A polyhedron is a body whose boundary is the union of a finite number of polygons.
\nThe first mention of polyhedra is known as early as three thousand years BC in Egypt and Babylon.
\nBut the theory of polyhedra is also a modern branch of mathematics. It is closely related to
\ntopology, graph theory, and is of great importance both for theoretical research on geometry and for
\npractical applications in other branches of mathematics, for example, in algebra, number theory,
\napplied mathematics – linear programming, optimal control theory.
\nPolyhedra have beautiful shapes, for example, regular, semi-regular and stellate polyhedra. They
\nhave a rich history, which is associated with the names of such scientists as Pythagoras, Euclid, and
\nArchimedes. Polyhedra are distinguished by unusual properties, the most striking of which is
\nformulated in Euler’s theorem on the number of faces, vertices and edges of a convex polyhedron:
\nfor any convex polyhedron, the ratio G + B-P = 2 is valid, where G is the number of faces, B is the
\nnumber of vertices, P is the number of edges of this polyhedron. Historians of mathematics call
\nEuler’s theorem the first theorem of topology, a major branch of modern mathematics.
\nSince ancient times, our ideas about beauty have been connected with symmetry. Perhaps this
\nexplains the human interest in polyhedra – amazing symbols of symmetry that attracted the attention
\nof outstanding thinkers.
\nThe history of regular polyhedra goes back to ancient times. Pythagoras and his disciples were
\nregular polyhedra. They were struck by the beauty, perfection, harmony of these figures. The
\nPythagoreans considered regular polyhedra to be divine figures and used them in their philosophical
\nwritings: the fundamental principles of existence – fire, earth, air, water – were given the shape of a
\ntetrahedron, cube, octahedron, icosahedron, respectively, and the whole universe had the shape of a
\ndodecahedron. Later, the Pythagorean doctrine of regular polyhedra was expounded in his writings
\nby another ancient Greek scientist, the idealist philosopher Plato. Since then, regular polyhedra have
\nbeen called Platonic solids.<\/p>\n
A regular polyhedron is a polyhedron whose all faces are regular equal polygons, and all dihedral
\nangles are equal. But there are also such polyhedra, in which all polyhedral angles are equal, and the
\nfaces are regular, but dissimilar regular polygons. Polyhedra of this type are called equiangularsemiregular polyhedra. For the first time, polyhedra of this type were discovered by Archimedes.
\nHe described in detail 13 polyhedra, which were later named Archimedes bodies in honor of the
\ngreat scientist.
\nThese are truncated tetrahedron, truncated oxahedron, truncated icosahedron, truncated cube,
\ntruncated dodecahedron, cuboctahedron, icosododecahedron, truncated cuboctahedron, truncated
\nicosododecahedron, rhombocuboctahedron, rhomboicosododecahedron, “flat-nosed” (snub-nosed)
\ncube, “flat-nosed\u201d (snub-nosed) dodecahedron.
\nIn addition to semi-regular polyhedra from regular polyhedra – Platonic solids, it is possible to
\nobtain so-called regular stellate polyhedra. There are only four of them; they are also called KeplerPoinsot bodies. Kepler discovered the small dodecahedron, which he called the prickly, or
\nhedgehog, and the large dodecahedron. Poinsot discovered two other regular stellate polyhedra,
\ndual respectively to the first two: the large stellate dodecahedron and the large icosahedron.
\nA pyramid is a body formed by a flat polygon (base), a point that does not lie in the plane of this
\npolygon (vertex), and all segments connecting the base points to the vertex. The sweep of the
\nsurface of an irregular pyramid will consist of irregular triangles of the lateral surface and an
\nirregular triangle lying at the base, combined in one plane, and their mutual arrangement on the
\nsweep should correspond to their mutual arrangement on orthogonal projections. Since the sides of
\nthe base of an irregular pyramid are different and the edges of the side surface are not equal to each
\nother, first we find the full size of all the side edges. To do this, we use one of the ways to
\ndetermine the full size of a straight line segment of the general position. In this case, the rotation
\nmethod is used. We rotate the side edges around the axis drawn through the vertex of the pyramid S
\nperpendicular to the plane H.
\nA conical surface is formed by the movement of a rectilinear generatrix along a curved guide. In
\nthis case, the generatrix passes through some fixed point S, which is called a vertex.
\nA cylindrical surface is formed by the movement of a rectilinear generatrix parallel to a given
\nstraight line 1 along a curved guide. Point N belongs to these surfaces, since it belongs to the
\ngeneratrix \/ of these surfaces.<\/p>","protected":false},"excerpt":{"rendered":"
A polyhedron is a body whose boundary is the union of a finite number of polygons. The first mention of polyhedra is known as early as three thousand years BC in Egypt and Babylon. But the theory of polyhedra is also a modern branch of mathematics. It is closely related to topology, graph theory, and […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[145],"tags":[],"sdg_goal":[],"class_list":["post-39469","post","type-post","status-publish","format-standard","hentry","category-ommaviy_maqolalar"],"acf":[],"views":1051,"_links":{"self":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts\/39469","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/comments?post=39469"}],"version-history":[{"count":1,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts\/39469\/revisions"}],"predecessor-version":[{"id":39470,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts\/39469\/revisions\/39470"}],"wp:attachment":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/media?parent=39469"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/categories?post=39469"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/tags?post=39469"},{"taxonomy":"sdg_goal","embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/sdg_goal?post=39469"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}