Euclidyne

Instrument-Lab Explorations into Euclid, Ratios, and Rhythm

Core Algorithm

Fibonacci Race

Why Consecutive Fibonacci Numbers Drag Euclid Out the Longest

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Your Pair

Pair

89/55

Steps

9

Fibonacci Pair?

yes

1 1 1 1 1 1 1 1 2

Benchmark Pair

Pair

89/55

Steps

9

Step Gap

0

1 1 1 1 1 1 1 1 2

Claim Status

Rigour stays visible

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  • Running the Euclidean algorithm on positive integers yields the finite continued fraction of a/b.
  • Rectangle-to-squares dissection follows the same quotient sequence as the Euclidean algorithm.
  • The logarithmic-spiral story belongs to the special golden-rectangle case, not to every Euclidean rectangle dissection.

Fun Fact

1844

Lamé's 1844 Result

Gabriel Lamé's Euclid analysis is a classic early complexity result: Fibonacci pairs force the slowest standard run.