Interesting patterns emerge when primes are represented as white blocks in a series of n-width left-right lines laid vertically, one atop the other. When the line is five blocks wide, the patterns look like this (the first green block is 1, followed by primes 2, 3 and 5, then 7 in the next line):

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Right at the bottom of the first column is an isolated prime diamond, or priamond (marked with a green block). It consists of the four primes 307-311-313-317, where the three latter primes equal 307 + 4 and 6 and 10, or 307 + 5-1, 5+1 and 5×2 (the last prime in the first column is 331 and the first prime in the second is 337). About a third of the way down the first column is a double priamond, consisting of 97, 101, 103, 107, 109 and 113. For a given n, then, a priamond is a set of primes, p₁, p₂, p₃ and p₄, such that p₂ = p₁ + n-1, p₃ = p + n+1 and p₄ = p₁ + 2n.

There are also fragments of pearl-necklace in the columns. One is above the isolated priamond. It consists of four prime-blocks slanting from left to right: 251-257-263-269, or 251 + 6, 12 and 18. A prearl-necklace, then, is a set of primes, p₁, p₂, p₃…, such that p₂ = p₁ + n+i, p₃ = p + 2(n+i)…, where i = +/-1. Now here are the 7-line and 9-line:

Above: 7-line for primes

Above: 9-line for primes

In the 9-line, you can see a prime-ladder marked with a red block. It consists of the primes 43-53-61-71-79-89-97-107, in alternate increments of 10 and 8, or 9+1 and 9-1. A prime-ladder, then, is a set of primes, p₁, p₂, p₃, p₄…, such that p₂ = p₁ + n+1, p₃ = p + 2n, p₃ = p + 3n+1…

And here is an animated gif of lines 5 through 51:

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