Fibonacci and Padovan Spiral Identities

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 .
Similarly, starting with a unit square, place another unit square next to it; moving counterclockwise around the figure, construct a square with side length matching the current side of the resulting figure. The square side lengths match the Fibonacci numbers, traditionally defined with the relation , with initial conditions .


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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
[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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Related Curriculum Standards

US Common Core State Standards, Mathematics