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The properties of the Fibonacci numbers also provide some deeper insight, why the optical illusion works so well. A square whose side length is the Fibonacci number f n {\displaystyle f_{n}} can be dissected using line segments of lengths f n , f n − 1 , f n − 2 {\displaystyle f_{n},f_{n-1},f_{n-2}} in the same way the chessboard was ...
Hosoya's triangle or the Hosoya triangle (originally Fibonacci triangle; OEIS: A058071) is a triangular arrangement of numbers (like Pascal's triangle) based on the Fibonacci numbers. Each number is the sum of the two numbers above in either the left diagonal or the right diagonal.
A golden triangle. The ratio a/b is the golden ratio φ. The vertex angle is =.Base angles are 72° each. Golden gnomon, having side lengths 1, 1, and .. A golden triangle, also called a sublime triangle, [1] is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:
where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers. [6]
It is claimed that the upper level of 21 rows and the lower level of 34 rows of the Ancient Theatre of Epidaurus form an approximation of the Golden number since 21 and 34 are successive Fibonacci numbers with their ratio at / and a careful examination of the theatre's center reveals two back-to-back triangles balanced by the Golden number.
The golden angle plays a significant role in the theory of phyllotaxis; for example, the golden angle is the angle separating the florets on a sunflower. [2] Analysis of the pattern shows that it is highly sensitive to the angle separating the individual primordia, with the Fibonacci angle giving the parastichy with optimal packing density. [3]
A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the ...
A Fibonacci spiral starts with a rectangle partitioned into 2 squares. In each step, a square the length of the rectangle's longest side is added to the rectangle. Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous ...