The family of rectangles has the following nice properties:
- Every rectangle $R$ can be divided to two disjoint parts, $R_1 \cup R_2 = R$, such that both $R_1$ and $R_2$ are rectangles (i.e. belong to the same family).
- Moreover, for every number $p > 0$, it is possible to cut $R$ to two disjoint rectangles $R_1 \cup R_2 = R$, such that the ratio of their areas is $p$.
What other non-trivial families of geometric shapes have one or two of these properties?
- Triangles have properties 1 and 2 - a triangle can always be divided to two triangles using a ray from one of its vertices.
- Right-angled triangles have property 1, because they can be divided using a ray from the 90-degree vertex that is perpendicular to the opposite side, and the result will be two right-angled triangles. However, they don't have property 2 because the ratio between the two parts cannot be changed.
- Squares have none of these properties, as a square cannot be divided to two squares. Ditto for circles.
- L-shapes seem to have properties 1 and 2, as well as rings.
This raises several questions:
- Is it possible to give a complete characterization of such families (i.e. find all families with the above properties)?
- Given a family of shapes, how can we find a minimal containing family that has the above properties? (e.g. for squares, a possible extension is the rectangles; another extension is the L-shapes).
- (terminology) Is there a formal term for these two properties? If not, what could be a good term? Maybe "closed under division" / "closed under arbitrary proportion division"?
MOTIVATION: When designing algorithms for geometric shapes, if the shapes have property 1 (and possibly property 2), then it is possible to design a recursive algorithm. Otherwise, a recursive algorithm is not possible and we must think of other algorithms.
EDIT: A concept closely related to my property #1 is rep-tile. A rep-tile is a shape that can be divided to two or more smaller copies of itself. This concept was recently generalized to a rep-tile set by Sallows. Specifically, all rep-2 shapes and all rep-tile sets with 2 elements have property #1. But property #1 is more general, as it does not require that the parts will be copies of the original (for example, a rectangle can be divided into two rectangles that are rectangles, although they are not smaller copies of the original).
Property #2 is still open - I haven't found anything similar.