Generate Fibonacci numbers with spiral visualization. Explore Lucas numbers, golden ratio convergence, and Zeckendorf representations.
Follow one recurrence across several related views: Fibonacci terms, Lucas terms, adjacent-term ratios, a spiral visualization, and Zeckendorf decompositions. The explorer generates Fibonacci and Lucas sequences from their different starting pairs and shows how Fibonacci ratios approach the golden ratio. The common surprise is that Fibonacci indexing is not universal. One source may use F(0) = 0, F(1) = 1, while another presents the sequence as starting 1, 1. Read the displayed indices before comparing a term with a textbook, program, or database.
Fibonacci terms are formed by adding the two preceding terms. With F(0) = 0 and F(1) = 1, the values begin 0, 1, 1, 2, 3, 5. Lucas numbers use the same addition rule but start with L(0) = 2 and L(1) = 1, producing 2, 1, 3, 4, 7, 11.
The ratio view compares adjacent Fibonacci terms, typically F(n+1) divided by F(n), once the denominator is nonzero. Those ratios alternately fall above and below the golden ratio while approaching it. A Zeckendorf representation writes a positive integer as a sum of nonconsecutive Fibonacci numbers, using the convention that avoids treating the two occurrences of 1 as separate choices.
Two common presentations are 0, 1, 1, 2, 3 and 1, 1, 2, 3, 5. The recurrence is the same, but the indexing convention differs. Use the explorer's displayed F(n) labels when answering a question about a specific index. A request for the "tenth number" is ambiguous unless the starting convention is stated.
Yes, for a positive integer under the standard Zeckendorf convention: use distinct, nonconsecutive Fibonacci numbers and treat the usable sequence as 1, 2, 3, 5, 8, and so on. No, uniqueness is not preserved if both copies of 1 are treated as different selectable terms or if consecutive terms are allowed.
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