9846

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.
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 .

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DETAILS

Snapshots 1 and 2: just as the ratios of each rectangle's sides approach the golden ratio , the ratios of successive long cuboid sides approach a constant
Snapshot 3: both the 2D and 3D spirals maintain a similar "shape" at different scales
References
[1] M. Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, Chicago: The University of Chicago Press, 1987, p. 93.
[2] N. J. A. Sloane: A134816 in the Online Encyclopedia of Integer Sequences. oeis.org/A134816.
[3] I. Stewart, Math Hysteria: Fun and Games with Mathematics, New York: Oxford University Press, 2004, pp. 87–92.
[4] S. R. Finch, Mathematical Constants, New York: Cambridge University Press, 2003, pp. 6–9.
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