orthogonal polyhedron
{{Short description|Polyhedron in which all edges are parallel}}
File:Orthogonal polyhedron with no vertex visible from center.svg
An orthogonal polyhedron is a polyhedron in which all edges are parallel to the axes of a Cartesian coordinate system,{{r|o'rourke-2013}} resulting in the orthogonal faces and implying the dihedral angle between faces are right angles. The angle between Jessen's icosahedron's faces is right, but the edges are not axis-parallel, which is not an orthogonal polyhedron.{{r|jessen}} Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes and are three-dimensional analogs of planar polyominoes.{{r|gardner}} Orthogonal polyhedra can be either convex (such as rectangular cuboids) or non-convex.{{r|jessen|em}}
Orthogonal polyhedra were used in {{harvtxt|Sydler|1965}} in which he showed that any polyhedron is equivalent to a cube: it can be decomposed into pieces which later can be used to construct a cube. This showed the requirements for the polyhedral equivalence conditions by Dehn invariant.{{r|sydler|jessen}} Orthogonal polyhedra may also be used in computational geometry, where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.{{r|o'rourke-2008}}
The simple orthogonal polyhedra, as defined by {{harvtxt|Eppstein|Mumford|2014}}, are the three-dimensional polyhedra such that three mutually perpendicular edges meet at each vertex and that have the topology of a sphere.{{r|em}} By using Steinitz's theorem, there are three different classes: the arbitrary orthogonal polyhedron, the skeleton of its polyhedron drawn with hidden vertex by the isometric projection, and the polyhedron wherein each axis-parallel line through a vertex contains other vertices. All of these are polyhedral graphs that are cubic and bipartite.{{r|ch}}
References
{{reflist|refs=
| last1 = Christ | first1 = Tobias
| last2 = Hoffmann | first2 = Michael
| date = August 10–12, 2011
| title = 23d Canadian Conference on Computational Geometry, 2011
| url = https://2011.cccg.ca/proceedings/main.pdf
| contribution = Wireless Localization within Orthogonal Polyhedra
| contribution-url = https://cccg.ca/proceedings/2011/papers/paper18.pdf
| pages = 467–472
}}.
| last1 = Eppstein | first1 = David | author-link1 = David A. Eppstein
| last2 = Mumford | first2 = Elena
| year = 2014
| title = Stenitz theorems for simple orthogonal polyhedra
| journal = Journal of Computational Geometry
| url = https://jocg.org/index.php/jocg/article/view/2932
| volume = 5 | issue = 1
| pages = 179–244
}}.
| last = Gardner | first = Martin | author-link = Martin Gardner
| date = November 1966
| issue = 5
| journal = Scientific American
| jstor = 24931332
| pages = 138–143
| title = Mathematical Games: Is it possible to visualize a four-dimensional figure?
| volume = 215
| doi = 10.1038/scientificamerican1166-138
}}
| last = Jessen | first = Børge | author-link = Børge Jessen
| issue = 2
| journal = Nordisk Matematisk Tidskrift
| jstor = 24524998
| mr = 0226494
| pages = 90–96
| title = Orthogonal icosahedra
| volume = 15
| year = 1967
}}.
| last = O'Rourke | first = Joseph | author-link = Joseph O'Rourke (professor)
| contribution = Unfolding orthogonal polyhedra
| doi = 10.1090/conm/453/08805
| mr = 2405687
| pages = 307–317
| publisher = American Mathematical Society
| location = Providence, Rhode Island
| series = Contemp. Math.
| title = Surveys on discrete and computational geometry
| volume = 453
| year = 2008
| isbn = 978-0-8218-4239-3
| doi-access = free
}}.
| last = O'Rourke | first = Joseph | author-link = Joseph O'Rourke (professor)
| editor-last = Senechal | editor-first = Marjorie
| title = Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination
| contribution = Dürer's Problem
| contribution-url = https://books.google.com/books?id=kZtCAAAAQBAJ&pg=PA86
| page = 86
| publisher = Springer
| isbn = 978-0-387-92714-5
| doi = 10.1007/978-0-387-92714-5
}}
| last = Sydler | first = J.-P. | author-link = Jean-Pierre Sydler
| title = Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions
| journal = Commentarii Mathematici Helvetici
| language = fr
| volume = 40 | year = 1965 | pages = 43–80 | doi = 10.1007/bf02564364
| mr = 0192407
| s2cid = 123317371
| url = https://eudml.org/doc/139296
}}
}}
Further reading
- {{citation
| last1 = Biedl | first1 = Therese
| last2 = Genç | first2 = Burkay
| journal = International Journal of Computational Geometry & Applications
| volume = 21 | issue = 4 | pages = 383-391
| year = 2011
| title = Stoker's Theorem for Orthogonal Polyhedra
| doi = 10.1142/S0218195911003718
}}