Hyperbolic Rep-Tiles and Self-Similarity

ISEF Category: Mathematics

Subcategory: Geometry and Topology  ·  Difficulty: Advanced  ·  Setup: University Lab  ·  Time: Full Year

The Hook

What if one shape could copy itself forever, but only in a curved world? In the hyperbolic plane, triangles and polygons can fit together in ways that break your flat-paper instincts. That makes it a perfect place to study self-similarity, the same idea behind fractals and tessellations. You can turn that idea into a proof, a pattern, and an animation.

What Is It?

A rep-tile is a shape that can be cut into smaller copies of itself. On a flat surface, you might think of a square made from four smaller squares. In this project, the same idea happens in the hyperbolic plane, a geometry where space curves so that shapes behave very differently from the ones you draw on notebook paper.

The specific setting here uses the regular {7,3} tiling, which means seven-sided polygons meet three at a vertex. That tiling gives you a built-in pattern to explore substitution, which is a rule for replacing one shape with a collection of smaller shapes. Your job is to understand how the pieces fit, then prove that the replacement really recreates the original shape at a different scale.

Why This Is a Good Topic

This is a strong science fair topic because it has clear structure, but it still leaves room for original math. You can test one substitution rule, prove why it works, and compare different ways of describing the same self-similar pattern. The topic connects to geometry, symmetry, tilings, and algorithmic thinking, and you can present your results with diagrams and animation even if you do not have a lab.

Research Questions

• How does the chosen substitution rule change the number of tiles needed to recreate the original region?
• What is the effect of different seed shapes on whether a hyperbolic rep-tile construction closes cleanly?
• Does the {7,3} tiling admit more than one self-similar decomposition for the same base polygon?
• To what extent does the scaling factor determine the depth of the substitution tree?
• Which boundary patterns stay invariant under repeated substitution in the hyperbolic plane?
• How does the combinatorial growth rate of the tiling compare with the Euclidean case?

Basic Materials

• Graph paper or dot paper for sketching tilings.
• Straightedge and compass or a geometry ruler set.
• Colored pencils or markers for tracking tile generations.
• A notebook for recording proofs, diagrams, and conjectures.
• A computer with browser access for geometry visuals and writing.
• Optional printouts of hyperbolic tiling diagrams for annotation.

Advanced Materials

• University access to geometric modeling software or symbolic math tools.
• A computer with Python installed for combinatorial checks.
• Manim for creating the substitution animation.
• A LaTeX editor for writing the proof cleanly.
• Access to a mathematics library for books on hyperbolic geometry and tilings.
• Optional use of a drawing tablet for precise figure editing.

Software & Tools

• Manim: Animates the substitution rule so you can explain self-similarity step by step.
• GeoGebra: Helps you sketch and inspect geometric constructions before you formalize them.
• Python: Checks tile counts, recurrences, and branching patterns in your substitution system.
• LaTeX: Lets you write a polished proof with numbered lemmas and clean figures.
• Inkscape: Edits diagrams and labels so your poster and slides stay readable.

Experiment Steps

1. Define the exact hyperbolic region or tile family you will study, so your project has a precise object to prove things about.
2. Choose one substitution rule and write it in a way that a reader can follow without guessing missing steps.
3. Build the counting logic that shows how many smaller copies appear at each stage of the construction.
4. Test whether the pattern really reproduces the original shape after scaling, rotation, or both.
5. Check edge cases where the substitution may fail, overlap, or leave gaps.
6. Turn the proof into a visual story with an animation that matches each step of the argument.

Common Pitfalls

• Confusing a Euclidean tiling rule with a hyperbolic one, which breaks the geometry of the whole construction.
• Proving a picture instead of proving the substitution rule, which leaves the core argument incomplete.
• Losing track of orientation, which can make copies look right while the adjacency relations are wrong.
• Ignoring boundary behavior, which often hides gaps or overlaps that only appear after repeated substitution.
• Making the animation too polished too early, which can hide a weak proof or an inconsistent tile count.

What Makes This Competitive

A class-level version of this project might stop at a nice picture. A stronger version names the exact recurrence, proves the self-similarity carefully, and checks whether the construction is unique or one of several possible decompositions. You can raise the level by comparing the new family against known hyperbolic rep-tiles, then explaining what changes in the {7,3} setting. Clear definitions, airtight counting, and a clean visual proof will make the work feel serious.

Project Variations

• Study a different regular hyperbolic tiling, such as {p,3}, and see whether the same substitution idea still works.
• Focus on the animation alone and compare two substitution rules by how clearly they reveal the proof structure.
• Analyze the growth pattern of tile counts and compare it with a recurrence from a related Euclidean rep-tile family.

Learn More

• MIT OpenCourseWare: Search for courses on geometry, combinatorics, and discrete math to build the proof tools behind tilings.
• MathWorld: Read entries on hyperbolic geometry, tilings, and rep-tiles for concise definitions and examples.
• arXiv: Search for preprints on hyperbolic tilings, substitution systems, and self-similar geometry.
• SpringerLink books preview: Look for introductory books on hyperbolic geometry and geometric group theory, then use library access or previews.
• GeoGebra materials: Search the GeoGebra library for hyperbolic geometry applets and tiling explorations.
• NIST Digital Library of Mathematical Functions: Use it for general mathematical notation and reference style, especially if you write formal proofs.