Dividing a square into 7 parts of equal area and minimizing the perimeter-to-area ratio.

Greetings,

Here is something I enjoyed working on. I couldn't find the answer online, so I had to figure it out myself. The challenge was to divide a square into 7 equal parts. Each part had to be as compact as possible (minimum perimeter to area ratio), so I knew that the trivial solution of dividing it into 7 long, thin strips would not work. After drawing several iterations, I came up with this model. A hexagon is centered in the square, rotated at 45 degrees so that two of its vertices align with the diagonal. 

Each shape has an area of 7, in this 7 x 7 square. 

If this was a tile fastened to a surface, then the fasteners would be placed at the centroids (stars) to prevent uplift in the most efficient way, supposing that they want to use 7 fasteners and not any other number. Fastener patterns with other numbers are easier to figure out. The next-hardest one was with 5 fasteners, but that's just a simple square in the middle instead of a hexagon.

In this mathcad sheet, there are formulas for:

• The coordinates of any hexagon
• The area of any closed polygon, given vectors of its coordinates
• The centroid of any closed polygon, given vectors of its coordinates

The only thing I couldn't figure out with pure mathematics is the constant k, which is used in the coordinates of some quadrilaterals to make the area of that quadrilateral equal 7. If anyone can figure this out, I'd be grateful. Otherwise,

Enjoy 🙂