Spectral Gaps in Fractal Rectangles

ISEF Category: Mathematics

Subcategory: Analysis  ·  Difficulty: Advanced  ·  Setup: University Lab  ·  Time: Full Year

The Hook

A shape with more holes can change how a wave moves through it. That sounds simple, but the math gets strange fast. Your job is to measure how the first two vibration modes separate, then compare that gap to theory. Fractal cutouts turn a plain rectangle into a puzzle with real analysis behind it.

What Is It?

The Laplacian is a math operator that shows up when you study heat flow, vibration, and diffusion. On a flat rectangle, its spectrum, or list of vibration frequencies, is well behaved. The spectral gap is the distance between the first two eigenvalues, and it tells you how quickly the shape can move from one mode to the next.

Now change the rectangle by punching out holes in a self-similar pattern, like a Sierpinski-style cutout. The shape still looks like a rectangle from far away, but the holes block flow and reshape the modes. Think of it like a drum with a carefully designed pattern of missing wood. The notes do not just change, the spacing between notes can change too.

You can study this two ways. First, you can compute approximate eigenvalues with finite elements in FEniCS. Then you can compare those results with analytic bounds, such as Cheeger-type inequalities, which link geometry to the size of the spectral gap.

Why This Is a Good Topic

This makes a strong science fair topic because you can test a clear number, the spectral gap, across a family of shapes that you design. You are not just drawing pretty fractals, you are asking how geometry controls vibration and diffusion. That connects to acoustics, heat transfer, and shape optimization. You can also learn real math skills, like eigenvalue problems, numerical error, and inequality-based bounds.

Research Questions

• How does the spectral gap change as you increase the number of fractal cutout stages in a rectangle?
• What is the effect of hole size on the first two Laplacian eigenvalues?
• Does the spectral gap shrink faster for Sierpinski-like cutouts than for random hole patterns with the same area removed?
• To what extent do Cheeger-type lower bounds track the computed spectral gap across different perforation patterns?
• Which boundary condition, Dirichlet or Neumann, produces a larger gap on the same perforated shape?
• How does the location of holes, centered versus edge-biased, affect the gap for equal porosity?
• To what extent does mesh refinement change the computed gap in FEniCS?

Basic Materials

• Laptop or desktop computer with enough memory to run FEniCS.
• FEniCS installation with Python support.
• Python scientific stack, including NumPy, SciPy, and Matplotlib.
• Geometry sketching tool, such as a vector editor or CAD program.
• Graph paper or a digital notebook for tracking shape parameters.
• Spreadsheet or lab notebook for recording eigenvalues and gap values.

Advanced Materials

• University workstation or laptop with more RAM for finer meshes.
• FEniCS or FEniCSx with a solver backend for sparse eigenvalue problems.
• Python libraries for mesh generation and post-processing.
• Boundary integral or finite-element comparison tools for validating results.
• Access to journal articles on spectral geometry and Cheeger inequalities.
• Optional symbolic math software for checking derivations.

Software & Tools

• FEniCS: Solves the Laplacian eigenvalue problem on your perforated domains.
• Python: Organizes shape parameters, runs batch simulations, and graphs spectral gaps.
• Matplotlib: Plots eigenvalue trends, convergence checks, and comparison curves.
• ImageJ: Helps inspect mesh screenshots or geometry images when you need quick visual checks.
• QGIS: Can help if you turn the project into a geometry or domain-analysis workflow.

Experiment Steps

1. Define one family of perforated rectangles and choose a single geometric parameter to vary first.
2. Build a clean computational model of each shape, then decide how you will represent the holes and the outer boundary.
3. Set up a mesh refinement plan so you can tell numerical error apart from real spectral change.
4. Compute the first few Laplacian eigenvalues for each shape, then extract the spectral gap as a derived quantity.
5. Derive or collect a Cheeger-type bound that matches your boundary conditions and domain type.
6. Compare the numerical gaps with the theoretical bounds, then test whether the trend survives when you change the perforation pattern.

Common Pitfalls

• Using a mesh that is too coarse near the holes, which can distort the first eigenvalues.
• Comparing shapes with different removed areas, which makes the spectral gap trend hard to interpret.
• Forgetting to match the boundary condition in the code to the one assumed in the theory.
• Treating a single computed eigenvalue as exact, which hides solver error and mesh dependence.
• Changing several geometry features at once, which makes it impossible to tell which feature caused the gap change.

What Makes This Competitive

A strong version of this project goes beyond one shape and one plot. You can compare multiple hole patterns, track mesh convergence, and test whether a theoretical bound stays informative as the domain gets more fractal. The best entries also explain why one geometry changes the gap more than another, not just that it does. Clear math, careful numerics, and a sharp comparison between theory and computation make the project feel serious.

Project Variations

• Compare Dirichlet and Neumann boundary conditions on the same perforated rectangle to see how the spectral gap shifts.
• Replace the Sierpinski-style pattern with random holes of equal area, then compare whether self-similarity changes the gap trend.
• Study higher eigenvalue gaps, not just the first gap, to see whether the fractal geometry affects low and mid-spectrum behavior differently.

Learn More

• MIT OpenCourseWare: Search for lectures on partial differential equations, numerical methods, and eigenvalue problems.
• NIST Digital Library of Mathematical Functions: Use it for background on special functions and spectral ideas when needed.
• MathSciNet or zbMATH: Search for review articles on spectral geometry, Laplacian eigenvalues, and Cheeger inequalities.
• SIAM Journal on Applied Mathematics: Browse papers on numerical eigenvalue computation and shape effects on spectra.
• arXiv: Search for recent preprints on Laplacian spectra of fractal or perforated domains.