Manhattan Polygon Inequalities in Geometry

ISEF Category: Mathematics

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

The Hook

A shape can hide extra complexity even when every edge runs straight up, down, left, or right. That makes Manhattan polygons a great test case for geometry. You can ask when area, perimeter, and corner counts must obey a fixed rule. You can also search for shapes that hit the limit exactly.

What Is It?

A Manhattan polygon is a polygon made only of horizontal and vertical edges. Think of it like a city block shape drawn on graph paper. Instead of smooth curves, you get right angles and corners. Some corners stick inward. Those inward corners are called reflex vertices.

This project studies an isoperimetric-type inequality. That is a rule that links area and perimeter. For ordinary shapes, a circle gives the most area for a given perimeter. For rectilinear polygons, the number of reflex vertices also matters. More inward dents usually mean less area for the same perimeter. Your job is to test that relationship, compare families of shapes, and see which ones come closest to the bound.

Why This Is a Good Topic

This topic works well for a science fair because you can define clear measurements, generate many test cases, and check a real mathematical claim. You do not need a wet lab, but you do need careful data collection and analysis. The project connects geometry, optimization, and computation. You can learn how mathematicians look for patterns, turn them into conjectures, and test when a bound is sharp.

Research Questions

• How does the area-to-perimeter relationship change as the number of reflex vertices increases?
• What is the effect of adding one reflex vertex to a rectilinear polygon with fixed perimeter?
• Does a family of grid-based polygons approach the proposed inequality bound more closely than random shapes?
• To what extent do symmetric Manhattan polygons maximize area for a fixed perimeter and reflex vertex count?
• Which polygon generation rules produce shapes that are closest to the sharpness examples?
• How does the inequality behave for polygons with the same perimeter but different edge counts?

Basic Materials

• Graph paper or square-grid paper.
• Pencil and ruler.
• Geometry software with polygon tools, such as GeoGebra.
• Spreadsheet software, such as Google Sheets or Excel.
• Digital camera or smartphone for recording hand-drawn shapes.
• Calculator.

Advanced Materials

• Computer with integer programming software or a modeling language such as Python plus PuLP or OR-Tools.
• Python with NumPy and SciPy for geometric calculations.
• GeoGebra or similar geometry software for visual checks.
• ImageJ for measuring rasterized shape area from screenshots if needed.
• University access to optimization software such as Gurobi or CPLEX, if available.
• TeX or LaTeX for writing proofs and figures.

Software & Tools

• GeoGebra: Lets you draw rectilinear polygons and measure side lengths, area, and perimeter.
• Python: Helps you generate polygon families, compute statistics, and test conjectures.
• Google Sheets: Organizes measurements and compares candidate inequalities across many shapes.
• PuLP: Solves integer programming models when you want to search for sharp examples.
• ImageJ: Measures area from saved images when you want a pixel-based check on hand-made figures.

Experiment Steps

1. Define the exact polygon class you will study, including how you count reflex vertices and how you handle holes or self-intersections.
2. Choose one inequality form to test first, so you have a single target instead of a vague pattern hunt.
3. Build a clean dataset of many rectilinear polygons, using both hand-made examples and generated examples on a grid.
4. Measure area, perimeter, and reflex vertex count in a consistent way, then check whether the data match the proposed bound.
5. Search for near-extreme shapes with optimization or systematic enumeration, so you can compare your examples against the sharpness claim.
6. Stress-test the result with special families of polygons and record where the bound gets tight, weak, or fails.

Common Pitfalls

• Counting reflex vertices inconsistently, which makes the same polygon appear to change class.
• Mixing polygons with holes and simple polygons, which breaks the assumptions behind the inequality.
• Using grid resolution that is too coarse, which hides small changes in area or perimeter.
• Comparing shapes with different scaling rules, which makes the data useless for an isoperimetric test.
• Only testing symmetric examples, which can make a weak conjecture look stronger than it really is.

What Makes This Competitive

A strong version of this project would not just restate the inequality. It would test many shape families, explain why the bound should hold, and look for the exact cases where equality or near-equality happens. You can also make it stronger by using optimization to generate counterexample candidates, then proving why they fail or why they match the bound. Clear definitions, careful computation, and a sharp comparison against known geometry results can push the work much higher.

Project Variations

• Study only lattice polygons with vertices on integer grid points, then compare your results with the general rectilinear case.
• Test how the inequality changes for polygons with a fixed number of edges but varying reflex vertex counts.
• Use an algorithmic search to find the polygons that maximize area for each perimeter class, then compare them with the theoretical bound.

Learn More

• MIT OpenCourseWare, geometry and optimization courses: Search MIT OpenCourseWare for undergraduate materials on combinatorial geometry and optimization.
• Discrete and Computational Geometry: Search for review chapters or textbook excerpts through Google Books, library catalogs, or course reading lists.
• arXiv: Search for preprints on isoperimetric inequalities, rectilinear polygons, and discrete geometry.
• MathWorld: Read background entries on polygons, perimeters, area, and isoperimetric inequalities.
• Wikipedia references and cited papers: Use the reference lists at the bottom of rectilinear polygon and isoperimetric inequality pages to find original papers, then follow the citations.