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Telstar Balls Gone Wild

This paper describes an artistic project involving the fabrication of all 3,532 unique soccer balls that can be created by randomly reassembling the 32 pieces of a classic Telstar ball.

Original authors: Thomas Fernique

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas Fernique

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a classic black-and-white soccer ball, the kind used in the 1970 World Cup, known as the "Telstar." It's made of 32 pieces of leather: 20 white hexagons (six-sided) and 12 black pentagons (five-sided). When sewn together in the standard way, they form a perfect, round sphere.

Now, imagine taking those exact same 32 pieces of leather and sewing them together in every possible different way.

That is the heart of this project. The author, Thomas Fernique, is creating an art collection of 3,532 unique soccer balls. Each one is made from the same ingredients as the original Telstar, but the pattern of how the pieces are connected is different. Some will be nearly round, while others will look lumpy, bumpy, or strangely twisted.

Here is a breakdown of how this works, using simple analogies:

1. The Puzzle of Curvature

Why can't we just make a ball out of one giant piece of leather? Because leather is flat (like a sheet of paper), and a ball is round. You can't turn a flat sheet into a sphere without crumpling it or cutting it.

Think of it like a map of the Earth. If you try to flatten a globe onto a piece of paper, the continents get distorted. To make a real ball, you need to cut the leather into smaller pieces. By sewing these pieces together, you concentrate the "bendiness" (curvature) at the seams. The classic Telstar is special because it distributes this bendiness perfectly evenly, creating a smooth sphere.

2. The "Wild" Variations

The author realized that while the Telstar is the most famous arrangement, it's not the only way to connect 20 hexagons and 12 pentagons.

The Rule: You can never sew two black pentagons directly next to each other. They must be separated by white hexagons.
The Result: Even with this rule, there are 3,532 different ways to assemble the ball.
- Some of these new balls are "mirror images" of each other (like a left hand and a right hand).
- Some are perfectly symmetrical.
- Some are so distorted they look like they've been squashed or stretched.

The project's goal is to physically sew and display every single one of these 3,532 variations.

3. Ranking the Balls

Since there are so many, the author needed a way to organize them. They decided to rank them from "Roundest" to "Least Round."

They didn't just guess; they used a mathematical formula called the "isoperimetric quotient."
Think of it like this: If you have a fixed amount of leather (surface area), which shape holds the most air (volume)? A perfect sphere holds the most. Any weird shape holds less.
So, the author calculated the volume of every possible ball and lined them up from the biggest volume (roundest) to the smallest volume (weirdest).

4. The "Unfolding" Challenge (How to Sew Them)

How do you tell a human being how to sew a weird, lumpy ball? You can't just hand them a 3D model; they need a flat pattern to cut and sew.

The Net: Imagine peeling an orange. You can peel it in a spiral or in strips to lay it flat on a table. In math, this is called an "unfolding" or a "net."
The Problem: For some of these weird balls, it's hard to find a way to peel them flat without the pieces overlapping or tearing.
The Solution: The author wrote a computer program to find a "compact" way to unfold each ball. They created a flat map for every single one of the 3,532 balls, complete with numbers and markers to tell the sewer exactly where to stitch.

5. The Computer Simulation

Before sewing a single stitch, the author had to know which ball was rounder than the other. You can't measure the volume of a ball you haven't built yet!

Step 1: The computer takes the flat "peeling" of the ball and stretches it onto a sphere.
Step 2: It uses a physics simulation. Imagine the edges of the leather pieces are springs. The computer lets these springs push and pull until the shape settles into a stable form.
Step 3: The computer smooths out the sharp corners (using a technique called Catmull-Clark subdivision) to make it look like a real, slightly flexible leather ball.
Step 4: It calculates the volume of this digital model to determine its rank in the lineup.

The Current Status

This is an ongoing artistic and mathematical marathon.

The author has already hand-stitched about a dozen of these unique balls using real leather.
They have created a digital gallery where you can spin and examine all 3,532 computer models online.
The ultimate goal is to have a physical museum of every possible soccer ball that can be made from the Telstar's pieces, showing the incredible variety hidden within a simple set of shapes.

In short, this paper describes a quest to exhaust every possible combination of a soccer ball's pieces, turning a mathematical curiosity into a tangible, hand-stitched art collection.

Technical Summary: Telstar Balls Gone Wild

Problem Statement
The paper addresses the geometric and combinatorial challenge of fabricating all possible variations of the classic Telstar soccer ball. A standard Telstar is constructed from 32 pieces: 12 pentagons and 20 hexagons, arranged to form a truncated icosahedron. The project aims to physically construct every distinct isomer obtainable by randomly assembling these 32 pieces. While the truncated icosahedron is the most famous configuration, the mathematical space of fullerenes (structures composed of hexagons and pentagons with degree-3 vertices) allows for 3,532 distinct topological arrangements (isomers) using exactly 12 pentagons and 20 hexagons. The core problem involves describing, ordering, modeling, and physically assembling these 3,532 unique structures, each possessing a distinct shape and volume due to the varying distribution of curvature across the surface.

Methodology
The project employs a multi-stage pipeline combining graph theory, computational geometry, and physical fabrication:

Enumeration and Ordering:
The authors utilize existing algorithms (specifically Buckygen) to enumerate the 3,532 isomers. These are categorized into 92 achiral structures and 1,720 pairs of enantiomers (mirror images). To order the collection, the authors use the isoperimetric quotient ( $q = 36\pi V^2 / S^3$ ). Since the surface area ( $S$ ) is constant for all balls (composed of the same number of pentagons and hexagons), ordering by $q$ is equivalent to ordering by volume ( $V$ ). This allows the balls to be ranked from "roundest" (highest volume) to "least round."
Fabrication Strategy (Unfolding):
To facilitate hand-stitching, the authors determine an edge unfolding (net) for each isomer. This process involves:
- Constructing the dual graph of the isomer (where nodes represent faces and edges represent shared boundaries).
- Selecting a spanning tree of this dual graph.
- Cutting the edges of the polyhedron corresponding to the non-tree edges of the dual graph.
- Challenge: While "Dürer's Problem" (the existence of a non-overlapping edge unfolding for convex polyhedra) remains an open mathematical question, the authors found valid, non-overlapping unfoldings for all 3,532 isomers.
- Optimization: To assist the seamster, the authors employ a heuristic to generate "compact" unfoldings. Starting from an arbitrary face, they iteratively add adjacent faces that minimize the distance to the initial face, avoiding overlaps. The resulting nets are numbered, and markers are added to indicate seams that are distant in the 2D net but adjacent in the 3D form.
Computational Modeling and Volume Estimation:
Since physical fabrication is too slow to determine the ordering, the authors developed a computational model to estimate the volume of each isomer:
- Step 1 (Stereographic Lifting): The 2D unfolding is mapped onto a unit sphere using the inverse of the stereographic projection. This provides a topological sphere but with irregular, non-planar faces.
- Step 2 (Force-Based Relaxation): An iterative algorithm applies "spring forces" to the edges and non-adjacent vertices within faces. This simulates the limited elasticity of leather, pushing the structure toward a more realistic, rounded shape while preventing self-intersection.
- Step 3 (Surface Subdivision): To address the non-coplanarity of vertices within a face, the Catmull-Clark algorithm is applied (three iterations) to subdivide the surface into quadrilaterals, rounding the geometry.
- Volume Calculation: The final volume is calculated by triangulating the faces from their centers and summing the volumes of the resulting tetrahedra formed with the ball's center of mass.
Physical Construction:
The project involves hand-stitching natural leather pieces according to the generated nets. The process includes inserting an air bladder before closing the ball. The "Isolated Pentagon Rule" (IPR)—never sewing two pentagons together—is maintained, as the Telstar is the only fullerene with 20 hexagons that satisfies this rule.

Key Results

Total Count: The project identifies and plans to fabricate 3,532 unique Telstar isomers.
Chirality: The set includes 92 superimposable (achiral) structures and 1,720 pairs of enantiomers (totaling 1,812 distinct structures up to isometry).
Unfolding Success: Valid, non-overlapping edge unfoldings were successfully generated for every isomer, resolving the practical aspect of Dürer's Problem for this specific class of convex polyhedra.
Progress: As of the paper's writing, approximately a dozen balls have been physically hand-stitched, with the full fabrication and modeling pipeline ongoing.

Significance and Claims
The paper presents an artistic and mathematical project that bridges the gap between abstract graph theory (fullerene isomers) and physical reality (fabricated soccer balls).

Mathematical Context: The work highlights the richness of the fullerene isomer space, noting that while the truncated icosahedron is the standard for soccer balls, it is merely one of thousands of possible configurations.
Practical Application: The study demonstrates a viable method for generating manufacturing instructions (nets) for complex, non-regular polyhedra, solving the "which unfolding to choose" problem via a compactness heuristic.
Modest Scope: The authors frame the project as an artistic endeavor to "fabricate" these balls, rather than a study proposing new chemical applications or material science breakthroughs. The significance lies in the realization of the theoretical 3,532 variations and the visualization of how different assembly rules alter the global geometry (roundness) of the object. The paper explicitly notes that the study of isomers poses numerous mathematical problems but focuses here on the enumeration and physical instantiation of the Telstar family.