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Equiangular Polygon
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral polygon, equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. For clarity, a planar equiangular polygon can be called ''direct'' or ''indirect''. A direct equiangular polygon has all angles turning in the same direction in a plane and can include multiple turn (angle), turns. Convex equiangular polygons are always direct. An indirect equiangular polygon can include angles turning right or left in any combination. A skew polygon, skew equiangular polygon may be isogonal figure, isogonal, but can't be considered direct since it is nonplanar. A spirolateral ''n''θ is a special case of an ''equiangular polygon'' with a set of ''n'' integer edge lengths repeating sequence until returning to the start, with vertex internal angles θ. Construc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equiangular Rectangle2x3
Equiangular may refer to: *Equiangular lines, a set of lines where every pair of lines makes the same angle *Equiangular polygon, a polygon with equal angles *Logarithmic spiral or equiangular spiral, a type of geometric spiral *Unicode symbol represents the equiangular relation {{Disambiguation sv:Likvinklig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Figure
In geometry, a polytope (e.g. a polygon or polyhedron) or a Tessellation, tiling is isogonal or vertex-transitive if all its vertex (geometry), vertices are equivalent under the Symmetry, symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face (geometry), face in the same or reverse order, and with the same Dihedral angle, angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope Map (mathematics), mapping the first isometry, isometrically onto the second. Other ways of saying this are that the automorphism group, group of automorphisms of the polytope ''Group action#Remarkable properties of actions, 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 n-sphere, -sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Gazette
''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journal was established in 1894 by Edward Mann Langley as the successor to the ''Reports of the Association for the Improvement of Geometrical Teaching''. William John Greenstreet was its editor-in-chief for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors-in-chief The following persons are or have been editor-in-chief: Abstracting and indexing The journal is abstracted and indexed in EBSCO databases, Emerging Sources Citation Index, Scopus Scopus is a scientific abstract and citation database, launched by the academic publisher Elsevier as a competitor to older Web of Science in 2004. The ensuing competition between the two databases has been characterized as "intense" and is c ..., and zbMA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Polygon
In geometry, a set (mathematics), set of point (geometry), points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertex (geometry), vertices are concyclic is called a cyclic polygon, and the circle is called its ''circumscribing circle'' or ''circumcircle''. All concyclic points are equidistant from the center of the circle. Three points in the Euclidean plane, plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied. Perpendicular bisectors In general the centre ''O'' of a circle on which points ''P'' and ''Q'' lie must be such that ''OP'' and ''OQ'' are equal distances. Therefore ''O'' must lie on the perpendicular bisector of the line segment ''PQ''. For ''n'' distinct points there are triangula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viviani's Theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world. Proof This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA 2010, , p. 96 () Let ABC be an equilateral triangle whose height is ''h'' and whose side is ''a''. Let P be any point inside the triangle, and ''s, t, u'' the perpendicular distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA. Now, the areas of these tria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density (polytope)
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets. The same calculation can be performed for any convex polyhedron, even one without symmetries, by choosing any point interior to the polyhedron as its center. For these polyhedra, the density will be 1. More generally, for any non-self-intersecting (acoptic) polyhedron, the density can be computed as 1 by a similar calculation that choo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density (geometry)
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets. The same calculation can be performed for any convex polyhedron, even one without symmetries, by choosing any point interior to the polyhedron as its center. For these polyhedra, the density will be 1. More generally, for any non-self-intersecting (acoptic) polyhedron, the density can be computed as 1 by a similar calculation that choose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can arise through truncation operations on regular simple or star polygons. Branko Grünbaum identified two primary usages of this terminology by Johannes Kepler, one corresponding to the regular star polygons with List of self-intersecting polygons, intersecting edges that do not generate new vertices, and the other one to the isotoxal Concave polygon, concave simple polygons.Grünbaum & Shephard (1987). Tilings and Patterns. Section 2.5 Polygram (geometry), Polygrams include polygons like the pentagram, but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having ''q'' ≥ 2 Turn (geometry), turns (''q'' is called the turning number or Density (polygon), density), like in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Angle
In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex (geometry), vertex. If every internal angle of a simple polygon is less than a straight angle (pi, radians or 180°), then the polygon is called convex polygon, convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a Extended side, line extended from an adjacent side.Posamentier, Alfred S., and Lehmann, Ingmar. ''The Secrets of Triangles'', Prometheus Books, 2012. Properties * The sum of the internal angle and the external angle on the same vertex is radians (180°). * The sum of all the internal angl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indirect Equiangular Polygon
Indirect, the opposite of direct, may refer to: *Indirect approach, a battle strategy *Indirect DNA damage, caused by UV-photons *Indirect agonist or indirect-acting agonist, a substance that enhances the release or action of an endogenous neurotransmitter *Indirect speech, a form of speech *Indirect costs, costs that are not directly accountable to a particular function or product *Indirect self-reference, describes an object referring to itself indirectly *Indirect effect, a principle of European Community Law *Indirect finance, where borrowers borrow funds from the financial market through indirect means *Indirection In computer programming, an indirection (also called a reference) is a way of referring to something using a name, reference, or container instead of the value itself. The most common form of indirection is the act of manipulating a value through ..., the ability to reference something in computer programming * Indirect transmission, infections passing from one hos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equiangular Hexagon-example
Equiangular may refer to: *Equiangular lines, a set of lines where every pair of lines makes the same angle *Equiangular polygon, a polygon with equal angles *Logarithmic spiral or equiangular spiral, a type of geometric spiral *Unicode Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ... symbol represents the equiangular relation {{Disambiguation sv:Likvinklig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supplementary Angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides. Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. The term ''angle'' is also used for the size, magnitude or quantity of these types of geometric figures and in this context an angle consists of a number and unit of measurement. Angular measure or measure of angle are sometimes used to distinguish between the measurement and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
