# Geometric textbook

**Instructions for working with transformable models on stereometry and planimetry '' Nane ''**

- Composition and purpose
- A means for the models
- Obtaining of the main basic models and their modifications
- The structure of models

* *

*Composition and purpose*

The educational geometric transformable manual "Nane" is a set of six geometric figures to study stereometry.

- Triangularpyramid
- Quadrangular pyramid
- Triangular prism
- Cube
- Cone
- Cylinder
- Six additional rods for different constructions and sections.

''Нанэ'' is intended for increase of learning efficiency of the school course geometry and for development of spatial thinking of pupils. It is reached by visual demonstration of various standard and non-standard geometrical figures. The game character which is the cornerstone of the models functioning allows considerably increasing the efficiency of their use.

*A means for the models*

Each of the sides of each pattern consists of three rods forming the edge and entering each other on the principle of telescopic antenna. Quality material combined with this device allows the ribs with sufficient ease and clarity to make the necessary manipulations to change the size of the parties. Places docking edges are at the top and bottom geometric bodies connected by the original ring that, in fact, with an unlimited degree of freedom allows you to rotate the rods by changing the lengths of the sides. Such moving parts can be obtained from a model of different new models and figures, both standard and non-standard. It should also be noted that the removable nature of the node (vertex) parts allows changing the number of sides at each vertex of the original figure, to open the way to search for new geometric figures.

**Obtaining the main models and their modifications**

**Triangular****pyramid**

From this model it is necessary to obtain all types of a triangular pyramid. Before starting working with the models it is necessary to be aware that for manipulations and transformations it is needed to work consistently only with each side separately. For the implementation of each of the following transformations it is needed to close consistently each of the sides up to the stop.

SB⊥(ABC)

AC⊥CS follows AC⊥BC и AC⊥BC follows AC⊥CS

With this type of model it is possible to solve numerous tasks. From this model it is possible to receive all types of quadrangles and a triangle for the construction of which it is necessary to make the following: all sides of the basis of a pyramid A, B and C should be pulled, and the edges SA, SB and SC should be decreased in length until they all fit in the plane of the triangle ABC.

*Quadrangular pyramid*

This model allows receiving of all types of the quadrangular pyramid and pentagon. We remind that the transformations are carried out with the help of change of each of the sides separately. Of particular interest is the case when the basement of ABDC quadrate and the lateral edge are perpendicular to the plane of the basement. Then according to the theorem of three perpendiculars, we receive: AC⊥CD from what it follows AC⊥SC.

It is possible to receive a triangular pyramid from a quadrangular pyramid as follows. All sides of a triangle ABD we extend up to the end, then we hold the top C and shortening SC gradually we put it inside (i.e. top of C) of the triangular pyramid ABDS until SC becomes ⊥ (ABD) and C ∈ (ABD). There is also the second option of transformation. The top of C ∈ (SAD). As a result we obtain a triangular pyramid ABDS (Fig. 4) in which it is possible to construct the height BC with the help of an auxiliary rod.

In the same way it is possible to obtain a triangular pyramid from the quadrangular pyramid. Shortening AD, we do so that the sides of DB and DC made one line to the line BC. It is clear that turning this pyramid, different types of triangular pyramids are created, which are used in the solutions of tasks.

For implementation of each of the following transformations it is necessary to close consistently each of the sides of the model up to the stop. The quadrangular pyramid turns into a regular pentagon with its diagonals.

**Triangular****prism**

We get this model from the box; it is in a set separately from all other models. We close all sides. Then we pull A1BA1C, and we obtain a regular triangular prism.

Opening the AB, AC and BC we obtain the truncated pyramid. Further, extending BB1 until the B_{1}A_{1}, A, B_{1}, C_{1}, C become straight lines, we obtain a triangular pyramid ABCB1, where different sections of a triangular pyramid are visible. This model is of interest that the simple turning from side to side generates more and more new pyramids with various sections.

We hold from top B1 and pull in order to obtain a pentagonal pyramid. All other sides are B_{1}A, B_{1}B, B_{1}C_{1}, B_{1}C_{1} and B_{1}A_{1}.

We increase the sides of the quadrangle ABCA_{1}, turning it into quadrate. We compress the C_{1}B_{1} till the end and move upwards and inwards the top C_{1} and bring it up to the osculation with CA. As a result we obtain a tetragonal pyramid with the height B_{1}C_{1}. For receiving a pentagonal pyramid again we close all sides.

The triangular prism turns into the regular hexagon, and you can obtain it yourself

To obtain an octahedron from this model (regular octahedron), we do the following. Strongly holding from the top C and twisting CA_{1} midstream completely separating from the top C. In the same way we separate CA_{1} from A_{1}. Then CA_{1} is strengthened in tops A and C1. Closing all sides, we obtain an octahedron.

*Cylinder.*

For creation of this model at first we fix those rods, which have no metal pins. We fix them against each other.

By means of separately taken circle and the diameter thereof is possible to construct an inscribed triangle, quadrangle.

**Creation of sections**

The auxiliary rods are included in the kitfor sections construction, which allow to obtain any sections. It is easy to be convinced, if we notice that the rods can be fixed in any combinations. In a triangular pyramid by means of auxiliary rods, as shown in figure, we demonstrate that section which represents a quadrangle. By the way, it is possible to construct with their help the height, the median, the bisector, etc. Auxiliary rods can also serve for replacement of the broken rods models.