How often have you been eating a meal in one of life's eating establishments, and thought, "I wish my napkin was more aesthetically sized."?

I thought so, which is why I bring you 5 easy steps to make your eating experience that little bit more perfect.

  • Start with a square portion of the napkin.
          +-----------+
          |           |
          |           |
          |           |
          |           |
          +-----------+
  • Divide the square in half.
          +-----+-----+
          |     |     |
          |     |     |
          |     |     |
          |     |     |
          +-----+-----+
  • Draw a diagonal across one of half of the square.
          +-----+-----+
          |     |   / |
          |     |  /  |
          |     | /   |
          |     |/    |
          +-----+-----+
  • Use the diagonal as the radius of a circle, and complete an arc to the baseline of the square.
          +-----+-----+-
          |     |   / |  \
          |     |  /  |    \
          |     | /   |     \
          |     |/    |      |
          +-----+-----+------+
  • Complete the rectangle from this point on the baseline.
          +-----+-----+------+
          |     |   / |  \   |
          |     |  /  |    \ |
          |     | /   |     \|
          |     |/    |      |
          +-----+-----+------+
You now are the proud owner of a napkin that is a golden rectangle. What a pleasure wiping your mouth will be.

But, I hear you ask, how do you know that this is a golden rectangle? Well, dear reader, read on.

Lets say the original square was x by x. Given that, the length of the diagonal can be found using the school boy favourite, Pythagoras:

           c2 = a2 + b2
          
                x2
              = -  + x2
                4
          
                   __________
                  | x2
           c   =  | -  + x2
                 \| 4
          
                   _____
                  | 5x2
           c   =  | ---
                 \|  4
           
With that said, the bottom edge of the golden rectangle is:
                 _____
           x    | 5x2
           - +  | ---
           2   \|  4
          
                 _____
           x + \| 5x2
           -----------
                2
          
           x ( 1 + √5 )
           -------------
                2
So, if we now consider this in terms of the ratio of short to long edge we have:
               x ( 1 + √5 )
          x :  ------------
                    2
          
               1 + √5
          1 :  ------
                 2
Which, my fellow geometrist, is the golden ratio.