Paper handicrafts are not only differentPostcards 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.
The ancient mathematical science leaves itsroots 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 Greekmathematicians, 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 the 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.
The icosahedron consists of the same sizeisosceles 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 be simultaneously 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 paper, close the extreme rows between each other. In this case, the vertices of the triangles are joined to one point.
All the figures differ from each other in differentnumber of faces and their shape. In addition, some models can be composed of a single sheet (as described in the example of manufacturing the icosahedron), others - only by collecting from several modules. Classical regular polyhedra are considered. 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 about their names, number and shape of faces:
Name | Number of faces | The shape of each face |
tetrahedron | 4 | triangle |
hexahedron | 6 | square |
octahedron | 8 | triangle |
dodecahedron | 12 | pentagon |
icosahedron | 20 | triangle |
Based on the five species presented, usingskill and imagination, craftsmen easily design many different models of paper. A 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 thick, 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 with the example of a dodecahedron (photo 3).
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 folding, which produces a finished product. In addition to the main faces, the drawing also contains connecting parts for gluing, 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.
Make two main templates (photo 5):
- First. Draw a circle on the sheet and divide ittransversely 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 semi-circle, only in the mirror image. The resulting part looks like two cones when folded. Manufacture of such similar patterns only six pieces. For their gluing a second part is used, which will be placed inward.
- 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 glued to the faces.
A complete collection of the figure is obtained by connectingdouble blocks with additional pieces of paper, winding them inward. Modeling products, it is quite problematic to make them different in size. Ready-made polygon models from paper 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 blanks for which the required number of individual parts are already being made.