Encoding-Decoding Constellations

Many classic designs feature periodic arrangements of interlocking stars. Crafting these motifs is more than an aesthetic exercise—it is a geometric puzzle in which every element must dovetail precisely with its neighbors. When the stars exhibit “standard” four-fold or six-fold symmetry—those that align naturally with square or hexagonal grids (Fig. 1a–b)—the puzzle has well-known solutions [3].

Difficulty escalates as soon as one ventures beyond these canonical lattices. Five-pointed stars, for example, can tile the plane only if artists slip in auxiliary “bow-tie” tiles (Fig. 1c). On the other hand,  nine- or eleven-pointed stars require a delicate redistribution of geometric error across the pattern (Fig. 1d) [4]. Despite ingenious historical workarounds, designing motifs that depart from standard symmetries—or that combine stars of differing point counts—remains labor-intensive and error-prone.

This challenge raises a central question: Is it possible to develop a systematic way of designing Islamic art–inspired constellation patterns that mix stars of arbitrary symmetries into a single, seamless composition? We answer with MotifMap.

Overview of Method

Craig S. Kaplan and I developed MotifMap, an encoding–decoding framework that transforms any triangulated graph into a constellation pattern. Users specify only the combinatorial structure—stars as vertices and their connections as edges—while the system automatically computes the geometry: precise radii, positions, and orientations that allow the stars to interlock seamlessly. The core insight lies in decoding the graph into a circle-packing scaffold (via the circle packing theorem) and then applying classical compass-and-straightedge construction principles [5, 6].

The video below demonstrates MotifMap in our custom design tool. Users specify only simple graphical descriptions, and the system automatically generates the corresponding patterns.

By combining mathematical abstraction with computational tooling, our approach provides several advantages, including: 

1. Automatic resolution of geometric inconsistencies: The circle-packing construction naturally distributes geometric errors and irregularities across the pattern, even when motifs have highly unconventional symmetries. This approach produces seamless baseline patterns, including those that combine stars with unusual point counts.
2. Rapid design and exploration: Users can work with simpler graph-based descriptions, and because decoding is fast, they can easily turn what-if questions into visual prototypes. By modifying the graph (e.g., as shown in the video above), users can see the resulting effects almost instantly.

Results and Discussion

We demonstrate possible applications of MotifMap through a series of results and ongoing experiments.

Understanding classical patterns. The system facilitates the study and reconstruction of historical designs. By reverse-engineering existing patterns in MotifMap, I was able to interactively explore and gain deeper insight into traditional construction rules.

Designing new patterns. MotifMap supports the discovery of novel constellation patterns. By manipulating the encodings and instantly observing the resulting constellations in our design tool, I was able to quickly explore new designs, such as the one shown below.

Creating freeform or generative designs. Because the framework allows stars of any symmetry, I was able to create forms ranging from structured to chaotic, including generative compositions such as the artworks shown below. Disintegrating (State of Mind) was exhibited in the Mathematical Art Exhibition at the 2025 Joint Mathematics Meetings, where it received an Honorable Mention.

Spiraling (2025) appeared in public art exhibition Intersections, jointly produced by the Seattle Universal Math Museum and the Mercer Island Visual Arts League, and then Bridges, a mathematical art conference, in Eindhoven, Netherlands. 

The sculpture below explores the interaction between a freeform pattern and three-dimensional form, laser-cut from printmaking paper.

Beyond static forms, MotifMap has the potential to support diverse modes of expression, including live  pattern-making and performances. We will share some of these possibilities in future posts.

We thank the Council for the Arts at MIT for partly funding our fabrication.