23.4.2. Recurrence Relation for Triangulations

Let T denote the number of triangulations of a convex polygon with n + 2 sides. A triangulation is the process of dividing a convex polygon by non-intersecting diagonals. For example, if n = 3 (5 sides), there are 5 ways of triangulating it by non-intersecting diagonals.

A triangulation involves drawing diagonals in a polygon to divide it into triangles. For a polygon with n + 2 sides, the goal is to calculate how many unique ways we can draw these diagonals without them crossing each other. Triangulating adds more structure to our understanding of polygons and helps in computations related to them.

Imagine you have a large piece of paper shaped like a pentagon. If you wanted to connect the corners with lines without crossing any lines and ensure that every section formed is a triangle, that’s exactly what triangulation is. It’s like slicing a pizza where each slice must be triangular.

To solve the problem of triangulating a convex polygon with n + 2 sides, we start by considering an arbitrary edge — for simplicity, let's choose the edge between vertices v_n+1 and v_n+2. This side can form a triangle with any other vertex of the polygon as the third vertex of that triangle.

By fixing one edge, we restrict the triangulation to two smaller polygons. When we pick a vertex as the third point in forming a triangle with the fixed edge, we split our polygon into two smaller polygons, P and Q. Each of these smaller polygons can be triangulated independently, so the number of ways to triangulate the original polygon is the product of the number of triangulations of P and Q.

Think of a triangular canvas you want to fill with paints. Instead of painting the whole triangle at once, you could first paint one half and then the other half separately. This is similar to how we can tackle the triangulation by breaking the problem down into smaller parts.

Once the third vertex is fixed, we can express T(n) as a summation over all possible choices for the third vertex. Therefore, the total number of triangulation for an n + 2 sided polygon is given as T(n) = Σ[T(k - 1) * T(n - k)], where k varies from 1 to n.

This expression shows that the total number of triangulations is the sum of products of the number of ways to triangulate each of the newly formed polygons. As we vary the choice of the third vertex (k), we calculate all possible ways of forming valid triangulations.

Imagine making a sandwich. You can first choose what kind of bread you want (this is like choosing your third vertex), and then from that sandwich, you decide on toppings from the smaller portions of filling you have. Each combination gives a different sandwich just like each combination of vertices gives a different triangulated shape.

The initial conditions are as follows: T(0) = 1 (no way to triangulate a polygon with 2 vertices), T(1) = 1 (a triangle is already a triangle), and T(2) = 2 (a rectangle can be triangulated in two ways).

Establishing initial conditions is crucial for understanding how the recurrence works. It provides base cases upon which we can build our understanding of triangulations for larger polygons. These conditions show that simple polygons like triangles and rectangles have known triangulation counts.

Consider the simplest meal preparation — making toast with butter. There's only one way to prepare a piece of bread with butter (T(0)), and it becomes a straightforward step from there for any additional ingredients you want to add later.

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