Mixolydian Hex Knight Lattice

This simulation is a generative artwork and procedural music sequencer driven by graph theory, greedy algorithms, and chiral geometry. What you are seeing and hearing is not pre-composed. It is the real time result of autonomous mathematical entities trying to solve a spatial puzzle. This construction does not crash or self interfere, instead it renders a continuous expansion outward to construct an emergent lattice. To the musicians, listen to how the chaos of spatial movement is anchored into harmony. Every time an entity leaps to a new hexagon, the numerical value of that space is converted into a musical pitch. To keep the rapid jumping from turning into harsh dissonant noise, the raw numbers are passed through a modulo function and mapped strictly to a Mixolydian scale. This anchors the generative melodies into a slightly bluesy, twelve tone equal temperament key. As the entities drift toward the outer edges of the spiral, the pitch climbs the octaves. If the frequency exceeds one thousand Hertz, the math wraps the pitch back down to the bass registers, creating beautiful cascading arpeggios. The rhythm, however, is entirely dictated by geometry. The drums only trigger when an entity lands on a rare central node within the emergent lattice. The cyan generation triggers kicks and snares, while the orange generation triggers hats and claps. Because the movement algorithm makes these specific spots incredibly difficult to land on, the resulting beat is an unpredictable and highly syncopated groove completely derived from the geometric paths of the pieces. To the mathematicians, you will recognize the foundational inspiration here. This is a heavy modification of the classic Trapped Knight puzzle. However, instead of a standard square chessboard, this universe operates on G3 triangular symmetry plotted with a non linear axial coordinate system. The numbers radiate outward from the center in a hexagonal spiral. This creates a gravitational pull, constantly forcing the entities to seek the lowest available number inward. The real magic happens through chiral slicing. A standard hex knight has twelve symmetrical jumping moves. In this two group mode, those moves are split in half based on rotational direction. Generation zero can only veer left, and generation one can only veer right. Because they are forced into these biased leaps, they weave a highly structured super lattice. The entities build perfect six sided rings that isolate the unvisited hexagons inside them. Those isolated central nodes, where the percussion hits, are mathematically defined by an equivalence class where q minus r is congruent to zero modulo three. As the simulation expands outward, dropping additional points at the end of the video demonstrates how the internal geometry of the lattice can still be meticulously filled. To the programmers, you are watching the precise orchestration of automata painting their own expansion. These entities operate in a self avoiding pathfinding loop, always prioritizing the lowest unvisited integer on the board. Because their movement rules are perfectly symmetrical and complementary, they never crash or trap themselves in a corner. Instead, they lock into a permanent, non interfering flow, mathematically guaranteed to weave an endlessly expanding tapestry. The visual simulation is directly tied to a procedural audio engine built on Tone.js. The engine runs a continuous check against the spatial math. The percussive triggers and melodic arpeggios are not a static backing track. They are dynamically fired by the state machine, turning the continuous path generation into a real time sequencer.