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

[KevinFinity]

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 🙂

[DJF]

That's pretty cool.  But the more important question is: How did you format it like that?  I've never seen that before.  i.e. the equations are in neat centered rows and immovable up or down.  And select doesn't work like it normally does.   

Thanks.

Edit: Nevermind.  I see now it's all just a big text box.  I never thought of using a text box like that.  Neither was I aware that I could do math inside a text box.  I'd only ever put math regions in text boxes to insert greek letters or variables.  huh, learn something everyday.

[KevinFinity]

First, create a text field (ctrl+shift+T). Then, within the text field create a math field (ctrl+shift+M). There's no need to use the mouse, and everything is formatted nicely inline. 

[VladimirN]

 Hi Kevin,

By the way, you may be interested in future challenges - "PTC Mathcad Community Challenge Index and Guidelines": https://community.ptc.com/t5/PTC-Mathcad-Tips/PTC-Mathcad-Community-Challenge-Index-and-Guidelines/m-p/766913#M171