Extend every side of a given regular polygon can start a construction of another regular polygon, provided that the extensions intersect.  Such a construction and its result are called a stellation of the initial polygon, if this polygon and the result are respectively convex and not convex.  On the image is given a convex regular dodecagon.  Twelve straight lines in black are extensions of its sides.  Each black line cuts ten other black lines.  Each intersection point is counted two times if we multiply ten by twelve, so there are sixty intersection points, including the vertices of the initial polygon.  Among the sixty points, twelve points are the vertices of the polygon, the only one up to similarity, being regular and not convex and having twelve sides.  To construct such a dodecagon from the given regular dodecagon, another way is to draw the segment joining <i>A</i>  and  <i>B</i>,  and all the diagonals equal to this one.
<p>To construct the stellation, we choose a vertex of the given polygon as start point:  point <i>A</i>,  and then a rotation sense around the center of the polygon: anticlockwise on the image.  In other words, we choose a ray (a half-line) with origin <i>A</i>,  containing a side of the convex polygon.  Running along this ray from point <i>A</i>,  our pencil stops at the fifth intersection with a black line:  point <i>F</i>,  first vertex of the star.  And then from point <i>F</i>,  our pencil runs along the new black line up to a second vertex of the convex polygon:  point <i>B</i>,  start point of the second step.  The process consists of twelve steps, repeated until we come back to point <i>A</i>,  after passing through all the intersection points:  <span style="white-space:nowrap">12 × 5 = 60 points. </span> To draw attention on the intersection points of the first steps, there are two times five bicolour disks on the intersections.
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If an integer lower than 5 replaces 5 in the process, a convex regular polygon is constructed, not a stellation.  Each side of the result contains a side of the initial polygon, of course.  With 4 instead of 5, we construct an equilateral triangle.  With 3 instead of 5 we get a square, with 2 a regular hexagon.  We construct nothing but the initial polygon if we replace 5 with 1.  In Schläfli's notation, where the first integer is the number of sides of the regular polygon, the results are denoted by {12;5} for the stellation, {3;1}, {4;1}, {6;1}, {12;1} for the other constructions.
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