From Wikipedia, the free encyclopedia
Construction of a stellated dodecagon: a regular polygon with Schläfli
symbol {12/5}.
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three
dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the
process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they
meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original.
The word stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star".
Stellation is the reciprocal or dual process to faceting.
Kepler's definition[edit]
In 1619 Kepler defined stellation for polygons and polyhedra as the process of extending edges or faces until
they meet to form a new polygon or polyhedron.
He stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated
dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella
octangula, a regular compound of two tetrahedra.
Stellating polygons[edit]
Regular convex and star polygons with 3 to 12
vertices labelled with their Schläfli symbols
Stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These
polygons are characterised by the number of times m that the polygonal boundary winds around the centre of
the figure. Like all regular polygons, their vertices lie on a circle. m also corresponds to the number of vertices
around the circle to get from one end of a given edge to the other, starting at 1.
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 coprime (have no common factor). The
case m = 1 gives the convex polygon {n}. m also must be less than half of n; otherwise the lines will either be
parallel or diverge, preventing the figure from ever closing.
If n and m do have a common factor, then the figure is a regular compound. For example {6/2} is the regular
compound of two triangles {3} or hexagram, while {10/4} is 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 (assuming compounds of multiple degenerate digons are not
considered), and n – 3/2 stellations if n is odd.
The hexagram, {6/2},
the stellation of
The pentagram, {5/2},
a hexagon and a
is the only stellation
compound of two
of a pentagon
triangles.
The enneagon (nonagon) {9} has
3 enneagrammic forms:
{9/2}, {9/3}, {9/4}, with {9/3} being a compound
of 3 triangles.
The heptagon has two heptagrammic forms:
{7/2}, {7/3}
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.
Stellating polyhedra
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