## Table of contents:

- History of figures
- Making a twentyhedron
- Regular polyhedra
- Variety of shapes
- How to make a polyhedron with twelve vertices out of paper: the first way
- How to make a paper polyhedron: the second way

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

## 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 same type of faces, each of which has three vertices. The earth is symbolized by a hexahedron of squares. And the dodecahedron, thanks to the special pentagonal faces, plays a decorative role and is the prototype of harmony and peace.

It is also known that one of the Greek mathematicians, Euclid, proved in his doctrine of the "Beginnings" the uniqueness of the mentioned Platonic solids and their property to "fit" into the sphere (photo 2). The polyhedron shown from paper is made by folding twenty isosceles triangles closed together. The diagram clearly demonstrates a pattern for making a figure. Let's take a closer look at all the stages of creating an icosahedron.

## Making a twentyhedron

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

## Regular polyhedra

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

Name |
Number of faces |
Shape of each face |

tetrahedron |
4 |
triangle |

hexahedron |
6 |
square |

octahedron |
8 |
triangle |

dodecahedron |
12 |
pentagon |

icosahedron |
20 |
triangle |

## Variety of shapes

Based on the five types given, using skill and imagination, craftsmen can easily design many different paper models. A polyhedron can be completely different from the five figures described above, being formed simultaneously from faces of different shapes, for example, from squares and triangles. This is how Archimedean solids are obtained. And if you skip one or more faces, you get an open figure, viewed both from the outside and inside. For the manufacture of three-dimensional models, special patterns are used, cut out of fairly dense, well-shaped paper. They also make special polyhedrons from paper. Schemes of such products providethe presence of additional, protruding modules. Let's look at ways to construct a very beautiful figure using the dodecahedron as an example (photo 3).

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

Such a figure is also called a stellated dodecahedron. Each of its vertices is a regular pentagon at its base. Therefore, such paper polyhedra are made in two ways. Schemes for manufacturing will be slightly different from each other. In the first case, this is a single part (photo 4), as a result of which the finished product is folded. In addition to the main faces, the drawing contains connecting parts for gluing, thanks to which the figure closes into a single whole. To make a polyhedron in the second way, you need to make several templates separately. Let's consider the work process in more detail.

## How to make a paper polyhedron: the second way

Make two main templates (Pic 5):

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

- Second. This pattern is a five-pointed star. Perform the same twelve blanks. Forming a polyhedron, each of the stars with the ends bent up is placed inside the cone-shaped parts and glued to the edges.

The complete assembly of the figure is obtained by connecting double blocks with additional pieces of paper, turning them inward. Modeling products, it is quite problematic to make them different in size. Ready-made models of paper polyhedra are not so easy to enlarge. To do this, it is not enough just to make allowances for all external borders. You need to scale each of the faces separately. This is the only way 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 blanks, on which the required number of individual parts are already being made.