The Ulam Spiral is one of the most intriguing of all mathematical images. The spiral is a graphing of all real numbers in a clockwise direction, starting with the number one. A peculiar pattern emerges when you look at it from a distance. It turns out that the primes aren’t arranged randomly, as one might expect, but along diagonal clusters that resemble the design of folded circuitry, or a fractal of swastikas.
This begs the question: is this diagram of prime numbers analogous to spiral-like dynamical systems in nature? A colleague of mine came up with the idea of using a transform algorithm on the Ulam Spiral to depict a typical galaxy, which is apparently something that others have already done. I took it even further and suggested that the Ulam Spiral might be a concrete dynamical system for the entire universe: one that serves as a blueprint for the scattering of matter after the Big Bang. Perhaps the dispersion of primes in the Ulam Spiral represents how matter would have assembled if the force of gravity had never existed to bring matter into globular clusters. Instead of the clusters being globular, they might have remained linear, like the primes on the Ulam Spiral.
The origin of the swastika is of some importance to it. The Rig Veda, an ancient Indian text full of cosmological hymns, is the earliest known source of the swastika. Swastika literally means “mark of the sun” in Sanskrit, but Wikipedia says that “the Hindus represent (the Swastika) as the Universe in our own spiral galaxy in the fore finger of Vishnu.” The Vedas were spiritual texts, but they introduced many mathematical concepts to the world, including the number pi. If the progenitors of Vedic tradition had used the Ulam Spiral to spiritualize the swastika, I wouldn’t be surprised.
It also begs a comparison to the game of chess and the I-Ching philosophy of causality. Geometrically, the 8 hexagrams of the I Ching revolve around a five-sided pentagram. The hexagrams make up the inner arm of the mandala, and they culminate in one of sixty-four possible outcomes on its outer arm. The directional changes depicted on eight swastikas can represent the 64 possible paths of I-Ching causality.
It may seem like the prime numbers are too arbitrary to be a legitimate model of causality in the universe; the only special property about them is that they can only be divided by the number one and themselves. Even if there aren’t any mystical qualities about the Ulam Spiral, it’s still pretty interesting that an infinite amount of odd numbers can create such an orderly, virtual chess-board on something as chaotic as a spiral.
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