# Padovan's Spiral Numbers

Padovan's Spiral Numbers

In two dimensions, start with two vertically adjacent squares. Next, draw a square to the left of the figure, creating a rectangle. Repeat the process by placing a square above the rectangle, then a square to the right of the resulting rectangle, and so on, continuing around the figure, placing a square with side length matching the current edge of the figure. At each stage, the rectangle lengths are the Fibonacci numbers, which can be calculated recursively using the relation with initial conditions . The rectangles approximate the golden rectangle, and one can connect opposite corners of each new square to approximate the golden spiral.

1×1

2×2

2×3

3×3

5×5

1,1,2,3,5,8,13,…

f(n)=f(n-1)+f(n-2)

f(1)=f(2)=1

A similar process can be carried out in three dimensions, beginning with two cubes: continue around the figure, repeatedly placing cuboids to the left, back, top, right, front, and bottom of the resulting figure. At each stage, the large cuboid sides form a sequence called Padovan's spiral numbers, and connecting corners of one side of each new block creates an approximation to a logarithmic spiral. The sequence begins with the terms ; terms can be computed recursively using the relation , with .

1×1

1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,…

p(n)=p(n-2)+p(n-3)

p(1)=p(2)=p(3)=1