  The first aperiodic monotiling, introduced by Taylor, was based on a
trapezoidal prototile equipped with 14 distinct decorations. A presentation of
the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal
prototile equipped with 7 decorations. This paper gives decoration-free
algebraic descriptions equivalent to each of these presentations. It also shows
how the monotilings and Taylor triangles pattern that characterizes the
aperiodicity can be obtained from just one algebraic equation.
