Define the Fibonacci word sequence as follows:
f_0 = 0, f_1 = 01, f_n = f_(n-1)f_(n-2) (the concatenation of f_(n-1) and f_(n-2)), for n ≥ 2. For example, f_2 = 010, f_3 = 01001, f_4 = 01001010, f_5 = 0100101001001, ...
The infinite Fibonacci word is the "limit" of these sequences. Let w_n be the n-th symbol in the infinite Fibonacci word.
The Fibonacci curve is a sequence of segments, drawn as follows:
- at step 0, draw a segment of length 1 from left to right;
- at step n,
- if w_n = 0, then draw segment of length 1 in the same direction as the previous segment
- if w_n = 1, then draw segment of length 1, turning 90º right if n is even and 90º left if n is odd.
Which are the properties of the given curve (e.g.: Does it intersect itself, does it have autosimilarities...). What happens if we change the angle of rotation? What if we change the Fibonacci word with a Sturm word?
f_0 = 0, f_1 = 01, f_n = f_(n-1)f_(n-2) (the concatenation of f_(n-1) and f_(n-2)), for n ≥ 2. For example, f_2 = 010, f_3 = 01001, f_4 = 01001010, f_5 = 0100101001001, ...
The infinite Fibonacci word is the "limit" of these sequences. Let w_n be the n-th symbol in the infinite Fibonacci word.
The Fibonacci curve is a sequence of segments, drawn as follows:
- at step 0, draw a segment of length 1 from left to right;
- at step n,
- if w_n = 0, then draw segment of length 1 in the same direction as the previous segment
- if w_n = 1, then draw segment of length 1, turning 90º right if n is even and 90º left if n is odd.
Which are the properties of the given curve (e.g.: Does it intersect itself, does it have autosimilarities...). What happens if we change the angle of rotation? What if we change the Fibonacci word with a Sturm word?