Fractional Harmonics and Star Polygons

This book has its roots in a previous publication of mine, Spherical Harmony: A Journey of Geometric Discovery. I will introduce the concept of spherical harmonics, but the main focus will be on fractional harmonics and their associated polygram, or star polygon. There exists a unique fractional harmonic of each of the infinite number of star polygons. Much has been written about the pentagram and the golden ratio that is embedded in its geometry. However, it was the heptagon and associated heptagram that really piqued my interest. So I decided to explore the heptagram since I didn't know much about its geometric properties or characteristics and there wasn't much written about them either. The prime number seven has a special place in the ancient world and was referred to as the virgin number and was the most sacred number in many religions. some amazing ratios of proportion.