# Circumscribed sphere

In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices.[1] The word circumsphere is sometimes used to mean the same thing, by analogy with the term circumcircle.[2] As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed around a polyhedron P is called the circumradius of P,[3] and the center point of this sphere is called the circumcenter of P.[4]

## Existence and optimality

When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of the vertices of the polyhedron.[5]

In De solidorum elementis (circa 1630), René Descartes observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for the existence of a circumscribed sphere is sufficient, but it is not true: some bipyramids, for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever a simple polyhedron has a circumscribed circle for each of its faces, it also has a circumscribed sphere.[6]

The circumscribed sphere is the three-dimensional analogue of the circumscribed circle. All regular polyhedra have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all vertices lie on a common sphere. The circumscribed sphere (when it exists) is an example of a bounding sphere, a sphere that contains a given shape. It is possible to define the smallest bounding sphere for any polyhedron, and compute it in linear time.[5]

Other spheres defined for some but not all polyhedra include a midsphere, a sphere tangent to all edges of a polyhedron, and an inscribed sphere, a sphere tangent to all faces of a polyhedron. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric.[7]

When the circumscribed sphere is the set of infinite limiting points of hyperbolic space, a polyhedron that it circumscribes is known as an ideal polyhedron.

## Point on the circumscribed sphere

There are five convex regular polyhedra, known as the Platonic solids. All Platonic solids have circumscribed spheres. For an arbitrary point ${\displaystyle M}$ on the circumscribed sphere of each Platonic solid with number of the vertices ${\displaystyle n}$, if ${\displaystyle MA_{i}}$ are the distances to the vertices ${\displaystyle A_{i}}$, then [8]

${\displaystyle 4(MA_{1}^{2}+MA_{2}^{2}+...+MA_{n}^{2})^{2}=3n(MA_{1}^{4}+MA_{2}^{4}+...+MA_{n}^{4}).}$

## References

1. ^ James, R. C. (1992), The Mathematics Dictionary, Springer, p. 62, ISBN 9780412990410.
2. ^ Popko, Edward S. (2012), Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, CRC Press, p. 144, ISBN 9781466504295.
3. ^ Smith, James T. (2011), Methods of Geometry, John Wiley & Sons, p. 419, ISBN 9781118031032.
4. ^ Altshiller-Court, Nathan (1964), Modern pure solid geometry (2nd ed.), Chelsea Pub. Co., p. 57.
5. ^ a b Fischer, Kaspar; Gärtner, Bernd; Kutz, Martin (2003), "Fast smallest-enclosing-ball computation in high dimensions", Algorithms - ESA 2003: 11th Annual European Symposium, Budapest, Hungary, September 16-19, 2003, Proceedings (PDF), Lecture Notes in Computer Science, vol. 2832, Springer, pp. 630–641, doi:10.1007/978-3-540-39658-1_57, ISBN 978-3-540-20064-2.
6. ^ Federico, Pasquale Joseph (1982), Descartes on Polyhedra: A Study of the "De solidorum elementis", Sources in the History of Mathematics and Physical Sciences, vol. 4, Springer, pp. 52–53
7. ^ Coxeter, H. S. M. (1973), "2.1 Regular polyhedra; 2.2 Reciprocation", Regular Polytopes (3rd ed.), Dover, pp. 16–17, ISBN 0-486-61480-8.
8. ^ Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340. doi:10.26713/cma.v11i3.1420 (inactive 31 January 2024).`{{cite journal}}`: CS1 maint: DOI inactive as of January 2024 (link)