Spirals and Polygons
Join every k-th of n equally spaced points around a circle instead of each neighbour, and see what star-shaped polygons appear.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
Place n points at equal angles around a circle, the same points you would use to draw a regular n-sided polygon. Instead of joining each point only to its immediate neighbour, join every k-th point instead - point 1 to point 1+k, point 1+k to point 1+2k, and so on. For most choices of n and k this traces out a star-shaped polygon rather than a simple n-gon, and the pattern depends on whether n and k share a common factor. This exploration asks you to construct these star polygons, classify when they form one connected path or several, and find their side lengths and areas as functions of n and k.
Guiding Questions
- Place n points equally spaced around a circle, then join every k-th point instead of every neighbouring point (for example, point 1 to point 3, point 3 to point 5, and so on). What pattern of star polygon appears for different values of n and k?
- Generate this construction precisely for several choices of n and k, starting with n = 5 and n = 7. When does the path close into a single connected star, and when does it split into several separate smaller polygons?
- What curve do the vertices trace as n grows very large while k stays a fixed fraction of n? Investigate the limit.
- Find lengths or areas in your pattern as functions of n and k.