The structure of natural snowflakes can be abstracted mathematically and can be reproduced in paper and fabric art. Mathematically, natural snowflakes are beautiful examples of objects with a combination of both rotation symmetry and reflection symmetry, known as dihedral symmetry. Designs with dihedral symmetry can be easily constructed from regular polygons; so we provide a brief summary of the classic methods for constructing regular polygons including Euclidean constructions (compass and straightedge), paper folding constructions, as well as a practical method that uses a protractor. Finally, we provide several examples of how artists can use regular polygons made from paper to produce art with dihedral symmetry.