Kobayashi Art Studio | Generative Math Art

Kobayashi’s legacy

Shoshichi Kobayashi spent most of his career at UC Berkeley, making groundbreaking contributions to the understanding of complex manifolds and geometric spaces.

• The Kobayashi metric: His most famous contribution introduced a sophisticated way to measure “distance” on curved mathematical spaces that fold and curve in ways that are difficult or impossible to map and measure.
• Mappings and transformations: What made Kobayashi’s work powerful was his focus on mappings between spaces, understanding how one geometric structure can transform into another while preserving essential properties.

The Fibonacci connection

The heart of our tool’s visual attractiveness lies in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, an arrangement thought by some to be pleasing to the eye, though research is unclear on the point.

This sequence appears throughout nature, in flower petals, and pinecone scales, because it embodies a kind of optimal growth and structure. Our eyes find these proportions inherently pleasing. The tool uses what we call the “Kobayashi sequence” (Fibonacci-based) to determine color relationships, ensuring shapes at different depths and positions share a mathematical kinship rather than clashing randomly.

Kobayashi found a way to measure distance on complex spaces that cannot be measured normally. He used holomorphic discs: small, perfectly round patches of flat space, each one bent into place along the curved space without tearing or folding back on itself. To find the distance between two points, you lay a chain of these discs from one point to the other and measure across each disc with the disc’s own built-in distance. The Kobayashi distance is the smallest total any chain can achieve.

In other words, he measured something complicated by covering it with copies of one simple shape. Our tool works similarly. Every image is built from a few simple shapes repeated across a grid, and then the Fibonacci-based sequence sets a steady, measured step from one color to the next, so the pieces relate to each other instead of feeling random. The result is a harmonious piece that feels put together.

From recursive spaces to recursive art

The tool offers two modes that reflect different approaches to understanding geometric structure:

• Fixed Grid mode creates uniform cells: a regular tessellation of space where every shape gets equal importance. This mirrors how mathematicians often begin studying complex spaces: by imposing a regular coordinate system.
• Recursive mode echoes Kobayashi’s work more directly. The algorithm subdivides space organically, creating a hierarchy where each region contains smaller regions.

Just like Kobayashi worked with large, complicated spaces by looking closely at small pieces of it and how they fit together, each split in our recursive math generates a smaller piece that still fits back into the whole.

How to use the tool

Choose a palette

Click any color circle to select a palette. Each contains five harmonious colors distributed according to the Fibonacci-based sequence:

Terra | Ocean | Forest | Gold | Noir | Berry | Pastel | Sunset

Select a mode

• Fixed Grid creates uniform cells: every shape gets equal space. Choose grid sizes from 2×2 up to 24×24.
• Recursive mode subdivides space organically. The algorithm decides whether to split each cell into four smaller cells, which creates natural variation in shape and grid sizes.
• Adjust the depth to control design complexity. Lower depths give you larger shapes with subtle detail; higher depths create dense, intricate patterns.

Set your print size

Choose dimensions that match your intended use, from 8×8 to 30×36.

Choose your DPI

DPI (dots per inch) determines print quality:

• 150 DPI: Drafts and screen viewing
• 200 DPI: Good quality prints
• 300 DPI: Professional, high-resolution output
• Higher DPI means larger file sizes but crisper prints.

The shapes

Each cell can contain one of several shapes, chosen by an algorithm using weighted randomness.

• Squares and circles
• Quarter arcs and diagonal triangles
• Concentric shapes and outlined squares

Just as Kobayashi’s mathematics maintained consistent measurement even as underlying spaces curved and changed, our tool maintains visual coherence even as shapes and colors vary.

Coloring pages

Every composition can be exported as a coloring page: the same design rendered as black outlines on white. These make wonderful:

• Mindful coloring activities for all ages
• Classroom art projects exploring mathematical patterns
• Therapeutic exercises for focus and relaxation

The coloring page matches your current design exactly, so you can compare your colored version to the generated palette, or create something entirely your own.

Tips for best results

• Generate multiple times. Each click creates a unique composition. Keep clicking until one speaks to you.
• Match grid size to print size. A 4×4 grid on a 30×30″ print creates bold, chunky shapes. A 16×16 grid on an 8×8″ print creates delicate detail.
• Try recursive mode for an organic feel. The variable cell sizes create more natural-looking compositions that echo patterns found in nature.
• Use Noir for versatility. Black-and-white prints match any décor and frame beautifully.
• Print on quality paper. Matte cardstock or fine art paper.