The webpage Visualizing Prime Numbers [primalchaos.yolasite.com] does exactly that. It contains a number of consecutive visualization experiments titled "Maps of Factors". Each "map" reveals the factors for each number, so that when only 2 circles intersect the horizontal axis at their right quadrants, a prime number is revealed (highlighted by green vertical lines in the graphs). Since factors in general are geometrically chaotic in nature, prime numbers are also geometrically chaotic in nature, as the resulting image seem dependent from initial conditions (much like "atoms in Quantum Mechanics").
If I understand this correctly, in an interesting twist, the author attempts to find large prime numbers in a more efficient way through analyzing and predicting the resulting visualizations, instead of working the other way around, that is to visually represent prime numbers which were derived purely algorithmically.