A Euclidean Construction of a Regular Triacontagon

Steps of this construction:

  1. Use the green circles and blue lines to construct the yellow pentagon, along with its green inscribed pentagram.
  2. Construct the equilateral triangle shown in gray. This is needed to obtain a twelve degree angle. The triangle is needed for its sixty degree angle, because 72 – 60 = 12. (The 72 degree angle is found inside the pentagon.)
  3. Identify the twelve degree angle shown in bold. A twelve degree angle is needed because 360 / 30 = 12.
  4. Use the red circles to complete the thirty sides of the regular triacontagon, which is shown with bold black segments, inscribed inside a large, bold, red circle.

Eyes

This collection of curves was built around a tessellation of the plane using regular hexagons. To make the second version, I inverted the colors, except for the black circles and arcs.