Editing Polyhedron (section)

=== Curved polyhedra ===
Some fields of study allow polyhedra to have curved faces and edges. Curved faces can allow [[digon]]al faces to exist with a positive area.
* When the surface of a sphere is divided by finitely many [[great arc]]s (equivalently, by planes passing through the center of the sphere), the result is called a [[spherical polyhedron]]. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron.  However, the reverse process is not always possible; some spherical polyhedra (such as the [[hosohedron|hosohedra]]) have no flat-faced analogue.<ref>{{citation|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|first=Edward S.|last=Popko|publisher=CRC Press|year=2012|isbn=978-1-4665-0430-1|page=463|url=https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA463|quote="A hosohedron is only possible on a sphere"}}.</ref>
* If faces are allowed to be concave as well as convex, adjacent faces may be made to meet together with no gap. Some of these curved polyhedra can pack together to fill space. Two important types are bubbles in froths and foams such as [[Weaire–Phelan structure|Weaire-Phelan bubbles]],<ref>{{citation
 | last1 = Kraynik | first1 = A.M.
 | last2 = Reinelt | first2 = D.A.
 | editor-last = Mortensen | editor-first = Andreas
 | contribution = Foams, Microrheology of
 | edition = 2nd
 | pages = 402–407
 | publisher = Elsevier
 | title = Concise Encyclopedia of Composite Materials
 | year = 2007}}. See in particular [https://books.google.com/books?id=zs_lGeGsuaAC&pg=PA403 p.&nbsp;403]: "foams consist of polyhedral gas bubbles ... each face on a polyhedron is a minimal surface with uniform mean curvature ... no face can be a flat polygon with straight edges".</ref> and forms used in architecture.<ref>{{citation|last=Pearce|first=P.|title=Structure in nature is a strategy for design|publisher=MIT Press|year=1978|page=224|url=https://books.google.com/books?id=sfc2OEuE8oQC&pg=PA224|contribution=14 Saddle polyhedra and continuous surfaces as environmental structures|isbn=978-0-262-66045-7}}.</ref>
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