Exploring Graph Theory through Tile Patterns

What is Graph Theory?

Graph theory is the study of networks made up of points (called vertices) and lines connecting them (called edges). While graphs might sound abstract, they exist all around us — in road networks, computer networks, and even the tile patterns on the floor!

Tiles as a Graph

Take a closer look at the tile pattern in the image. Instead of viewing the tiles as shapes, let’s think about them as a graph:

1. Vertices: Each tile is represented as a vertex (a point in the graph).
2. Edges: If two tiles share a boundary (a side), draw an edge between their vertices.

This turns the pattern into a graph where the connections between tiles become edges.

Euler’s Formula and the Tile Graph

Leonhard Euler discovered a cool relationship between vertices (V), edges(E), and faces/regions(R) of a connected graph. It is called the Euler’s Characteristic:

V−E+R=2

Here:

• V is the number of vertices.
• E is the number of edges.
• R is the number of faces/regions.

How Does This Apply ?

In the image:

• There are 8 tiles, so we have 8 vertices (V=8).
• Tiles that share boundaries are connected, creating 11 edges (E=11).
• Since this graph is drawn on a flat surface, there is 1 outer region that surrounds the entire graph and there are 4 inner regions. (R=5).
• Plug these numbers into the Euler’s formula to see if V−E+R=2

Try It Yourself!

1. Step 1: Pick a part of the tile pattern. You can draw it on paper or trace it.
2. Step 2: Count the vertices, edges, and faces/regions.
3. Step 3: Check Euler’s formula. Does V−E+R equal 2?

This fun activity will help you explore graph theory and Euler’s characteristic while observing real-world patterns!

Why this matters ? and what is the relevance ?

Euler’s characteristic is not just about polygons or tile patterns — it’s a fundamental concept in math and science. This simple idea of treating tiles as vertices and shared edges as connections has far-reaching applications:

1. Network Design: Like internet connections or social networks — vertices are computers/people, and edges are connections between them.
2. City Planning: Roads (edges) connect locations (vertices).
3. AI and Data Structures: Graphs are at the heart of building systems like recommendation engines or path finding algorithms.

It’s like finding order in chaos! Euler’s characteristic shows that even complex patterns have structure and follow rules. It connects math to the real world. Euler’s rule is used everywhere, from bridges to video games.

Conclusion

The tile pattern in the image isn’t just a design —it’s a graph waiting to be explored. By treating tiles as vertices and connections as edges, we can uncover the hidden order and structure in what looks like a simple floor design. Math is everywhere — sometimes, you just have to look down to see it!

Whether it’s exploring connections, solving puzzles, or building AI systems, graph theory gives us tools to see patterns and relationships in ways that go far beyond the classroom. So, the next time you walk across a tiled floor, remember — you’re walking on a graph!

Now it’s your turn to explore and discover how math is hidden all around us!

Kannan K

(Volunteer at RAM Foundation)

Director of Technology, ConcertIDC