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Vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope '' acts transitively'' on its vertices, or that the vertices lie within a single '' symmetry orbit''. All vertices of a finite -dimensional isogonal figure exist on an -sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. The pseudorhombicuboctahedronwhich is ''not'' isogonaldemonstrates that simply asserting that "all vertices look the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotoxal Polygon
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. Isotoxal polygons An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. Isotoxal 4n-gons are centrally symmetric, so are also zonogons. In general, an isotoxal 2n-gon has \mathrm_n, (^*nn) dihedral symmetry. For example, a rhombus is an isotoxal "2×2-gon" (quadrilateral) with \mathrm_2, (^*22) symmetry. All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular n-gon has \mathrm_n, (^*nn) dihedral symmetry. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Square Gyrobicupola
In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its Face (geometry), faces consist of regular polygons that meet in the same pattern at each of its Vertex (geometry), vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that map every vertex to every other vertex (though Branko Grünbaum, Grünbaum has suggested it should be added to the traditional list of Archimedean solids as a 14th example). It strongly resembles, but should not be mistaken for, the rhombicuboctahedron, which ''is'' an Archimedean solid. It is also a Midsphere#Canonical polyhedron, canonical polyhedron. This shape may have been discovered by Johannes Kepler in his enumeration of the Archimedean solids, but its first clear appearance in print appears to be the work of Duncan Sommerville in 1905. It was independently rediscovered by J. C. P. Mille ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a dual polyhedron is the dual polygon of the corresponding vertex figure. Duality in quadrilaterals As an example of the side-angle duality of polygons we compare properties of the cyclic and tangential quadrilaterals.Michael de Villiers, ''Some Adventures in Euclid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Star Polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygons
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Apeirogon
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder. Regular infinite skew polygons exist in the Petrie polygons of the affine and hyperbolic Coxeter groups. They are constructed a single operator as the composite of all the reflections of the Coxeter group. Regular zig-zag skew apeirogons in two dimensions A regular zig-zag skew apeirogon has (2*∞), D∞d Frieze group symmetry. Regular zig-zag skew apeirogons exist as Petrie polygons of the three regular tilings of the plane: , , and . These regular zig-zag skew apeirogons have internal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings: Isotoxal skew apeirogons in two dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon2d
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon2c
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon2b
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon2a
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon2
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
