Paper crafts - is not only a variety of cards and applications made in the form of flat products.Very original figures obtained three-dimensional model (photo 1).For example, can be constructed from paper polyhedron.Consider some of the ways of its implementation, using diagrams and pictures.
history shapes
ancient mathematical science is rooted in the distant past, in times of prosperity of ancient Rome and Greece.Then it was decided to link the technical aspects of the philosophical.Therefore, according to the teachings of Plato (one of the ancient Greek philosophers), each of the polyhedrons, consisting of a number of identical planes, symbolizes one element.Figures from triangles - octahedron and tetrahedron icosahedron - associated with air, fire and water, respectively, and may be converted into each other due to the uniformity of faces, each of which has three apexes.Earth symbolizes hexahedron squares.A dodecahedron, thanks to a special pentagonal faces, performs a decorative role, and is the prototype of harmony and peace.
also known that one of the Greek mathematician Euclid proved in his teaching "Elements" originality referred to the Platonic solids and their property "fit" into the sphere (photo 2).It is shown by the polyhedron of paper folding serried together twenty equilateral triangles.The scheme demonstrates a pattern for making the figures.Let us consider all the stages of work on the creation of an icosahedron.
Do
icosahedron icosahedron consists of identical size isosceles triangles.It can be easily folded, using the representation in Figure 2 sweep.Take a rectangular sheet of paper.Draw on it twenty identical in size and shape of triangles, placing them in four rows.Thus, each face one will simultaneously be a party to the other.The resulting pattern is used for the manufacture of blanks.It will be different from the base-sweep presence allowances for bonding on all external lines.Cut paper blank, fold it along the lines.Forming paper polyhedron extreme short the ranks together.At the same time the top of the triangle is connected to one point.
Regular polyhedra
All figures are different faces and different amounts of their form.Furthermore, some models can be folded from a single sheet (as described in Example manufacture icosahedron) others - only through collection of several modules.Classic considered regular polyhedra.Because they are made of paper, sticking to the main rules of symmetry - the presence in the template completely identical faces.There are five main types of figures.The following table provides information about their names, numbers and shapes of faces:
| name | Qty faces | form each face |
| tetrahedron |
4 | triangle |
| hexahedron | 6 | square |
| octahedron | 8 | triangle |
| dodecahedron | 12 | pentagon |
| icosahedron | 20 | triangle |
Variety of shapes
On the basis of the five listed species, using skill and imagination, craftsmeneasy to design many different models out of paper.The polyhedron may be quite different from the above five figures, forming at the same time using different facets, such as squares and triangles.Thus obtained buoyancy body.And if one or more faces to miss, it will open a figure viewed as outside and inside.For the manufacture of solid models using special patterns cut from a dense, well keep in shape, paper.Make and special polyhedra out of paper.Schemes such products include the availability of additional acting modules.We examine ways to design a very beautiful figure as an example of a dodecahedron (photo 3).
How to make paper polyhedron with twelve peaks: the first method
This figure is called stellated dodecahedron.Each of its vertices at its base is a regular pentagon.Therefore, make two ways such polyhedra out of paper.Scheme for manufacturing will be somewhat different from each other.In the first case it is a single item (photo 4), which is obtained by folding the finished product.In addition to the main faces in the drawing there are connecting portions for bonding, by which the figure merges into a single whole.For the production of the second polyhedron way to do separate multiple patterns.Consider the process works in detail.
How to make a polyhedron made of paper: the second method
Fabricate two main template (photo 5):
- first. Draw a circle on a sheet and divide it across in two parts.One will be the basis for a pattern, a second arc immediately wipe off for convenience.Divide the piece into five equal parts and limit the radius of the cross all segments.These result in coupled together five identical isosceles triangles.Draw near adjacent to the middle segment of the exact same semicircle, but in mirror image.Receipt of the item when minimized looks like two cones.Make a similar pattern of only six pieces.For their gluing use the second part, which will be placed inside.
- second. This pattern - a five-pointed star.Follow the same twelve pieces.By forming the polyhedron, each of the stars from the bent-up ends placed inside the cone-shaped parts and glued to the faces.
complete collection figure is obtained by combining the double block additional segments of paper, setting the stage for them inside.Modeling products is problematic enough to make them different in size.Ready-made models of polyhedra of the paper is not so easy to increase.It is not enough simply to make allowances for all external borders.We need to scale separately each of the faces.The only way possible to get an enlarged copy of the original model.Using the second method of manufacturing a polyhedron, it makes it much easier, as it will be enough to increase the initial preform, which has carried out the desired number of individual parts.
