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Canonical Polyhedron


A polyhedron is said to be canonical if all its polyhedron edges touch a sphere and the center of gravity of their contact points is the center of that sphere. In other words, a canonical polyhedron is a polyhedron possessing a midsphere.

A dual polyhedron can be constructed from a canonical polyhedron that possesses these properties as well. Moreover, the edges of the canonical polyhedron and its dual cross at right angles.

Amazingly, there is a unique canonical version (modulo rotations and reflections) for each combinatorial type of (genus zero) convex polyhedron (Schramm 1992; Ziegler 1995, pp. 117-118; Springborn 2005). Many symmetric polyhedra are canonical in their "natural" form, including the Platonic solids as well as the Archimedean solids and their duals. Equilateral antiprisms and prisms are also canonical.


See also

Dual Polyhedron, Midsphere, Reciprocation

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References

Hart, G. W. "Calculating Canonical Polyhedra." Mathematica Educ. Res. 6, 5-10, Summer 1997.Hart, G. "Calculating Canonical Polyhedra." http://www.georgehart.com/canonical/canonical-supplement.html.Hart, G. "Canonical Polyhedra." http://www.georgehart.com/virtual-polyhedra/canonical.html.Sachs, H. "Coin Graphs, Polyhedra, and Conformal Mapping." Disc. Math. 134, 133-138, 1994.Schramm, O. "How to Cage an Egg." Invent. Math. 107, 543-560, 1992.Springborn, B. A. "A Unique Representation Theorem of Polyhedral Types: Centering Via Möbius Transformations." Math. Zeit. 249, 513-517, 2005.Wolfram Research Staff, based on content by George W. Hart. Wolfram Function Repository: PolyhedronCanonicalForm. https://resources.wolframcloud.com/FunctionRepository/resources/PolyhedronCanonicalForm/.Ziegler, G. M. Lectures on Polytopes. New York: Springer-Verlag, 1995.

Referenced on Wolfram|Alpha

Canonical Polyhedron

Cite this as:

Weisstein, Eric W. "Canonical Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CanonicalPolyhedron.html

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