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Magic Hexagram
A magic hexagram of order 2 is an arrangement of numbers in a hexagram with triangular cells with 2 cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant ''M''. Magic star hexagram Magic star hexagram or 6-pointed magic star is a star polygon with Schläfli symbol in which numbers are placed at each of the six vertices and six intersections, such that the four numbers on each line sum to the same magic constant. Magic star hexagram with triangular cell There are two solutions of magic star hexagram with 12 triangular cells. Magic star hexagram with more than 12 vertices Harold Reiter and David Ritchie calculated the solution of magic hexagrams with 19 vertices. See also *Magic square *Magic hexagon A magic hexagon of order ''n'' is an arrangement of numbers in a centered hexagonal pattern with ''n'' cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagram
, can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol , 2, or . Since there are no true regular continuous hexagrams, the term is instead used to refer to a compound figure of two equilateral triangles. The intersection is a regular hexagon. The hexagram is part of an infinite series of shapes which are compounds of two n-dimensional simplices. In three dimensions, the analogous compound is the stellated octahedron, and in four dimensions the compound of two 5-cells is obtained. It has been historically used in religious and cultural contexts and as decorative motifs. The symbol was used as a decorative motif in medieval Christian churches and Jewish synagogues. It was first used as a mystic symbol by Muslims in the medieval period, known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is, a magic square which contains the numbers 1, 2, ..., ''n''2 – the magic constant is M = n \cdot \frac. For normal magic squares of orders ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order n is (n^3+n)/2. The largest magic constant of normal magic square which is also a: *triangular number is 15 (solve the Diophantine equation x^2=y^3+16y+16, where y is divisible by 4); *square number is 1 (solve the Diophantine equation x^2=y^3+4y, where y is even); * generalized pentagonal number is 171535 (solve the Diophan ... [...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, certain notable ones can arise through truncation operations on regular simple and star polygons. Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons. The first usage is included in polygrams which includes 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 2 or more turns ( turning number and density), like in spirolaterals.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p.24 Etymology Star polygon names combine a numeral prefix, such as '' penta-'', with the Greek suffix ''-gram'' (in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. Definition The Schläfli symbol is a recursive description, starting with for a ''p''-sided regular polygon that is convex. For example, is an equilateral triangle, is a square, a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols contain irreducible fractions ''p''/''q'', where ''p'' is the number of vertices, and ''q'' is their turning number. Equivalently, is created from the vertices of , connected every ''q''. For example, is a pentagram; is a pentagon. A regular polyhedron that has ''q'' regular ''p''-sided polygon faces around each vertex is repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called " convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Hexagram-sum50
Magic or Magick most commonly refers to: * Magic (supernatural), beliefs and actions employed to influence supernatural beings and forces * Ceremonial magic, encompasses a wide variety of rituals of magic * Magical thinking, the belief that unrelated events are causally connected, particularly as a result of supernatural effects * Magic (illusion), the art of appearing to perform supernatural feats Magic(k) may also refer to: Art and entertainment Film and television * ''Magic'' (1917 film), a silent Hungarian drama * ''Magic'' (1978 film), an American horror film * ''Magic'' (soap opera), 2013 Indonesian soap opera * Magic (TV channel), a British music television station Literature * Magic in fiction, the genre of fiction that uses supernatural elements as a theme * ''Magic'' (Chesterton play), 1913 * ''Magic'' (short story collection), 1996 short story collection by Isaac Asimov * ''Magic'' (novel), 1976 novel by William Goldman * '' The Magic Comic'', a 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (''n''), and the constant sum is called the ' magic constant'. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a ''semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Hexagon
A magic hexagon of order ''n'' is an arrangement of numbers in a centered hexagonal pattern with ''n'' cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant ''M''. A normal magic hexagon contains the consecutive integers from 1 to 3''n''2 − 3''n'' + 1. It turns out that normal magic hexagons exist only for ''n'' = 1 (which is trivial, as it is composed of only 1 cell) and ''n'' = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof.Meng, F"Research into the Order 3 Magic Hexagon" '' Shing-Tung Yau Awards'', October 2008. Retrieved on 2009-12-16. The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is Ernst von Haselberg (1887). Proof of normal magic hexagons The numbers in the hexagon are consecutive, and run from 1 to 3n^2-3n+1. Hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Shapes
Magic or Magick most commonly refers to: * Magic (supernatural), beliefs and actions employed to influence supernatural beings and forces * Ceremonial magic, encompasses a wide variety of rituals of magic * Magical thinking, the belief that unrelated events are causally connected, particularly as a result of supernatural effects * Magic (illusion), the art of appearing to perform supernatural feats Magic(k) may also refer to: Art and entertainment Film and television * ''Magic'' (1917 film), a silent Hungarian drama * ''Magic'' (1978 film), an American horror film * ''Magic'' (soap opera), 2013 Indonesian soap opera * Magic (TV channel), a British music television station Literature * Magic in fiction, the genre of fiction that uses supernatural elements as a theme * ''Magic'' (Chesterton play), 1913 * ''Magic'' (short story collection), 1996 short story collection by Isaac Asimov * ''Magic'' (novel), 1976 novel by William Goldman * ''The Magic Comic'', a 1939–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
