Visualize the golden ratio in rectangles, spirals, pentagons, and Fibonacci sequences with interactive constructions.
Phi appears in this explorer as a shared proportion rather than only a decimal. Interactive constructions connect golden rectangles, nested squares, spiral-like curves, regular pentagons, and ratios of consecutive Fibonacci numbers. These views help distinguish exact identities from visual approximations. The most important distinction concerns spirals: quarter-circle arcs drawn through adjoining Fibonacci squares form a useful Fibonacci spiral, but they are not the exact logarithmic curve commonly called the golden spiral.
A golden rectangle has long-to-short side ratio phi, where phi = (1 + sqrt(5)) / 2, approximately 1.6180339887. Removing a square whose side equals the rectangle's short side leaves a smaller rectangle with the same proportions. For sides phi and 1, the remainder has sides 1 and phi - 1, and 1 / (phi - 1) equals phi.
In a regular pentagon, the ratio of a diagonal to a side is exactly phi, so drawing all diagonals exposes repeated golden proportions. The Fibonacci view approaches the same number differently: ratios of successive terms in 1, 1, 2, 3, 5, 8, and so on alternate around phi and converge toward it, although every finite ratio remains rational.
No. The exact value is (1 + sqrt(5)) / 2, which is irrational and has a nonterminating, nonrepeating decimal expansion beginning 1.6180339887. Writing 1.618 is convenient for comparison, but it is rounded. Repeated geometric construction or measurement using that shortened value can accumulate visible differences.
No, not when its side lengths are consecutive Fibonacci numbers. Its aspect ratio is a quotient of two integers and therefore rational, while phi is irrational. Larger consecutive terms give progressively closer approximations. Likewise, quarter-circle arcs fitted to Fibonacci squares approximate the appearance of a golden spiral without becoming the exact logarithmic curve.
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