The Ulam Spiral is not a spiral but isolated segments of the number sequence that get larger 8 numbers per segment.

Why not write it down in segments of 6 that stay six? Behold, the Amazing Abarim Pillar.

The base-structure of Ulam's Rose's revealed by the underlying rythm of the number sequence (where P is a possible prime:
1 2 3 * P * P * * * P * P * * * P * P * * * P * P * * * P * P

The entire number sequence came forth at once and wholly when Euclid noted its axioms. Though infinitely vast, it remains ultimately predictable and therefore incapable of representing the most basic principle of the universe: freedom.