In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular. There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case. These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol or r{p,q}, to represent that their faces are all the faces (turned differently) of both the regular {p,q} and the dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)2). More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing r (2 or more) sequences of the faces around the vertex. Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)2. Or more generally: (p.q)2, with 1/p + 1/q < 1/2. Regular polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure with Schläfli symbol {p,q} can be considered quasiregular, with vertex configuration (p.p)q/2, if q is even. Examples: The regular octahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), with vertex configuration (3.3)4/2 = (3a.3b)2, alternating two colors of triangular faces. The square tiling, with vertex configuration 44 and 4 being even, can be considered quasiregular, with vertex configuration (4.4)4/2 = (4a.4b)2, colored as a checkerboard. The triangular tiling, with vertex configuration 36 and 6 being even, can be considered quasiregular, with vertex configuration (3.3)6/2 = (3a.3b)3, alternating two colors of triangular faces. (Wikipedia).
Boris Apanasov: Non-rigidity for Hyperbolic Lattices and Geometric Analysis
Boris Apanasov, University of Oklahoma Title: Non-rigidity for Hyperbolic Lattices and Geometric Analysis We create a conformal analogue of the M. Gromov-I. Piatetski-Shapiro interbreeding construction to obtain non-faithful representations of uniform hyperbolic 3-lattices with arbitrarily
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
On the dyadic Hilbert transform – Stefanie Petermichl – ICM2018
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
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From playlist Classify Polygons
Live CEOing Ep 186: Polyhedra in Wolfram Language
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From playlist Workshop: Tropical geometry and the geometry of linear programming
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From playlist MIT 6.849 Geometric Folding Algorithms, Fall 2012
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From playlist Classify Polygons
Lecture 6 | Convex Optimization II (Stanford)
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From playlist Classify Polygons