Network from the circles of the same radius for sustainable energy-efficient roof structures

. In this research paper, we investigated one of the methods of formation of geometric networks of arches of the same radius using regular spherical polyhedra. The variants of cutting sustainable energy-efficient coatings of buildings in the form of spherical domes are proposed. The task conditions of placing the specified network on the sphere are set. The criterion for evaluating the effectiveness of solving the problem is the minimum number of standard sizes of segments of the dome arches, the possibility of using pre-assembly technologies. The solution of one variant of the problem as placing the network on a spherical icosahedron and, accordingly, on a sphere is given. The placement of arches of one radius on the sphere, different from the location in the form of meridians, has an effective solution in the form of a network with minimal dimensions of arch segments and with nodes of paired arches comprised on the basis of circles of the same radii formed on the ground of regular spherical polyhedra. The problem is solved by constructing and combining in a system of regular spherical polyhedral with independent frameworks of arches of the same radius on the basis of paired circles of equal radius .


Introduction
In a geodesic dome, on the basis of a regular spherical polyhedron (icosahedron), there is a possibility of such placement in the form of a network with the minimum sizes of arch segments and with pair arches unites comprised on the basis of circles of the same radii formed on the ground of the correct spherical polyhedra, with preservation of the minimum number of standard sizes which will provide an effective arrangement of support unites much lower or higher than the equator and at one, quite certain level [1][2][3][4][5][6][7][8][9][10][11][12][13]. In the investigated regular spherical polyhedron (Fig. 1), the vertices of the planes are denoted as , and the centers of the planes о and the radius of the edges of one radius as .
If we analyze the possibility of such a variant of the geometric network on the sphere, we conclude that if we choose as the poles for the construction the centers of the planes о , the correct spherical twenty-triangle (icosahedron), and for the geometric network (figure 1), a simplified scheme of figure 2 should be shown, where the arcs are cut off on the edges and * Corresponding author: asf.arch_i_dizain@mail.ru bisectors of the faces of the arc x and the arc , which determine the position of the main figures of the network. On the circles of the same radius, the arc is cut off, the arc x and the arc . New radius of the equator is shown for illustration purposes and is denoted as о .

Decision 1
The diagrams in figures 1 and 2 show the placement of paired circles of the same radius [1][2][3][4][5][6][7][8][9][10][11][12][13] on the basis of a spherical icosahedron. On the known location conditions of the centers of the circles in the centers of opposite planes, the problem of forming a geometric network on a sphere with the centers of unites located on the circles parallel to the equator circle (i.e., at one level) is reduced to solving a system of equations for spherical triangles shown in figure 1. We tentatively identify the parameters of a spherical face of a regular twentytriangle inscribed in a sphere as a spherical triangle. According to the problem, we take the interior angle at the Zenith considered the large triangle in figure1 О о = 20°, total arc arc tightens triangles in the polar angles equal to x and y. Interior angles of triangles around the vertices of the faces of the icosahedron, the subtending arc is equal to 108 0 and,respectively, with the sides x and is equal а = 108 0 and with sides and yа = 144 0 . Using the known Napier's expressions [14] for the sides and angles of right-angled spherical triangles, we obtain cos = cos cos + sin sin cos 108 0 .
10 0 = 0,5 . ( Let us transform the system of equations (1 and 2) using the formulas [14-22] We use the same formulas for the parameter . On the circles of the same radius, the arc, which tightens the x and arcs, is also cut off. The radius of the equator is shown for clarity and is denoted as .
The scheme of the figure 2 shows the placement of paired circles of the same radius [1][2][3][4][5][6][7][8][9] on the basis of a spherical icosahedron. Under condition of the known location of the centers of the circles in the centers of opposite faces, the problem of forming a geometric network on a sphere with the centers of nodes located on the circles parallel to the equator circle (ie, at one point), will be reduced to solving a system of equations for spherical triangles, shown in figure 1.
We predetermine the parameters of spherical triangles formed on a spherical face of a regular icosahedrons. To solve our problem by means of spherical trigonometry, we can determine the parameters of arch arcs cut off by large circles forming a regular polyhedron. Then according to the problem, we need to define a new received interior angle at the Zenith considered the large triangle in figure 2, the total arc arc also tightens triangles in the polar angles equal to and , ℎ is the height of this triangle.
Using the known Napier's expressions [14] for the sides and angles of right-angled spherical triangles, respectively, we obtain again a system of equations (1). Ie, in the end, we can use the unchanged equations (1)(2)(3)(4)(5) with partial replacement of the interior angles of the sectors that intercept arcs of the arched elements = 63,434966 0 . cos = cos cos + sin cos 108 0 .
Taking into account the substitutions of the formula (4) cos = cos 1− 2 1+ 2 cos + sin 2 1+ 2 cos 108 0 ; The resulting equation takes the same form as (5). Taking into account that the values of and , ℎ vary with the new value of , the simplified equation of the known radius is (cos + ) 2 + + + 2 108 0 + cos − = 0 .
Finding the inner angle centered on the pole , which draws the arc , which is equal to Similarly, we find the value of . The scheme of figure 4 shows the placement of circles of the same radius, which allows to create an effective design of the dome of two independent systems of frames in the form of the same arches [14][15][16][17][18][19][20][21].
The chaotic arrangement of the arches of two combined variants of the schemes in figure  4 actually contains two regular simple framework of paired arches.

Summary
The proposed cutting solution is the geometric basis of the geodesic dome formed by paired circles of the same radius. It allows us to greatly simplify the solutions of all the shell unites in which no more than four elements converge; including the support unites, since the base is made with nodes at the same level. These sustainable energy-efficient structures can be assembled from individual arches, and the technology of integration assemblies of the dome elements can also be used.