Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. This phenomena can be found in many fields from mathematics and linguistics to social sciences. An interesting occurrence of recursion is usually referred to as a strange loop - not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction to another, which feels like an upwards movement in a hierarchy, like the paradoxical hierarchies in the fictions of Jorge Luis Borges, or the recursive computational loops that can be found in quines: tiny computer programs that take no input, and output a copy of their source code as the only output. It's like a picture inside a picture, inside a picture, and so on, forming a loop that keeps going in a curious, repeating manner. In short, a strange loop is a paradoxical level-crossing feedback loop.

Recursion can also be found in natural language, such as in structures like syntactic categories, like a sentence. A sentence can be constructed in a way in which what follows the verb is another sentence: Dorothy thinks witches are dangerous, in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion. These systems are extremely useful in artificial languages, such as programming, where a series of instructions can be compiled and executed in ways that can achieve complexity very easily based on a simple set of rules. Repetition and loops based on certain conditions make a program dynamic, scaleable and reusable within many scenarios and environments. If using recursion, a function can call itself again and again, while considering changes in the environment during its repetitive executions. I use this method for example to control robotic arms and servo motors where the system needs to adjust its position along each step in a sequence, based on the feedback from the motors. Loops can be achieved using recursion, as the following pseudocode demonstrates:

function countFingers( hand ):
    if hand has no fingers:
        return 0
    else:
        return 1 + countFingers( hand with one less finger )

Apart of recursive calls in concepts and texts, the intersection of hierarchical abstraction layers is evident in various visual environments, where the medium takes on a taxonomical role closely tied to the represented content itself. Noteworthy instances of this artistic phenomenon occur when technology influences and shapes artistic concepts. A case in point for example is the indirect impact of physics and optics in post-impressionism, pointillism and similar, scientifically developed painting styles. In these movements, the dots and separated paint strokes cease to be mere components but instead become independent entities within the image architecture. They play a pivotal role in formulating meaning and colours within the perceptual domain. The resulting colours are often not mixed directly on canvas, but formulate an impression of the desired shade during the process of perception, in the visual cortex and the brain. This blending of world hierarchies is also observable in numerous op-art pieces, visual and auditory illusions, movies, performances or even games as theatre pieces. In such cases, the play incorporates different aspects of reality through chance, improvisation and abstraction leaks, creating a dynamic interplay between artistic expression and the unpredictable facets of existence.

Meta-narratives

Many of these medium-focused critical expressions can be found in the experimental glitch scene of the early 2000s. Starting around that time, bugs, application errors, system crashes, clipping, aliasing, distortion, quantization failure, and even the noise floor of computer sound cards became used and served as the raw materials that composers aim to incorporate into their music and computational visual environments. Similar to how post-impressionists responded to the taxonomical changes in their technological apparatus, post-digital artists, a few decades ago, had to shift their focus from the well-defined, figurative concepts of the medium to its meta-narratives. These artists began seeking and exploring aesthetic constellations within the mechanics, temporality, and reconfigurable aspects of these novel authoring systems. Much like visual artists who transitioned their focus from the foreground to the background, this shift helped to expand their perceptual boundaries. It allowed them to capture the enigmatic character of the background, finding inspiration in the often overlooked intricacies of the medium's underlying workings.

In today's art scene, many artists are exploring the idea of meta-narratives, especially in connection with digital media. With the rise of decentralized computation and the emergence of network-native digital culture, there's a broader interest in using the medium with its underlying mechanics as a tool for world-building. There are myriad examples in the blockchain-native and crypto-critical art scenes, where the conceptual ground of the pieces are directly related with digital transactions, shared and distributed consensus mechanisms and composable identities. An early, critical piece is called Harvest, by Julian Oliver, which is a work of critical engineering and computational climate art. It is an outdoor physical installation that uses wind-energy to mine cryptocurrency, the earnings of which are used as a source of funding for climate-change research. A poetic reflection on the abstraction layers of the blockchain ecosystem, also an example of a conceptual feedback loop, where the output of the process has been fed back to its input as an investigation of the laws of thermodynamics and the unsustainable nature of the proof of work crypto mining ecosystem.

Another, more recent playful piece is called You Are Here from the artist 0xfff, which is an on-chain, conceptual generative artwork that plays with the act of bridging between blockchains. The individual tokens of the collection are bridgeable between various blockchains. Every bridge transfer creates a visual trace between the chain ids, which results to minimalistic, conceptual graphs that are resembling the continuous transitions of the digital artefact itself.

Recursive Grids

The idea of grids, serving as the most basic forms of representing worlds, is prevalent in various scenarios, ranging from straightforward simulations to intricate narrative structures. Grids consist of values distributed across a two-dimensional surface, functioning as positions in a board game, pixel configurations on screens, proportions of financial values in two-dimensional datasets, or geometric references on real-world maps. At its core, the grid acts as a metaphor for the world, making it useful to simulate objects with complexities either smaller or larger than itself. This serves as a valuable tool for investigating abstraction leaks and recursive ontologies. When appropriate rules are applied to its cells, a grid can transform into an operating system. Furthermore, recent attempts have demonstrated that a grid can become a Turing-complete universal computer, by utilizing Conway's Game of Life to construct the machine. Grids are also employed to navigate addressable space, representing chunks and clusters of memories that colonize the abstract digital space. Given that digital information lacks inherent spatial qualities, visualizing data coverage in such spaces requires specialized efforts and algorithms.

By employing recursion, space-filling, geometrical subdivision, and packing algorithms, it becomes possible to arrange space according to factors beyond just size and position. In essence, any comparable feature can be utilized to depict the relationships between entities within a space. The example mentioned above provides a practical everyday illustration of this concept, showcasing how the amount of allocated space on a hard drive can be visualized using such techniques.

The idea of recursive grids however is not limited to literal grids but can encompass any repeated pattern or structure that recursively appears within the artwork. As it has been shown earlier, this concept aligns with mathematical and geometrical principles, as well as with artistic explorations of visual complexity and optical illusions. Recursive grids are recurring motives within my long-form generative art practice, these structures served as the basis for my pieces Nil, k3rnel, and sh3ll respectively.

State Flow

In gr1dflow, the visual grid system serves as the foundational architecture for the scenes. The depth of recursion and the grid alignment parameters are derived directly from blockchain specific components, like mint iteration, minting timestamp and the minted block number on the chain. Custom GLSL shaders are applied to the grid cells, enabling swift and seamless colour transitions using a set of manually chosen palettes. The gradients employed in this process draw inspiration from techniques found in retro video games, specifically dithering, which allows for the display of a limited set of colours. Dithering is a deliberate introduction of noise to randomize quantization error, preventing noticeable patterns like colour banding in images. gr1dflow also contemplates the limitations of the medium by employing this technique. Dithering is a common practice in processing digital audio and video data, often applied in the final stages of mastering audio for CDs. Human perception has already incorporated this limited aesthetic into daily life, evoking peculiar nostalgic sensations reminiscent of VHS cassette error stripes, the faded outlines of Xerox photocopies, or the distorted, blurred colour fields seen in Polaroid photographs.

Computers operate in real-time, allowing for changes, dynamics, and movements that extend beyond static images. Interfaces of computers have their own limitations, aesthetic qualities, and occasional shortcomings. The temporal aspects of an interface play a crucial role in the human experience. Digital worlds created with computers are fundamentally different from traditional concepts of space, especially when viewed through feature-agnostic, recursive spaces. They also lack traditional notions of time: temporality, choreography, and dramaturgy. Artists and creators must explicitly incorporate these elements through animations, delays, rhythmic responses, and similar techniques into these systems. At times, the transitions between different states within an interface can convey more meaning than the content of the states themselves. The speed, dynamics, and density of movements can effectively communicate messages and evoke feelings that might be challenging, if not impossible, to express with words or static images. These changes can mentally be experienced through a continuity of transitional flow. The concept of "flow" refers to a mental state of complete absorption and focus in an activity, to the point where time seems to pass quickly, and individuals experience a sense of energized focus and enjoyment. Coined by psychologist Mihaly Csikszentmihalyi, flow is characterized by a deep immersion in a task, where the individual is fully engaged and concentrated, often losing a sense of self-awareness. The transitions in gr1dflow are intended to convey these altered states of the mind - constant flux of a system, with no explicit purpose, yet showing references to the concepts above.

Thanks to Nat Emodi for providing insightful suggestions that contributed to making the text more comprehensible. Cover image contains elements from M.C. Escher's Relativity and a recursive elementary automation from Stephen Wolfram's New Kind of Science (p. 274).