Reptile trapezoids

Abstract

A geometric figure is a reptile if it can be dissected into at least two similar copies congruent to each other. We prove that if a trapezoid is a reptile and not a parallelogram, then the length of each base is a linear combination of the lengths of its legs with rational coefficients. We then rule out isosceles trapezoids and right trapezoids which are not reptile. In particular, we prove that, up to similarity, there are at most six reptile right trapezoids, not a parallelogram, whose acute internal angle is a rational multiple of $\pi$. Finally, we present a rep-25 right trapezoid that is not a parallelogram and is not similar to any of the known reptile trapezoids.