In a paper I am writing, I rely on the following definition
Given a geometric shape $C$ and a family of geometric shapes $S$, The division number of $C$ relative to $S$, denoted $DivNum(C,S)$, is the smallest number $n$ such that:
- $C$ can be divided to $n$ disjoint pieces from family $S$, $C=s_{1}\cup...\cup s_{n}$;
- For every $i\in\{1,...,n\}$ and every factor $f\in(0,1)$, the piece $s_{i}$ can be scaled (shrinked) by a factor $f$ to a piece $s_{i}^{f}\in S$, such that the remainder $C\setminus s_{i}^{f}$ can be covered by at most $n-1$ shapes from family $S$ (possibly overlapping).
For example: $DivNum(Square,\, Squares)=4$, because any square can be divided to 4 squares in a 2-by-2 grid and each square in this grid can be shrinked by any factor (towards the corner) such that the remainder can be covered by 3 squares. See figure:
However, it is not true that: $DivNum(1\times 2\ rectangle,\, Squares)=2$, because, although a $1\times 2$ rectangle can be divided to two disjoint squares, if one of the squares is shrinked to a smaller square, the remainder can no longer be covered by a single square:
I feel that the definition is not clear/formal enough. How would you improve it to make it clearer and more formal? (I hope these two requests are not contradictory..)