A complete collection of the figure is obtained by connecting the double blocks with additional pieces of paper, leading them inwards. When modeling products, it is quite problematic to make them different in size. Finished models of polyhedra made of paper are not so easy to enlarge. To do this, it is not enough just to make allowances at all external borders. It is necessary to scale separately each of the faces. Only in this way is it possible to get an enlarged copy of the original model. Using the second method of manufacturing a polyhedron, it is much easier to do this, since it will be enough to increase the initial workpieces for which the required number of individual parts is already being performed.