![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
A star polygon , with
Positive Integers, is a figure formed by connecting with straight
lines every
th point out of
regularly spaced points lying on a Circumference. The number
is called the
Density of the star polygon. Without loss of generality, take
.
The usual definition (Coxeter 1969) requires and
to be Relatively Prime. However, the star
polygon can also be generalized to the Star Figure (or ``improper'' star polygon) when
and
share
a common divisor (Savio and Suryanaroyan 1993). For such a figure, if all points are not connected after the
first pass, i.e., if
, then start with the first unconnected point and repeat the procedure.
Repeat until all points are connected. For
, the
symbol can be factored as
![]() |
(1) |
![]() |
![]() |
![]() |
(2) |
![]() |
![]() |
![]() |
(3) |
If , a Regular Polygon
is obtained. Special cases of
include
(the Pentagram),
(the Hexagram, or Star of David),
(the Star of Lakshmi),
(the
Octagram),
(the Decagram), and
(the Dodecagram).
The star polygons were first systematically studied by Thomas Bradwardine.
See also Decagram, Hexagram, Nonagram, Octagram, Pentagram, Regular Polygon, Star of Lakshmi, Stellated Polyhedron
References
Coxeter, H. S. M. ``Star Polygons.'' §2.8 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 36-38, 1969.
Frederickson, G. ``Stardom.'' Ch. 16 in Dissections: Plane and Fancy. New York: Cambridge University Press,
pp. 172-186, 1997.
Savio, D. Y. and Suryanaroyan, E. R. ``Chebyshev Polynomials and Regular Polygons.'' Amer. Math. Monthly 100, 657-661, 1993.
![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
© 1996-9 Eric W. Weisstein