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Mathematics Final Year Topic: Counting Polygon Colourings under Rotation and Reflection with Burnside's Lemma

This Mathematics final year project compares raw colour assignments with equivalence classes under a specified symmetry group.

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Why choose this project topic?

A focused study of symmetry counting gives you a specific question in combinatorics and group actions. It compares raw colour assignments with equivalence classes under a specified symmetry group. The bounded comparison creates room to explain how your evidence supports an interpretation and where the method has limits.

How does the choice of rotation-only or full dihedral symmetry change the number of distinct polygon colourings?

Agree the apparatus or dataset, comparison range and feasible measurement schedule for symmetry counting with your supervisor.

Proposed project objectives

  1. 01Define the materials, variables and comparison conditions for symmetry counting.
  2. 02Define the colouring rules and group action, enumerate fixed colourings for each symmetry type and apply Burnside's lemma.
  3. 03Evaluate the measurements or model outputs in relation to this question: How does the choice of rotation-only or full dihedral symmetry change the number of distinct polygon colourings?

A suggested research approach

Define the colouring rules and group action, enumerate fixed colourings for each symmetry type and apply Burnside's lemma. Verify small cases by exhaustive enumeration with a clearly bounded computational scope. Agree the available resources and record uncertainty, deviations from the protocol and any observations that challenge the initial interpretation.

What you will need

  • Specified polygon sizes and colouring rules
  • References on group actions and Burnside's lemma
  • A finite-enumeration check for small cases

Keep your project scope clear

The count changes when adjacency restrictions, labelled positions or allowable symmetries change, so equivalence must be defined before counting.

Mathematics project chapter outline

Use this outline as a starting point. You can edit the chapter titles to match your department’s format during setup.

  1. Chapter 1Introduction
  2. Chapter 2Literature Review
  3. Chapter 3Theory and Methodology
  4. Chapter 4Results and Applications
  5. Chapter 5Summary, Conclusion and Recommendations

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