# Fibonacci and Padovan Spiral Identities

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Start with an equilateral triangle with side length 1, and place a second unit triangle next to it. At each stage, moving counterclockwise around the resulting figure, construct an equilateral triangle whose base falls along the current side. The triangle side lengths match a sequence called the Padovan numbers, which are normally defined with the relation , with initial conditions .

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Contributed by: Robert Dickau (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Two other relations are evident by matching up triangles or squares on opposite sides of a side.

Snapshot 1: by matching up triangle sides at each stage, the figures show that

Snapshot 2: similarly, by matching up square sides, the figures show that

Snapshots 3, 4: hiding the relations highlights the overall construction of each shape

References

[1] M. Gardner, *The Second Scientific American Book of Mathematical Puzzles and Diversions*, Chicago: The University of Chicago Press, 1987, p. 93.

[2] I. Stewart, *Math Hysteria: Fun and Games with Mathematics*, New York: Oxford University Press, 2004, pp. 87–92.

[3] S. R. Finch, *Mathematical Constants*, New York: Cambridge University Press, 2003, pp. 6–9.

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