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The word "star" is used in a number of different ways in mathematics. The term is commonly used to voice an asterisk when appearing in a mathematical expression. For example, a^* is voiced "a-star". The "star" is something used to denote the adjoint a^*, or sometimes the complex conjugate. In common usage, a star is a star polygon or star figure (i.e., regular convex polygon or polygon compound) such as the pentagram or hexagram

StarCoxeter

In plane and solid geometry, a star, sometimes called a sheaf (Ball and Coxeter 1987, p. 141) is defined as a set of n line segments with a common midpoint (Coxeter 1973, p. 27). The figures above show the first few regular n-stars in the plane. A star is called nonsingular if no three of the lines comprising it are coplanar. Every convex polyhedron bounded by parallelograms determines a nonsingular star, having a single line segment for each set of 2(n-1) parallel edges of the polyhedron (Coxeter 1973, p. 27).

In formal geometry, a star is a set of 2n vectors +/-a_1, ..., +/-a_n which form a fixed center in Euclidean 3-space.

In algebraic topology, if v is a vertex of a simplicial complex K, then the star of v in K, denoted Stv or St(v,K), is the union of the interiors of those simplices of K that have v as a vertex (Munkres 1993, p. 11).


See also

Closed Star, Cross, Eutactic Star, Hexagram, Pencil, Pentagram, Polar Zonohedron, Sheaf of Planes, Simplicial Complex Link, Star Discrepancy, Star Figure, Star Graph, Star Polygon, Zonohedron

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 27, 1973.Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub., 1993.

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Cite this as:

Weisstein, Eric W. "Star." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Star.html

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