If we could see or build the Sieve of Eratosthenes the way a tree grows, from the bottom up, we might have no doubt, and much greater understanding, of its beauty and of the structure of the prime numbers. But its roots extend infinitely down, its branches growing up and up, and we cannot hold it in our finite mortal minds.
The screenshot below is of an attempt at grasping the symmetry. I had to work backward and forward, due to imperfect (as of yet) understanding of the structure. I also stopped at 65, because I will turn 65 in just a few days. It just seemed right.
“. . . for you yourself, sir, should grow old as I am, if like a crab you could go backwards.” – Hamlet (William Shakespeare)
(or if like a tree you could seek the sky and the center at the same time)
LEGEND FOR THE TREE BELOW:
- Each row corresponds to its integer.
- X = a divisor
- . = a non-divisor
- 0 = the ceiling square, unless it is a divisor
- Space = added for the ceiling square “trunk” and the “leaves” of the tree to be aligned