Quadrature
Systems Summoned into Motion.
— By Darien Brito
Have you ever wondered why most digital works of art are static, and displayed and treated as if they were analogue paintings? Quadrature is a digital piece, and exists to relay the beauty of systemic forms in a manner only possible in the digital realm. If algorithmic art is essentially the definition of a system, then Quadrature is designed as a generator of systems. Each mint opens an endless expanse of shifting forms to be discovered and explored interactively: the seed’s initial image soon morphing in tandem with its owner’s desire, as you alter its behaviour and appearance via numerous controls.
As a creative coder, I find that the discovery process of an algorithm and its potential variations is most enjoyable, and one of the aspects that makes generative art so engaging. Unfortunately, collectors are often far removed from that process once they mint a new collection. Quadrature is my attempt to bestow agency to the collector, whose every mint can be re-configured within a specified set of features, and whose structures and reconfigurations are dynamic and enjoyable parts of the composition.
Deterministic Serendipities
Under this concept, a mint is not just a single piece, but a generator of artworks capable of yielding variations that the collector can discover and curate. But unlike projects that allow the collector to tweak parameters at will, the parameters of possibilities for a Quadrature are bounded by the features it acquires upon its minting. Its algorithm is embodied between an open-ended parametric set-up and a fixed art piece. Due to the constraints of on-chain generative art, every morphing has to be deterministic; a mint will always have the same initial state and morph across the same transformations. The amount of possible transformations, however, is virtually infinite.
This deterministic nature of on-chain art inherent to NFT technology is highly desirable in the context of provenance within both digital, and traditional, art markets. Yet the use of determinism is atypical in the creation of generative systems which are based on aleatoric processes. In the long tradition of computer-driven art, many artists have in fact employed indeterminism as an inherent property of their work. That involves a different kind of compositional process where the unknown, or even irreproducible, becomes part of the core language of the piece. I ascribe to that approach: my work outside the blockchain rarely specifies a seed on initialization and embraces indeterminism wholeheartedly as a driving force instead. To this end, I have worked with opera productions, theatre groups, and in my own electronic music performances to create real-time graphics which react to live inputs, such as changes in sound.
Quadrature may be seen as an attempt to capture the essence of these transcendent, indeterminate moments, suggesting motion over stillness and encouraging serendipity within a purely deterministic environment. I’m well aware that this aim remains ultimately ironic, given that the current on-chain paradigm for NFTs does not facilitate it.
The earliest sketches of this project were experiments in trying to understand what was geometrically intriguing about Theo van Doesburg’s work, as seen in Composition VIII (1918). How could I make sense of it? I wasn’t so interested in its representational aspect- to what extent it looked (or didn’t look!) like a cow- but more in how geometry was employed in the abstraction of its subject. My early experiments in trying to replicate this style of the distribution of squares in space developed into deeper research into recursion and grids, far from van Doesburg’s motivations and methods. This effectively became a study into ratios: of how to subdivide space in such a way that I could preserve big spaces composed from small forms in a perpetual balancing act. The best choice was to use rectangles which perfectly fit into the regions that were just cut. Nonetheless, some variations in Quadrature are left intentionally minimal as a nod to the initial inspiration of van Doesburg.
I have experimented with morphing artwork before- most notably with the ArtBlocks collection Pigments- but whilst both use shaders to implement this, the respective collections share little else in common. Quadrature has an almost two-dimensional aesthetic despite the fact it morphs across three dimensions. The speed of its morphing is ‘curved’, so that each element has its own timing within a discrete phase. This technique of using ‘smooth-outs’ and ‘smooth-ins’ is a core principle of animation, employed to make movement feel organic. These transitions between states in Quadrature are thus not cuts, but smooth, non-linear interpolations which reveal the individual elements that constitute the whole.
Above all, I was concerned with how minimalist aesthetics might help lay bare the underlying mechanics of the algorithm itself. To that end, I wanted the pieces to retain the quality of technical drawings, with a style inspired by Dutch graphic design. I am Ecuadorian, but the Netherlands has been my home since arriving many years back as a student, and so the palettes and aesthetics of de Stijl loomed large.
Algorithmic Tension
In the mathematics of ancient Greece, quadrature meant determining an area in geometry using squares. I've adopted the term to illustrate the algorithm in this project, where specific areas of the canvas are computed with various ratios and either filled with solid colours, gradients and patterns, or left empty.
The compositions build on the tension between two underlying recursive systems. These independently compute their sub-spaces and are ultimately combined:
- System A subdivides the virtual grid without keeping track of initial areas (destructive process).
- System B partitions space and keeps track of computed parts (constructive process).
It is often easy to spot the two systems at play since one tends to result in large, ordered blocks, and the other in granular, chaotic elements. There is a simple purity in the reconfiguration of spaces, with the act producing a myriad of interesting elements, even when using simple components to manifest itself. It is often forgotten that generative coding produces systems, and systems are beautiful to dissect, revealing the algorithm itself.
Every time a re-configuration is triggered, the system updates its properties and morphs between the current state and a newly found one. Given the deterministic nature of on-chain art, this process proved to be technically challenging since the history of all the random numbers in the code needs to be accounted for at all times, as they are swapped seamlessly between transformations.
I was drawn to apply to the ‘Blind’ curation process at gm. studio for multiple reasons. I experienced a great degree of critical and financial acclaim after my launches of 2021, which ultimately left me feeling rather empty; a common occurence, I’ve learned, among artists who experience rapid success. At this juncture of my artistic career, the anonymity of blind curation served as an ego-check which allowed me to validate the merit of my work, helping to quell my own self-doubt. As I work in a diverse style or form with each new project, the critical feedback I received helped immensely as I worked to improve Quadrature towards acceptance by the panel.
With Quadrature, I appeal to a deeper appreciation of digital generative artforms: a sphere in which interpretation is often drowned out by market chatter, and where concentration is measured in seconds spent glancing at gallery thumbnails. I want to motivate people to engage and play with their mints dynamically until something catches their eye. In that special moment, I believe the feeling of discovery enriches the appreciation of the finding, and invites a more careful contemplation of the piece, the medium and perhaps even personal taste.
I hypothesise that in this mortal world, the impermanence of a configuration enhances its desirability. A Quadrature owner may not find the default state of their mint particularly interesting. But within it, there is potential for much more if they are willing to modify the piece. I foresee a situation where someone will find a variation they particularly like, but to get there, they will have to remember how many steps they took in interacting with the artwork. That process can be as enjoyable as it may be frustrating since it requires an investment of attention and engagement with the piece. I want Quadrature to evoke that universal beauty of the ephemeral, and the fragility of desired forms.
Traits
| Assembly | A|BA |
| Surface | Solid|Patterns|Gradients |
| Set | 8 possible sets from 1-8 |
| Palette | (88) Aramee|Ackern|Terracotta[abc]|Modario|Blane|Mint|Fluoric|Cyricle|Seurat|Nolenic|Rhye[abcde]|Grise[ab]|Azur[abc]|Corio|Torque[abcde]|Vefem[abc]|Ripe celeste[abc]|Rosa[ab]|Nivel|Kelenic|Askaton[ab]|Ceales|Kinse[ab]|Turquoise rose|Nerman|Venthe[abc]|Watermelon[ab]|Ladrillo|Abneme|Micelan[abcd]|Esper|Archipielago|Dark fluoric|Nitee[ab]|Delata[abc]|Carnaval|Charemica[ab]|Oggol|Bosque[abc]|Elaene|Linolee|Malegenta|Fragilem[ab] Vithe[ab]|Hemes|Circo|Dark orange|Golden opacity|Apritz|Soberel[ab]|Purpurine|Monochrome[ab] |
| Decoupled | True|False |
| Minimized | True|False |
Controls
Animation
| shift + A | enable auto-morphing |
| shift + S | static view |
| shift + → | increase morph speed |
| shift + ← | decrease morph speed |
| shift + ↑ | increase morph wait |
| shift + ↓ | decrease morph wait |
| shift + Space bar | trigger a single morph |
Aspect ratio
| shift + 1 | 3/4 (default) |
| shift + 2 | 1/1 |
| shift + 3 | 1/2 |
| shift + 4 | 2/3 |
| shift + 0 | adaptive aspect ratio |
Exporting
| shift + B | use background texture |
| shift + N | remove background texture (use to print on paper) |
| shift + E | export image at the current resolution |
| shift + O | export image at 6k resolution |
| shift + P | export image at 12k resolution (exported in parts) |
Collection Size 300
Special thanks go to the Hyphen collective, for their generous insights, suggestions and criticism.
Darien Brito,
with Yorks (@generativepoet)
