Orthogonal polyhedron
An orthogonal polyhedron is a polyhedron in which all edges are parallel to the axes of a Cartesian coordinate system,[1] 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.[2] Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes and are three-dimensional analogs of planar polyominoes.[3] Orthogonal polyhedra can be either convex (such as rectangular cuboids) or non-convex.[2][4]
Orthogonal polyhedra were used in (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.[5][2] 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.[6]
The simple orthogonal polyhedra, as defined by (Eppstein Mumford), are the three-dimensional polyhedra such that three mutually perpendicular edges meet at each vertex and that have the topology of a sphere.[4] 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.[7]
References
- ↑ Senechal, Marjorie, ed. (2013), "Dürer's Problem", Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 86, doi:10.1007/978-0-387-92714-5, ISBN 978-0-387-92714-5, https://books.google.com/books?id=kZtCAAAAQBAJ&pg=PA86
- ↑ 2.0 2.1 2.2 "Orthogonal icosahedra", Nordisk Matematisk Tidskrift 15 (2): 90–96, 1967.
- ↑ "Mathematical Games: Is it possible to visualize a four-dimensional figure?", Scientific American 215 (5): 138–143, November 1966, doi:10.1038/scientificamerican1166-138
- ↑ 4.0 4.1 "Stenitz theorems for simple orthogonal polyhedra", Journal of Computational Geometry 5 (1): 179–244, 2014, https://jocg.org/index.php/jocg/article/view/2932.
- ↑ "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions" (in fr), Commentarii Mathematici Helvetici 40: 43–80, 1965, doi:10.1007/bf02564364, https://eudml.org/doc/139296
- ↑ "Unfolding orthogonal polyhedra", Surveys on discrete and computational geometry, Contemp. Math., 453, Providence, Rhode Island: American Mathematical Society, 2008, pp. 307–317, doi:10.1090/conm/453/08805, ISBN 978-0-8218-4239-3.
- ↑ Christ, Tobias; Hoffmann, Michael (August 10–12, 2011), "Wireless Localization within Orthogonal Polyhedra", 23d Canadian Conference on Computational Geometry, 2011, pp. 467–472, https://cccg.ca/proceedings/2011/papers/paper18.pdf.
Further reading
- Biedl, Therese; Genç, Burkay (2011), "Stoker's Theorem for Orthogonal Polyhedra", International Journal of Computational Geometry & Applications 21 (4): 383-391, doi:10.1142/S0218195911003718
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