How to make a polyhedron from paper. Polyhedrons from paper - schemes

Paper crafts are not only different postcards and applications made in the form of flat products. Very original three-dimensional models of figures are obtained (photo 1). For example, you can design a polyhedron from paper. Let's consider some ways of its implementation, 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 associate 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. Figures from triangles - octahedron, icosahedron and tetrahedron - are associated with air, water and fire respectively and can be transformed into each other due to the uniformity of the faces, each of which has three vertices. The earth is also symbolized by a hexahedron of squares. And the dodecahedron, thanks to its special pentagonal facets, performs 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 teachings of the "Principle" the uniqueness of the mentioned platonic solids and their property "fit" into the sphere (photo 2). A polyhedron, shown from paper, is made by folding twenty isosceles triangles closed together. The diagram clearly demonstrates the pattern for making the figure. Let us consider in more detail all the stages of the creation of an icosahedron.

Making a dentagon

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

Regular polyhedra

All the figures differ from each other by a different number of faces and their shape. In addition, some models can be composed of a single sheet (as described in the example of manufacturing an icosahedron), others - only by collecting from several modules. The regular polyhedra are considered to be classical. Of paper they do, 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 provides information on their names, number and shape of faces:

Variety of figures

Based on the five species given, using skill and imagination, craftsmen easily design many different models of paper. The polyhedron can be completely different from the above five figures, forming simultaneously from different in form faces, for example from squares and triangles. So Archimedean bodies are obtained. And if you skip one or more faces, you will get an open figure, viewed both from the outside and inside. For the production of volumetric models, special patterns are used, cut out from a sufficiently dense, well-shaped paper. They make special polyhedrons from paper. Schemes of such products provide for 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 from a paper a polyhedron with twelve tops: the first way

Such a figure is also called the star dodecahedron. Each of its vertices in its base is a regular pentagon. Therefore, they make such polyhedrons of paper in two ways. Schemes for manufacturing will be slightly different from each other. In the first case, this is a single detail (photo 4), as a result of which a finished product is obtained. In addition to the main faces, the drawing also contains connecting parts for bonding, thanks to which the figure is joined together. To make a polyhedron in the second way, several templates must be made separately. Let's consider the work process in more detail.

How to make a polygon from paper: the second way

Make two main templates (photo 5):

- First. Draw a circle on the sheet and divide it across into two parts. One is the basis for the pattern, the second arc is immediately erased for convenience. Divide the part into five equal parts and limit all radii by transverse segments. As a result, five identical isosceles triangles are joined together. Draw near the middle segment adjacent to the middle segment exactly the same semicircle, only in the mirror image. The resulting part looks like two cones when folded. Produce such similar patterns only six pieces. For their gluing a second part is used, which will be placed inward.

- The second. This pattern is a five-pointed star. Perform the same twelve blanks. Forming a polyhedron, each of the stars with the ends bent upwards is placed inside the cone-shaped parts and pasted to the faces.

A complete collection of the figure is obtained by joining the double blocks with additional pieces of paper, winding them inward. Modeling products, it is quite problematic to make them different in size. Ready models of paper polyhedra are not so easy to increase. To do this, it is not enough just to make allowances for all external borders. It is necessary to scale separately each of the faces. Only in this way it is possible to obtain an enlarged copy of the original model. Using the second method of making a polyhedron, it is much easier to do this, since it will be sufficient to increase the initial preforms, according to which the required number of individual parts is already executed.