Polyhedron - tamentai

Japanese: 多面体 - ためんたい

A polyhedron is a solid body surrounded by a finite number of polygons, such as a prism or pyramid, whose interior is connected in the same way as the interior of a sphere. Its surfaces are connected in the same way as the surfaces of a sphere. A polyhedron is called a convex polyhedron when a line segment connecting two points inside it always belongs to the polyhedron. A convex polyhedron is a common part of a half space bounded by planes that contain each face. The polygons that surround a polyhedron, their edges, and vertices are called the faces, edges, and vertices of the polyhedron, respectively, and a polyhedron with n faces is called an n-hedron. A tetrahedron is the same as a triangular pyramid, but in addition to quadrangular pyramids, there are also pentahedrons that are made by cutting a triangular prism or a triangular pyramid with a single plane.

[Minoru Kurita]

Euler's polyhedron theorem

For a polyhedron, when the number of vertices is v, the number of edges is e, and the number of faces is f, v-e+f=2 is always true. This is called Euler's theorem. For a solid body surrounded by a finite number of polygons, with a hollowed-out interior like a torus, v-e+f=0 is true.

Generally, for a solid body surrounded by a finite number of polygons, v-e+f is a number determined by the degree of connectivity within the solid body, and is called Euler's characteristic.

[Minoru Kurita]

[Reference] | Regular polyhedron

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

角柱や角錐(かくすい)のように有限個の多角形で囲まれた立体で、その内部が球の内部と同じようなつながりぐあいになっているものを多面体という。その表面は球面と同じようなつながりぐあいをしている。多面体で、その内部の2点を結ぶ線分がつねにこの多面体に属するとき、この多面体を凸多面体という。凸多面体は、各面を含む平面を境とする半空間の共通部分になっている。多面体を囲む多角形、その辺、頂点を、それぞれ、多面体の面、辺、頂点といい、面の数がnの多面体をn面体という。四面体は三角錐と同じであるが、五面体には四角錐のほかに、三角柱や、三角錐を一つの平面で切ってできるものがある。

［栗田　稔］