Magic star

Last updated

An n-pointed magic star is a star polygon with Schläfli symbol {n/2} [1] in which numbers are placed at each of the n vertices and n intersections, such that the four numbers on each line sum to the same magic constant. [2] A normal magic star contains the consecutive integers 1 to 2n. No numbers are ever repeated. [3] The magic constant of an n-pointed normal magic star is M = 4n + 2.

Contents

No star polygons with fewer than 5 points exist, and the construction of a normal 5-pointed magic star turns out to be impossible. It can be proven that there exists no 4-pointed star that will satisfy the conditions here. The smallest examples of normal magic stars are therefore 6-pointed. Some examples are given below. Notice that for specific values of n, the n-pointed magic stars are also known as magic hexagram etc.

Magic6star-sum26.svg Magic7star-sum30.svg Magic8star-sum34.svg
Magic hexagram
M = 26
Magic heptagram
M = 30
Magic octagram
M = 34

See also

Related Research Articles

In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In elementary geometry, a polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon.

Magic square arrangement of numbers (usually integers) in a square grid

In recreational mathematics and combinatorial design, a magic square is a square grid filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the magic constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n.

In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.

Star polygon regular non-convex polygon

In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncation operations on regular simple and star polygons.

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions. The term can mean

In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be either convex or star. 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, if the edge length is fixed.

Diagonal In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron

In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus.

In mathematics, a P-multimagic square is a magic square that remains magic even if all its numbers are replaced by their kth power for 1 ≤ kP. Thus, a magic square is bimagic if it is 2-multimagic, and trimagic if it is 3-multimagic; tetramagic for 4-multimagic; and pentamagic for a 5-multimagic square.

Schläfli symbol notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

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. In general where is the side length of the square.

Centered triangular number centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers

A centeredtriangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for n is given by the formula

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer.

Heeschs problem

In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it. Heesch's problem is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch, who found a tile with Heesch number 1 and proposed the more general problem.

Skew polygon polygon whose vertices do not lie in a plane

In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The interior surface of such a polygon is not uniquely defined.

A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a symbol or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus.

Magic hexagram

A magic hexagram of order 2 is an arrangement of numbers in a hexagram with triangular cell 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.

References

  1. Weisstein, Eric W. "Star Polygon". MathWorld .
  2. Staszkow, Ronald (2003-05-01). Math Skills: Arithmetic with Introductory Algebra and Geometry . Kendall Hunt. p.  374. ISBN   9780787292966. magic star math.
  3. "Magic Stars Index Page". www.magic-squares.net. Retrieved 2017-01-14.