1 What it's about

A construction using equilateral triangles that leads to isosceles trapezoids in the context of a generalized Fibonacci sequence.

2 Generalized Fibonacci sequence

The starting values f₁ and f₂ are arbitrary. The recursion is the standard Fibonacci recursion:

(1) f_n₊₁ = f_n + f_n_–1

3 Visualizing the recursion

We draw an isosceles trapezoid with base angles of 60°. The upper parallel side has a length of f_n_–1, and the leg length is f_n (Fig. 1). Using a red dividing line parallel to one leg, we divide the trapezoid into an equilateral triangle with side length f_n and a parallelogram with side lengths f_n and f_n_–1. This shows that the lower parallel side has a length of f_n₊₁.

Fig. 1: Isosceles Trapezoid

We now rotate the trapezoid by +120° and by –120° and join the two rotated trapezoids at one vertex (Fig. 2).

Fig. 2: Iteration Step

The convex hull of the figure is again an isosceles trapezoid with base angles of 60°. The upper parallel side now has length f_n, and the leg length is f_n₊₁. The lower parallel side has length f_n₊₂.

These considerations are independent of the two initial values f₁ and f₂.

4 Example

4.1 Initial Values and Sequence

For the figures in the following example, we work with the two initial values f₁ = 2 und f₂ = 5. Table 1 gives the corresponding generalized Fibonacci sequence.

n	f_n
1	2	Starting value
2	5	Starting value
3	7
4	12
5	19
6	31
7	50
8	81
9	131
10	212

Table 1: Generalized Fibonacci sequence

4.2 Base trapezoid

We begin with an isosceles base trapezoid (Fig. 3) with the upper parallel side f₁ = 2 and leg length f₂ = 5. By tiling the base trapezoid with equilateral triangles of side length 1, the side lengths of the trapezoid can be read directly. The lower parallel side has a length f₃ = 7. The number of equilateral triangles is 45.

Fig. 3: Base trapezoid

4.3 Iterations

Using the base trapezoid, we perform the iteration step described above (Fig. 4.1). The new trapezoid has an area equivalent to 119 equilateral triangles with side length 1.