Abstract
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.
Highlights
On the basis of a regular spherical polyhedron, 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]
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 Oо, the correct spherical twenty-triangle, and for the geometric network, a simplified scheme of figure 2 should be shown, where the arcs r are cut off on the edges and
The proposed cutting solution is the geometric basis of the geodesic dome formed by paired circles of the same radius
Summary
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 O, and the centers of the planes Oо and the radius of the edges of one radius as r. 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 Oо, 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 r are cut off on the edges and. New radius of the equator is shown for illustration purposes and is denoted as rо
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