A Generalization of Montucla's Rectangle-to-Rectangle Dissection to Higher Dimensions

Dissections of polytopes are a well-studied subject by geometers as well as recreational mathematicians. A recent application in coding theory arises from the problem of parameterizing binary vectors of constant Hamming weight which has been shown previously to be equivalent to the problem of dissecting a tetrahedron to a brick. Applications of dissections also arise in problems related to the construction of analog codes. Here we consider the rectangle-to-rectangle dissection due to Montucla. Montucla's dissection is first reinterpreted in terms of the Two Tile Theorem. Based on this, a cube-to-brick dissection is developed in $\mathbb{R}^n$. We present a linear time algorithm (in $n$) that computes the dissection, i.e. determines a point in the cube given a point in a specific realization of the brick. An application of this algorithm to a previously reported analog coding scheme is also discussed.