# On Some Regular Toroids

- Published in 2001
- Added on
2018-07-26

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As it is known, in a regular polyhedron every face has the same number of edges and every vertex has the same number of edges, as well. A polyhedron is called topologically regular if further conditions (e.g. on the angle of the faces or the edges) are not imposed. An ordinary polyhedron is called a toroid if it is topologically torus-like (i.e. it can be converted to a torus by continuous deformation), and its faces are simple polygons. A toroid is said to be regular if it is topologically regular. It is easy to see, that the regular toroids can be classified into three classes, according to the number of edges of a vertex and of a face. There are infinitely many regular toroids in each class, because the number of the faces and vertices can be arbitrarily large. Hence, we study mainly those regular toroids, whose number of faces or vertices is minimal, or that ones, which have any other special properties. Among these polyhedra, we take special attention to the so called "Császár-polyhedron", which has no diagonal, i.e. each pair of vertices are neighbouring, and its dual polyhedron (in topological sense) the so called "Szilassi-polyhedron", whose each pair of faces are neighbouring. The first one was found by Ákos Császár in 1949, and the latter one was found by the author, in 1977.

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### BibTeX entry

@article{OnSomeRegularToroids, title = {On Some Regular Toroids}, abstract = {As it is known, in a regular polyhedron every face has the same number of edges and every vertex has the same number of edges, as well. A polyhedron is called topologically regular if further conditions (e.g. on the angle of the faces or the edges) are not imposed. An ordinary polyhedron is called a toroid if it is topologically torus-like (i.e. it can be converted to a torus by continuous deformation), and its faces are simple polygons. A toroid is said to be regular if it is topologically regular. It is easy to see, that the regular toroids can be classified into three classes, according to the number of edges of a vertex and of a face. There are infinitely many regular toroids in each class, because the number of the faces and vertices can be arbitrarily large. Hence, we study mainly those regular toroids, whose number of faces or vertices is minimal, or that ones, which have any other special properties. Among these polyhedra, we take special attention to the so called "Cs{\'{a}}sz{\'{a}}r-polyhedron", which has no diagonal, i.e. each pair of vertices are neighbouring, and its dual polyhedron (in topological sense) the so called "Szilassi-polyhedron", whose each pair of faces are neighbouring. The first one was found by {\'{A}}kos Cs{\'{a}}sz{\'{a}}r in 1949, and the latter one was found by the author, in 1977.}, url = {http://symmetry-us.com/Journals/visbook/szilassi/}, year = 2001, author = {Lajos Szilassi}, comment = {}, urldate = {2018-07-26}, collections = {Easily explained,Things to make and do,Geometry} }