# Fast Equipartition of Complex 2D Shapes with Minimal Boundaries +
## 4 2D-SEP Instances and Properties
### 2D-SEP of Square and Circle  
Next, we consider the case of the partitioning of a square and circle into two, three, four, and five equal-area regions. To provide comparable numerical results, we assume a unit area square and circle. Figure 5 depicts optimal solutions of the 2D-SEP for unit area (a-d) square and (e-h) circle under N = 2, 3, 4 and 5, while Figure 6 shows some suboptimal solutions of the 2D-SEP for unit area (a,b) square and (c,d) circle for \(N\in\{3,5\}\). Taking into account both figures, the optimality proof of most cases in Figure 5 is trivial; so, hereafter, we only provide the close form of the intrinsic boundary length (\(L\)), which is numerically reported in the caption of the figure.  
* When N = 2 (see Figure 5a,e), the optimal solution of 2D-SEP under the square and circle is given by the horizontal line that passes from the square centroid (\(L=1\)) and the diameter of the circle (\(L=2R=\frac{2}{\sqrt{\pi}}=1.128\)), respectively.
* When N = 3 (see Figure 5b,f), the optimal solution of 2D-SEP under the square is given by two suitable vertical lines that divide the square into three rectangles \(|R_{1}|=1\times\frac{1}{3}\), \(|R_{2}|=|R_{3}|=\frac{1}{3}\times\frac{2}{3}\), with \(L=1+\frac{2}{3}=1.667\). The optimal solution of 2D-SEP under the circle is given by the boundary of the three radiuses that passes from the center with \(L=3R=\frac{3}{\sqrt{\pi}}=1.693\).
* When N = 4 (see Figure 5c,g), the optimal solution of 2D-SEP under the square is given by two suitable vertical lines that cross at the square centroid and divide the square into four equal squares, with \(L=2\). The optimal solution of 2D-SEP under the circle is given similarly by two vertical diameters with \(L=4R=\frac{4}{\sqrt{\pi}}=2.257\).
* When N = 5 (see Figure 5d,h), the optimal solution of 2D-SEP under the square is given by a circle of the radius (\(\frac{1}{\sqrt{5\pi}}\)) plus four suitable vertical lines of length (\(\frac{1}{2}-\frac{1}{\sqrt{5\pi}}\)), with \(L=\frac{2\pi}{\sqrt{5\pi}}+4\cdot(\frac{1}{2}-\frac{1}{\sqrt{5\pi}})=2.576\). The optimal solution of 2D-SEP under the circle is given by five radiuses with \(L=5R=\frac{5}{\sqrt{\pi}}=2.821\), which is slightly lower than the corresponding solution of Figure 6d with \(L=2.833\).  
An interesting remark of the above examples is that, in any case, the \(L\) of the optimal solution of the square is lower than the corresponding \(L\) of the circle. The proof that this is true (or this is not true) for any value of \(N\) is an open problem.  
Figure 5: Optimal solutions of the 2D-SEP for unit area (**a–d**) square and (**e–h**) circle under \(N\in\{2,3,4,5\}\). The intrinsic boundary length (\(L\)) is reported in the caption of each shape.  
In Figure 7, we provide the corresponding results of the proposed method _SEP-FBC_ (_SEP-PSO FBC_ also provides the same results) under unit area square and circle and \(N\in{2,3,4,5}\) to compare with the optimal 2D-SEP solutions of Figure 5. We find that in any case of the circle, _the SEP-FBC_ results in the optimal solution, while in two out of five cases of the square (\(N=3\), \(N=5\)), the proposed segmentation of _SEP-FBC_ is suboptimal with slightly higher \(L\) than the optimal one:  
* For \(N=3\), _SEP-FBC_ yields \(L=1.755\), which is \(5.02\%\) higher than the corresponding optimal \(L\) of Figure 5b.
* For \(N=5\), _SEP-FBC_ yields \(L=2.632\), which is \(2.19\%\) higher than the corresponding optimal \(L\) of Figure 5d. In this case, a part of the intrinsic boundaries of the optimal solution is a circle. The reason why both proposed methods do not find it is that the intrinsic boundaries provided by the proposed method should be polygonal lines.