We study and catalog isoperimetric, planar tilings by unit-area Cairo and Prismatic pentagons. In particular, in counterpoint to the five wallpaper symmetry groups known to occur in Cairo-Prismatic tilings, we show that the five with order three rotational symmetry cannot occur.
Berry, John; Dannenberg, Matthew; Liang, Jason; and Zeng, Yingyi
"Symmetries of Cairo-Prismatic Tilings,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 17:
2, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss2/3