The Heart Of Doubling A Cube

Another of the great mysteries of Geometry is ‘doubling the cube’, perhaps the next step in the progression that began with ‘squaring the circle‘.  A cube has six surfaces and eight perpendicular angles in three dimensional space. Which of the six surfaces do you use to construct the cube’s double? Or does it matter?

The Greeks had known for a long time how to solve the problem of doubling the square. Take a square ABCD and draw the diagonal DB. Then construct a new square BDEF using the diagonal BD as one of the sides. Then it is easy to see that BDEF is double ABCD. A voila moment for Euclid.

Another way to look at it is to see a square envelope with its flap open. A card slipped in wishing your paramour a Happy Valentine’s Day. The square now becomes a testament of love, devotion of hearts and flowers until that moment the flap is ready for closure, then the thought, “Do I want my DNA floating around? Or is it better to get a self-adhesive envelope instead?” A mere tangential thought, flight of fanciful serendipity, attempting to con-temporize the imagery of the square, now that Valentine’s Day is approaching.

What Is So Mysterious About “Squaring The Circle”?