In geometry, polyhedra are three-dimensional objects where points are connected by lines to form polygons.
The points, lines, and polygons of a polyhedron are referred to as its vertices, edges, and faces, respectively.
[1] A polyhedron is considered to be convex if:[2] A convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid[3].
Some authors exclude uniform polyhedra from the definition.
A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal; examples include Platonic and Archimedean solids as well as prisms and antiprisms.
His conjecture that the list was complete and no other examples existed was proven by Russian-Israeli mathematician Victor Zalgaller (1920–2020) in 1969.
[5] Some of the Johnson solids may be categorized as elementary polyhedra, meaning they cannot be separated by a plane to create two small convex polyhedra with regular faces.
The criteria is also satisfied by eleven other Johnson solids, specifically the tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda.
Augmentation involves attaching the Johnson solids onto one or more faces of polyhedra, while elongation or gyroelongation involve joining them onto the bases of a prism or antiprism, respectively.
Some others are constructed by diminishment, the removal of one of the first six solids from one or more of a polyhedron's faces.
[7] The following table contains the 92 Johnson solids, with edge length
The table includes the solid's enumeration (denoted as
[8] It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area
Every polyhedron has its own characteristics, including symmetry and measurement.
preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.
[11] The mensuration of polyhedra includes the surface area and volume.
[12] A volume is a measurement of a region in three-dimensional space.
[13] The volume of a polyhedron may be ascertained in different ways: either through its base and height (like for pyramids and prisms), by slicing it off into pieces and summing their individual volumes, or by finding the root of a polynomial representing the polyhedron.