Inscribed sphere
Encyclopedia
In geometry
, the inscribed sphere or insphere of a convex polyhedron is a sphere
that is contained within the polyhedron and tangent
to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual
to the dual polyhedron
's circumsphere.
All regular polyhedra have inscribed spheres, but some irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes. For such cases, the notion of an insphere does not seem to have been properly defined and various interpretations of an insphere are to be found:
Often these spheres coincide, leading to confusion as to exactly what properties define the insphere for polyhedra where they do not coincide.
For example the regular small stellated dodecahedron has a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron. Which is the insphere? Important authorities such as Coxeter or Cundy & Rollett are clear enough that the face-tangent sphere is the insphere. Again, such authorities agree that the Archimedean polyhedra
(having regular faces and equivalent vertices) have no inspheres while the Archimedean dual or Catalan
polyhedra do have inspheres. But many authors fail to respect such distinctions and assume other definitions for the 'inspheres' of their polyhedra.
The radius of the sphere inscribed in a polyhedron P is called the inradius of P.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the inscribed sphere or insphere of a convex polyhedron is a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
that is contained within the polyhedron and tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual
Dual
Dual may refer to:* Dual , a notion of paired concepts that mirror one another** Dual , a formalization of mathematical duality** . . ...
to the dual polyhedron
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
's circumsphere.
All regular polyhedra have inscribed spheres, but some irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes. For such cases, the notion of an insphere does not seem to have been properly defined and various interpretations of an insphere are to be found:
- The sphere tangent to all faces (if one exists).
- The sphere tangent to all face planes (if one exists).
- The sphere tangent to a given set of faces (if one exists).
- The largest sphere that can fit inside the polyhedron.
Often these spheres coincide, leading to confusion as to exactly what properties define the insphere for polyhedra where they do not coincide.
For example the regular small stellated dodecahedron has a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron. Which is the insphere? Important authorities such as Coxeter or Cundy & Rollett are clear enough that the face-tangent sphere is the insphere. Again, such authorities agree that the Archimedean polyhedra
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
(having regular faces and equivalent vertices) have no inspheres while the Archimedean dual or Catalan
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
polyhedra do have inspheres. But many authors fail to respect such distinctions and assume other definitions for the 'inspheres' of their polyhedra.
The radius of the sphere inscribed in a polyhedron P is called the inradius of P.