To the problem 5 of emplacement of triangular geometric net on the sphere

The sphere creates the minimal surface of enclosing structures and has unique resource saving qualities which makes it indispensable in the construction of "smart buildings». One of the methods of formation of triangular networks in the spherewas investigated. Conditions of the problem of locating a triangular network in the area were established. The evaluation criterion of solution effectiveness of the problem is the minimum number of type-sizes of dome panels, the possibility of pre-assembly and prestressing. The solution of the problem of the triangular network emplacement in a compatible spherical triangle on the sphere variant was provided. The problem of the emplacement of regular and irregular hexagons on the sphere, inscribed in a circles, i.e., flat figures or composed ones of spherical triangles with minimum dimensions of the ribs, has an effective solution in the form of a network, formed on the basis of minimum radii circles, i.e., circles on a sphere obtained by the touch of three adjacent circles whose centers are at the shortest distance from each other. The optimization of triangular geometric network on a sphere on the criterion of minimum sizes of elements can be solved by emplacementinthe system the irregular hexagons inscribed in circles of minimal sizes, the maximum of regular hexagons.


Introduction
The sphere creates the minimal surface of enclosing structures and has unique resource saving qualities, which makes it indispensable in the construction of «smart buildings». The problem of the emplacement of regular and irregular hexagons on the sphere, inscribed in a circles, i.e., flat figures or composed ones of spherical triangles (See Fig. 1) with minimum dimensions of the ribs, has an effective solution in the form of a network, formed on the basis of minimum radii circles, i.e., circles on a sphere obtained by the touch of three adjacent circles whose centers are at the shortest distance from each other. [1][2][3][5][6][7][8]. The emplacement of irregular and regular hexagons inscribed in a circle, will be implemented at the example of cutting in the form of a 2000-poluhedron (Fig. 1).
For the solution of the problem (Fig. 2) a spherical triangle of ABC will be considered, and also a spherical rectangular triangle of O 2 NO 11 with a leg arch, equal to total radius ρ 1 and ρ 2 . There is a rectangular spherical triangle with internal corners of 90 and 60 degrees and a leg arch, equal ܽ.It is required to determine sizes of radii ρ 1 and ρ 2 between the O 11 , O 12 and O 2 centers of circles and, thus, position of all centers of circles of the first two rows of hexagons will be defined.
For definition of the centers of hexagons of equal radii some additional requirements to an arrangement of three circles will be entered. First, setting that the circle, with the center in a point of O 11 crosses the party arch perpendicularly and a bisector of O 2 of an isosceles triangle ofO 11 O 2 O 12 crosses the party arch of O 11 O 12 in two equal parts. Previously it is needed to determine parameters into this spherical triangle of ABC. It was set that internal corners of a triangle B=60 °, C=90°, СO 12 Using known expressions of the parts of a rectangular spherical triangle through tangents of legs and a hypotenuse, the hexagons will be received.
It is needed to enter the designation of O 11 C = x. Then O 11 B = а-x, from a triangle of O 11 BА connection of size x and required radius ρ 1 will be determined: From a triangle of O 11 O 12 C also the interrelation of sizes x and ρ 1 will be defined: 2 cos 2 ρ 1 = cos ρ 1 cos ‫ݔ‬ + 1.

Remark
It should be noted that the condition of a contact of two circles with the centers inO 11 points O 2 is respectively possible, perhaps in the case, if the angle O 2 O 11 B isa right angle. For finding of radius ρ 2 second circles a triangle of МO 2 O 11 will be considered. Application of the theorem of cosines gives the following ratio.

Conclusions
These solutions allow realising the algorithms of approximation the triangular geometric network with the maximum number of regular hexagons and preparing the options for optimization of sphere cutting.