PROOF OF {ART}WORK

Everything you see in these pieces is generated by mathematics and programming. At no point did I interact with graphic design software—only a code editor. The custom rendering software is responsible for precisely placing millions of points, their positions governed by the simple equation above.

While this equation is well-known, it has never been explored in this way before. A mainstay of the study of chaos, the sequence of points is at the whim of the butterfly effect. Small changes in \(z_0\)—the NFT ID—result in the vastly different forms that each series takes. Every piece combines macro-level gradients, reminiscent of organic forms, with intricate microstructures revealed when zooming.

This magnification lies at the heart of the rendering engine, capable of producing exapixel resolution on demand (kilo, mega, giga, tera, peta, exa). A billion billion pixels, if printed in full at 250 DPI, these images would be larger than the area of Jamaica.

n-Body Systems Note that the governing equation knows only of the current point \(z_n\) and the NFT ID \(z_0\). Despite this ignorance regarding its history, the inherent mathematical structure in the sequence is obvious when viewed globally as an entire image. The n-body systems (Binary Star, Styx, and Gemini) take this emergent behaviour one step further with multiple distinct objects, each sharing the same form despite a single \(z_0\) seed.

Scarcity The butterfly effect ensures that any change to \(z_0\) results in a different trajectory, so no two works are identical. Some pieces appear qualitiatively similar despite distinct sequences of \(z_n\)—these are sold as series of 3 editions although each member is provably unique. Artworks themselves are therefore non-fungible, much like the tokens that represent them.

Persistence Every work’s _tokenId contains all the information needed to extract \(z_0\) and generate the piece using the equation for \(z_n\), even if this site is taken down. The on-chain treasure hunt demonstrated this.

Rendering Bespoke algorithm with theoretically optimal memory-compute tradeoff. \(\mathcal{O}(k)\) memory and \(\mathcal{O}(\log{}k + m)\) compute for \(m\) average points per rendered region, regardless of zoom level.

Environment The effects of cryptocurrency on the environment are well documented, and proof of work is the culprit. In contrast, this art uses negligible energy—30 \(z_0\) values were found with 3.1kWh ≈ a half-hour charge of an EV.

Contract ERC-721 verified on Etherscan

@divergence_art

The genesis series of the collection, the feature image shown above was the first NFT ever sold. All editions have now found new homes, but may be available for purchase on the secondary market.

While the terraces of the three Wormhole pieces lead the eye deep into their centre, it is in fact the edges that are most fascinating upon closer inspection. As shown in the second view, a simple spiral begins with lone points only to form rich textures as it extends into the body of the piece—with points grouping together on the inner side to produce the frill effect seen from the distance.

The terraced steps are also interesting, an effect achieved by sudden changes in the density of the points. It is hard to believe these perfect borders could be achieved by such simple mathematics.

Mathematicians conjecture that 5 is the only odd untouchable number—the sum of all proper divisors of an integer. Similarly these editions, each with 5 arms, capture the untouchable nature of the butterfly effect’s sensitive dependence on initial conditions. No matter how small the deviation—the \(z_0\) values shown here are \(<1.25 \times 10^{-16}\) apart—the final outcome is vastly different in each of the three editions.

To collect an Untouchable NFT is to own something on the edge of chaos.

Like all of the pieces in this collection, this series displays shapes and patterns found in the natural world. As a diver, I am reminded of life found in and around coral reefs—in particular E. crispata, more commonly known as the lettuce sea slug. Also known as solar-powered slugs, they are efficient in their use of energy sources. This is something they have in common with this series, achieving exquisite aesthetic characteristics with as few as 700,000 points.

Binary stars are systems of two stars orbiting around a common barycentre—quintessential examples of two-body systems in nature. While these bodies appear to be distinct they are in fact generated from a single seed—much like their astronomical namesakes, which are theorised to result from a single star-formation process. The dense boundaries display a beautiful ridge-like effect that is reminiscent of pressure waves during atmospheric reentry.

This series was the first demonstration of an n-body system found while mining NFTs.

Named for the fifth moon of Pluto, Styx editions exemplify the beauty of n-body systems with a total of five inwardly oriented objects appearing to orbit a common epicentre. While each of these bodies differs in shape and size, they share common characteristics—including smooth density gradients and distinctive, wave-like edges.

Latin for twins, the Gemini two-body system displays remarkable finger-like projections as if two hands were reaching for one another—just as in its namesake constellation. Unlike any other series, the gradient waves seen in edition 0x10ae1ca895 are distinctly polygonal.

The macrostructure of Archipelago pieces belies their hidden micro allure—collections of almost identical islands dotted around the border, each of which could be a piece in its own right. Viewed as a whole, Archipelago editions are deceptively simple, but zoom in and you will discover the immense depth of NFTs.

The second view is a portion of the whole system shown in the first, simply magnified 512x.

Another magnificent “archipelago” series, Slartibartfast was named in honour of the character in The Hitchhiker’s Guide to the Galaxy. As a designer of planets with a penchant for elaborate coastlines, I think Slartibartfast would have been as proud of these editions as he was of his Norwegian fjords.

Again, the second view is a portion of the whole system shown in the first, magnified 512x.

In his eponymous sequence of galaxy classification, Edwin Hubble described two branches of spiral galaxies—the S classes. With no less than 20 million points in each edition, Hubble S combines macro gradients akin to Wormhole surrounded by scores of galaxies more in keeping with the archipelagos. New visual phenomena wait to be found as you continue to explore new depths, such as the spirals shown in this second image taken at 8192x.

Embodying all aspects of pieces, while also having the highest proof of work of any series, these are must-have editions for serious collectors.