Kobayashi Art Studio | Generative Math Art

What is Kobayashi Art?

My Kobayashi art tool is a tool that creates a pleasing geometric arrangement of shapes using mathematical principles. Each iteration combines simple shapes, squares, circles, triangles, and arcs, arranged in grids that feel both random and intentional. I think it looks like 60’s modern art, you know that there is a pattern. You can’t put your finger on it, but it is pleasing to the eye. I also created a Valentine’s Day Kabayashi heart, I think it would make a lovely card to give to your sweetie.

The tool is named after Shoshichi Kobayashi (1932–2012), a renowned Japanese mathematician whose work in differential geometry explored how complex spaces can be understood through their internal structure and the relationships between their parts.

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 particularly 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 tools’ visual attractiveness lies in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, an arrangement that is intrinsically pleasing to the eye.

This sequence appears throughout nature, in spiral shells, 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.

This isn’t arbitrary. Just as Kobayashi’s metrics measure the “optimal” paths through curved spaces, the Fibonacci sequence creates optimal visual rhythms. The result: compositions that feel discovered rather than designed, organic rather than mechanical.

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.

This reflects a fundamental principle of differential geometry: complex structures are understood by examining how they subdivide and relate to themselves at different subdivided scales. They mirror each other, no matter the complexity. Each recursive split creates a new “local geometry” that relates to the whole, much like how Kobayashi’s metrics. Each recursive split creates a new “local geometry” that relates to the whole, much like how Kobayashi’s metrics expand local properties to understand global structure.

How to Use the Tool

1. 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 slider to control design complexity. Lower depths give you larger shapes with subtle detail; higher depths create dense, intricate patterns.

3. Set Your Print Size

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

4. 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.