I used A,B,C in my example because the numbers 10, 11, and 12 don't fit in the same space as a single digit number.

Basically, just take a piece of graph paper and write down all the numbers (1 per square) in an outward spiral as described.

Then with a pencil and blot out all the ones that meet a given criteria.

Now imagine starting with a bunch of pieces of graph paper already taped together. That is what I am trying to achieve.
The effect as bart mentioned in the CB is tiling or making a mosaic.

Hope that helps clarify things - L~R

Update: It appears that what I had independently come up in my Sophmore year of high school had partially been thought of before me: article 1 & article 2
Thanks to Enlil for finding them. It seems that no one has been able to find the pattern for primes using this method, but no one has proved that they are not there either. I am in hopes that superimposing different "qualities" will turn up what I am looking for.