🔷🧠✨ “When Shapes Dance & Colours Echo: A Math Symphony of Symmetries & Polyominoes!” 🎨📐🔁

🔍 What’s This All About?

Imagine placing LEGO-like blocks on a colourful tiled floor… now twist them, flip them, and colour them… but wait! 🧊🎨🔄
How many truly different designs can exist, if we count only mathematically unique ones?

Welcome to a thrilling mathematical puzzle where symmetry meets colouring, and combinatorics sings in perfect harmony! 🎼📏

🧩 Polyominoes: The Math of Shapes That Stick Together

Polyominoes are shapes formed by connecting unit squares edge-to-edge:

Domino = 2 squares 🟦🟦
Triomino = 3 squares
Tetromino = 4 squares (Think: Tetris blocks! 🎮)
… and they go on!

These shapes are the building blocks of discrete geometry, and they’re fascinating mathematical objects used in tiling, game theory, and computational modeling.

🎨 C-Coloured Checkerboards: Painting the Playground

Take your classic checkerboard... now add C distinct colours 🔴🟢🔵🟡 in a repeating pattern. Each square has a colour, and that pattern matters! 🎨✨

Colour combinations affect how we distinguish polyominoes.
Two identical shapes with different colour overlays may not be the same mathematically!

It’s no longer just shapes — it’s coloured configurations!

🔁 Symmetry: The Invisible Artist in the Background

What if you rotate a shape 90°, or flip it across a mirror line? Should that count as the same shape?
In mathematics, we care deeply about symmetry!

🔄 Types of Symmetries Considered:

Rotations: 0°, 90°, 180°, 270° 🔃
Reflections: Across vertical, horizontal, and diagonal axes 🪞

All these transformations form a special symmetry group called Dihedral Group D₄.
👉 Polyominoes are considered the same if one turns into another using any of these.

But wait — the colour pattern must match too! So, a rotated shape may not be identical if the colour arrangement is altered.

🧮🔍 How Do Mathematicians Count All These?

✅ Burnside’s Lemma – Counting with Symmetries

This magical formula helps us avoid overcounting symmetrical duplicates!

\text{Distinct Patterns} = \frac{1}{|G|} \sum_{g \in G} \text{Fix}(g)

Where:

$G$ is the symmetry group (like D₄),
$\text{Fix}(g)$ counts configurations that remain unchanged under transformation $g$ .

🧠 It’s the mathematical equivalent of saying: “Let’s count only the truly unique!”

🌈 Pólya’s Enumeration Theorem – When Colours Join the Game

Now we add the C colours into our math potion!

\text{Unique Colourings} = \frac{1}{|G|} \sum_{g \in G} C^{c(g)}

Where:

$C$ = number of colours 🎨,
$c(g)$ = number of colour-preserving cycles under symmetry $g$ .

✨ This allows us to accurately count how many unique coloured polyominoes exist — even when colours twist and wrap around under rotations/reflections!

🧬 Why This Is More Than Just Counting Blocks

This problem connects deep mathematical fields:

🔹 Combinatorics – Counting with constraints
🔹 Group Theory – Studying symmetry operations
🔹 Geometry – Understanding shapes and structure
🔹 Algebraic Enumeration – Applying Burnside & Pólya magic
🔹 Applications – From physics to chemistry to AI tiling algorithms!

Whether you're designing a puzzle, analyzing crystal lattices, or modeling molecules, this elegant mathematical toolkit reveals the hidden structure beneath visual patterns.

🎯 Math Is More Than Numbers — It’s Structure, Colour & Imagination!

In the Symmetry-Based Enumeration of Polyominoes on C-Coloured Checkerboards, mathematics becomes a language of colourful patterns, invisible symmetries, and logical beauty. 🌟

It’s where shapes are more than just visuals — they are mathematical identities, ruled by symmetry and combinatoric logic.