- Single-seed
- \( z_n \rightarrow z_n^2 + z_0 \quad \)\(\forall \ |z_n| < 2 \)
- Composite
- \( \bigcup\limits_i z_{n,i} \rightarrow z_{n,i}^2 + z_{0,i} \quad \)\(\forall \ |z_{n,i}| < 2; \max\Re(z_{0,i})-\min\Re(z_{0,i}) < \epsilon \)
- Work
- \( \propto k^\alpha \quad \)\(k \in \mathbb{N^*} \)
- Mining
- \( \exists z_0 \in \mathbb{C}\ \)\( \ni |z_{n < k}|<2 \land |z_{n \ge k}|\ge2 \)

**Composites** Equal parts mathematical rigour and
aesthetic curation, layering multiple single-seed images
together results in beautiful composite forms. By restricting
the scope of seeds that can be overlaid, curation remains driven
by algorithms. The smaller the \( \epsilon \) the more
difficult it is to curate a piece—this value is listed in
metadata as *Reals Range* (the smaller the better). Note
that the relative positioning of each constituent element in a
composite is also mathematically governed.

** 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.

**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

Extending the theme of the logo, Composite I is anchored by a single seed near the Eclipse series of the Genesis collection. Each of the 18 scalloped hollows along the edge is decorated with a set of seeds, each with their own individual beauty. The 12 decorative seeds are not, however, individually placed but are n-body systems that naturally fit into the hollows.

A series of five n-body clusters, joined with sweeping pearl-threaded arches, Composite II captures the diversity of seeds that can be found despite having one of the tightest \( \epsilon \) ranges.

An ornate sweep to the central body creates three arcs, each filled with a cluster of seeds that themselves form delicate interlocking patterns.

One of only two Composites with “organic” symmetry, the sides are qualitatively mirrored but don’t match perfectly. Although the equations allow for perfect symmetry over the real axis, this instead achieves it through careful curation.

What do you get if you cross a 1970s time-warp animation with a robotic eye? Sorry, there’s no punchline, but there is a whimsical composite formed of seven overlapping trajectories. The distinctly quadralateral form around the central pupil contrasts with the otherwise flowing arcs of the others.

Surrounded by ghost-like forms reminiscent of Styx
`0x607711d926`, Composite VI is surrounded by pearl-threaded
arches that intersect sharply to form a jagged border not seen any any other
pieces.

The smallest \( \epsilon \) of all Composites, VII demonstrates the power of the butterfly effect with 20 tightly packed seeds that each follow slightly diverging trajectories. These result in clumps of Cantor-dust-like dots in the macro structure that culminate in the densest spiral amonst all the works, composite or otherwise.

As with IV, VIII is the only other composite to demonstrate “organic” symmetry—the sides are qualitatively mirrored but don’t match perfectly. Composite VIII, however, takes this a step further with the illusion of a third dimension as the butterfly wings are again mirrored in the background. Each of the four wings is ornately decorated as seen in the second image.