Orthogonal polyhedron

Nowadays, Orthogonal polyhedron is a topic that has captured the attention of many people around the world. With the advancement of technology and globalization, Orthogonal polyhedron has become an issue of universal relevance that impacts different sectors of society. Whether on a personal, professional or social level, Orthogonal polyhedron has generated widespread debate and has sparked the interest of experts and fans alike. In this article, we will thoroughly explore the impact of Orthogonal polyhedron and discuss its implications on our daily lives. From its origins to its current evolution, Orthogonal polyhedron represents a relevant phenomenon that deserves to be understood in its entirety.

An example of 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 (2014), 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

  1. ^ O'Rourke, Joseph, "Dürer's Problem", in Senechal, Marjorie (ed.), 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
  2. ^ a b c Jessen, Børge (1967), "Orthogonal icosahedra", Nordisk Matematisk Tidskrift, 15 (2): 90–96, JSTOR 24524998, MR 0226494.
  3. ^ Gardner, Martin (November 1966), "Mathematical Games: Is it possible to visualize a four-dimensional figure?", Scientific American, 215 (5): 138–143, doi:10.1038/scientificamerican1166-138, JSTOR 24931332
  4. ^ a b Eppstein, David; Mumford, Elena (2014), "Stenitz theorems for simple orthogonal polyhedra", Journal of Computational Geometry, 5 (1): 179–244.
  5. ^ Sydler, J.-P. (1965), "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions", Commentarii Mathematici Helvetici (in French), 40: 43–80, doi:10.1007/bf02564364, MR 0192407, S2CID 123317371
  6. ^ O'Rourke, Joseph (2008), "Unfolding orthogonal polyhedra", Surveys on discrete and computational geometry, Contemp. Math., vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 307–317, doi:10.1090/conm/453/08805, ISBN 978-0-8218-4239-3, MR 2405687.
  7. ^ Christ, Tobias; Hoffmann, Michael (August 10–12, 2011), "Wireless Localization within Orthogonal Polyhedra" (PDF), 23d Canadian Conference on Computational Geometry, 2011 (PDF), pp. 467–472.

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