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Fibonacci and Padovan Spiral Identities

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shape
triangle
square
iterations
6
highlight relation
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
(n)=(n-2)+(n-3)
, with initial conditions
(1)=(2)=1
.
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
(n)=(n-1)+(n-2)
, with initial conditions
(1)=(2)=1
.
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