What would you guess that this is?

This beautiful image was generated by Adrian Leatherland, at Monash University in Australia. Almost everyone who I have shown it to has suggested that it looks like something astronomical - like something you might expect to see through a powerful telescope. As a musician friend put it, "It looks like one of those things you see in an astronomy book, but you don't know what it is."

In fact, it is a pure mathematical image, generated by the first 80 million prime numbers and a remarkably simple algorithm.

When told this, a programmer I once met said jokingly: "That makes me seriously worried about the underlying nature of reality!"   To me, this image, with its almost physical beauty, could possibly illustrate what I see as an emerging and radical re-interpretation of the nature of number.

(Those of you who jump to the sceptical conclusion that similar images can be produced by applying the same algorithm to any suitably pseudo-random sequence of natural numbers should have a look at this.)

Adrian Leatherland's images may turn out to be irrelevant curiosities, but peoples' reactions to them interest me. The most common guesses as the nature of the image in question are:

(1) A distant astronomical object, a nebula or star cluster of some kind
(2) A satellite image of a city
(3) A fireball from an explosion of some kind
(4) A virus or something else microbiological
What do all of these have in common? We would certainly expect some kind of fractal geometry to be present in (1) and (2), and I'm fairly confident that in many cases this would be present in (3) and (4).

I'm suggesting that people shown the image in question instinctively detect a fractality. Importantly, this does not require their familiarity with the concepts of fractal geometry. This fractality suggests to them that they are looking at some kind of physical structure which has in some way grown, evolved, or self-organised.

Leatherland has produced analogous images for the first 100, 200,..., 500 million primes, and the progression certainly seems to suggest some sort of fractality (self-similarity). That is, the image expands and changes, but yet maintains something of its fundamental character.

Marek Wolf, who has already detected certain instances of fractality within the distribution of primes on an experimental basis, has pointed out that the mathematical object pictured above should be fractal in some sense simply because its generating algorithm is a kind of random walk. This of course is entirely independent of any properties of the distribution of prime numbers which might be somehow manifesting in the object.

However, the steps in the walk are not random - they are determined by the sequence of mod 9 residues of the primes. As we see here, genuinely random walks analogous to the one in question tend to produce noticeably more 'dispersed' images. Therefore, it seems reasonable to leave open the possibility that there is an unexplained phenomenon present here. Much like the Ulam spiral phenomenon, the reason for this may turn out to be nothing more than a minor, as-yet-undiscovered property of the prime distribution, or it may be due to something deeper.

Your opinions and insights are welcome.

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