Comment by AnotherGoodName

Start with the vertical patterns; The vertical patterns happen for any composite number. Take the number 10 for example. Obviously numbers 10n + [0,2,4,6,8] are divisible by 2 and 10n + 5 are divisible by 5. So you can understand the vertical gaps in primality whenever you display a composite number. Multiple's of 10 is one you can probably grasp quickly but there's nothing special about base 10.

Now when you take a vertical pattern like you see with column width=10 and add one more to the column size (so make it 11) the pattern is now offset by one pixel on each row. This makes the vertical pattern render diagonally as each row is offset by a pixel.

So you get the patterns going between vertical and diagonal. This can get really interesting when you have a bunch of composite numbers in a row. Each composite number has it's own vertical pattern as described above. So 435 has gaps every 3rd and 5th number (and for other factors) since those will be multiple of 3 or 5. Now the next number, 436 rotates this pattern 45degrees since it offsets each row by 1 pixel. But.. 436 has it's own patterns, eg. every second number is a multiple of 2 when offset from 436 since 436 is even. So it has it's own vertical patterns but also shows the diagonals from gaps in 435.

Anyway these well understood gaps in primality give a galaxy appearance for this simple reason. You're seeing well understood vertical patterns shifting to render diagonally with new vertical patterns from alignment on the next number and repeat.