Stellation

Stellation is the process of making a polygon into a star. This is done by drawing across the polygon rather than around its edge. Some polygons have several stellations, others have none.

The 'rules' for making a stellation of a polygon are as follows:

All lines must be the same length.
You must always turn through the same angle and in the same sense (clockwise or anticlockwise).
The number of lines that you draw must equal the number of vertices which is the same number as the sides of the original polygon.

Controls

Size

Use this to change the size of your star. The first line drawn is always horizontal.

Vertices Up/Down

The vertices are the points of the star or 'corners' of the polygon. The minimum number that you can choose is 3 which will draw a triangle. Their is no upper limit to the number of vertices that you can select.

Start Up/Down

This is the number of the vertex to which you will draw your first line. Your line starts from the origin which is vertex 0. If you use 'Start = 1' then you will simply draw the basic polygon. To construct a stellation you must use a 'Start' number greater than 1 but, be warned, some numbers will 'work' and others won't.

It might seem strange to begin numbering from zero but it will help you understand the relationship between the vertex number, the number of sides and the patterns that you are able to draw.

Tasks

Complete this table to show how many stellations there are for different polygons. Fill in the number of vertices and the start numbers which produce proper stellations. The first few rows have been done for you.

Vertices	Start numbers of working stellations
3	none
4	none
5	2
6	none
7	2,3
8
9
10
11
12

Questions

Why do some polygons allow stellations and not others?

Why do some polygons have more than one stellation?

Why are there some start numbers that don't work for a given polygon?

Which start numbers would produce stellations for a 19 sided polygon?

Harder Tasks

Write down the vertex order for the stellations of the heptagon. How does the start number relate to the number of orbits that you must make around the polygon before returning to the origin?

How could you use stellation to test for prime numbers?

What is the connection between the vertex number, the start number and their LCM (lowest common multiple)?

Fun Tasks

Use 'Print Screen' to drop your stellation into 'Paint' and then colour it in.