This student research project explores the properties of a family of matrices of zeros and ones that arises from the study of the diagonal lengths in a regular polygon. There is one family for each n greater than 2. A series of exercises guides the student to discover the eigenvalues and eigenvectors of the matrices, which leads in turn to formulas for products and inverses. For more advanced students there is a section on the connection to cyclotomic fields. A number of open-ended questions are given as avenues for further investigation.