Stellated Polygons
A stellation of a regular polygon is a regular star polygon or polygonal compound.
A regular star polygon is represented by its Schläfli symbol {n/m}, where n is the number of vertices, m is the step used in sequencing the edges around it, and m and n are co-prime (i.e. have no common divisor). Making m = 1 gives the convex {n}.
If n and m do have a common divisor, then a regular compound can be made. For example {6/2} leads to the regular compound of two triangles {3} or hexagram, while {10/4} leads to a compound of two pentagrams {5/2}.
Some authors use the Schläfli symbol for such regular compounds. Others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, for example 2{3} for the hexagram and 2{5/2} for the regular compound of two pentagrams.
A regular n-gon has (n-4)/2 stellations if n is even, and (n-3)/2 stellations if n is odd.
The pentagram, {5/2}, is the only stellation of a pentagon |
The hexagram, {6/2}, the stellation of a hexagon and a compound of two triangles. |
The enneagon (nonagon) has 3 enneagrammic forms: {9/2}, {9/3}, {9/4}, with {9/3} being 3 triangles. |
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Like the heptagon, the octagon also has two octagrammic stellations, one, {8/3} being a star polygon, and the other, {8/2}, being the compound of two squares.
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