3D-Printed Geodesics on Translation Surfaces

ISEF Category: Mathematics

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

The Hook

A flat surface can still behave like a maze. On some shapes, straight paths bounce around in patterns that reveal deep geometry. You can print those shapes, track a ball, and compare the motion to theory. That gives you a real chance to test advanced math with a hands-on setup.

What Is It?

A geodesic is the shortest path between two points on a surface. On a plane, that means a straight line. On a curved or folded surface, the shortest path can bend or wrap around. Translation surfaces are special shapes made by gluing flat pieces together in a way that creates corners, or singular points, where the rules change.

Think of it like traveling on a city map with no hills, but with weird street connections. You still move in straight lines on each block, but the connections between blocks change the route. Saddle connections are special straight paths that run from one singular point to another. Their lengths, directions, and counts form a pattern that mathematicians compare with predictions from the Veech group, which is a symmetry group tied to the surface.

With a home printer, you can build a physical model of a translation surface. Then you can roll a ball or track a marker path from overhead and study how the route lines up with the math. The project turns abstract geometry into data you can measure.

Why This Is a Good Topic

This is a strong science fair topic because you can change the surface design, measure the motion, and compare the results to a theory with clear predictions. It connects to pure math, but it also needs careful data collection, image analysis, and error checking. You can learn how to define variables, digitize paths, estimate lengths, and test whether observed statistics match a model. That makes the project both mathematical and experimental.

Research Questions

• How does the choice of printed translation surface change the distribution of saddle-connection lengths?
• What is the effect of surface symmetry on the directions of observed geodesic paths?
• Does increasing the number of singular points change how often a rolling ball follows a predicted saddle connection?
• To what extent do measured path counts match Veech-group predictions for the same surface?
• Which surface geometry produces the largest difference between observed and predicted geodesic statistics?
• How does print resolution affect the accuracy of tracked path length measurements?

Basic Materials

• Home FDM 3D printer.
• PLA filament.
• CAD or polygon-design software.
• Smooth rolling ball or small marble.
• Overhead camera or smartphone with tripod mount.
• Ruler or caliper for verifying printed dimensions.
• Flat table or board for test runs.
• High-contrast background sheet for tracking.
• ImageJ or similar image-analysis software.
• Spreadsheet software for organizing path data.

Advanced Materials

• Access to a higher-resolution 3D printer.
• Calibrated motion-tracking camera or high-frame-rate smartphone.
• Laser-cut or machined mounting base.
• Precision calipers.
• Surface profilometer or digital height gauge for print verification.
• MATLAB or Python for path extraction and statistics.
• Geometric modeling software for translation-surface construction.
• Statistical analysis software for comparing distributions.
• Optional optical marker system for automatic tracking.

Software & Tools

• ImageJ: Measures path coordinates, distances, and frame-by-frame motion from overhead video.
• Python: Cleans tracking data and runs statistical tests on path lengths and directions.
• GeoGebra: Helps you sketch surface layouts and check geometric relationships before printing.
• Blender: Lets you model complex printable shapes and inspect how the surface pieces connect.
• OpenCV: Extracts the rolling ball position from video frames for automated tracking.

Experiment Steps

1. Choose one translation-surface family and define the geometric feature you will vary first.
2. Design a printable model that preserves the math but still works on your printer.
3. Plan a tracking method that gives you repeatable path coordinates from overhead video.
4. Build a rule for turning each recorded route into a saddle connection, a miss, or a boundary case.
5. Set up a comparison between measured path statistics and the theoretical predictions you want to test.
6. Decide how you will handle print error, video noise, and repeated trials before you collect data.

Common Pitfalls

• Printing a surface with warped edges, which changes the geometry and breaks the theoretical model.
• Letting the ball slip or jump at seams, which makes a geodesic look like a random bounce.
• Changing camera angle between trials, which distorts measured lengths and directions.
• Using a surface design that is too complex to classify cleanly, which makes the saddle-connection data hard to interpret.
• Comparing raw counts without normalizing for different path lengths, which can fake a match or hide a real one.

What Makes This Competitive

A strong version of this project does more than say the print looks cool. You would need clean definitions, controlled geometry, and a careful comparison between observed paths and a real theorem. The best projects test more than one surface, quantify uncertainty, and explain when the theory works, and when it starts to fail. A new comparison angle, like print error versus statistical match, can make the work much stronger.

Project Variations

• Test the same geodesic idea on a different printable translation surface, such as a torus with slits or a polygon gluing model.
• Replace the rolling ball with digital path extraction from traced line segments on the printed surface to study the same statistics without motion.
• Compare observed saddle-connection data across coarse, medium, and fine print resolutions to measure how fabrication error changes the geometry.

Learn More

• Flat Surfaces and Dynamics by Anton Zorich: Look for survey chapters on translation surfaces through a university library or journal databases.
• Dynamics on Flat Surfaces lecture notes: Search for university lecture notes on translation surfaces, interval exchange maps, and Veech groups.
• arXiv: Search for preprints on translation surfaces, saddle connections, and Veech surfaces.
• Notices of the American Mathematical Society: Search for accessible articles on flat surfaces and billiards.
• MIT OpenCourseWare: Search for courses in geometry, dynamical systems, or mathematical physics that discuss geodesics and surface models.