*

A star polygon

*
, through
*
optimistic integers, is a number formed through connecting with straight lines every
*
th suggest out of
*
routinely spaced points lied on a circumference. The number
*
is referred to as the polygon density of the star polygon. Without loss the generality, take
*

The usual definition (Coxeter 1969) calls for

*
and
*
to be fairly prime. However, the star polygon can also be generalized to the star figure (or "improper" star polygon) as soon as
*
and
*
share a common divisor (Savio and also Suryanaroyan 1993). For such a figure, if all points space not associated after the very first pass, i.e., if
*
, then begin with the an initial unconnected point and repeat the procedure. Repeat until all points room connected. For
*
, the
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symbol have the right to be factored as




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*

where

*
*
*

*
*
*

to give

*
*
figures, each rotated by
*
radians, or
*
.

If

*
, a consistent polygon
*
is obtained. Special instances of
*
incorporate
*
(the pentagram),
*
(the hexagram, or star of David),
*
(the star of Lakshmi),
*
(the octagram),
*
(the decagram), and also
*
(the dodecagram).

*

Superposing all distinct star polygons

*
because that a given
*
offers beautiful trends such as those shown above. These numbers can likewise be obtained by wrapping thread roughly
*
nails spaced equally approximately the one of a one (Steinhaus 1999, pp.259-260).




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star polygon