4 ms·

a mind-rewiring video by numberphile concerning the golden ratio [0] made sense of all the spiral overlay art: yes, plants use it because it is an optimal solu

a mind-rewiring video by numberphile concerning the golden ratio [0] made sense of all the spiral overlay art: yes, plants use it because it is an optimal solution for minimizing overlap when sprouting leaves during growth, maximizing spatial coverage but also, as a number that is least able to be approximated by rational, erm, ratios - it is a pattern that most closely approximates random patterns. So more than any other spiral, a fibonacci spiral has the highest likelihood of overlapping random points in 2D space... (and fibonacci series converges on the golden ratio because they are equivelent expressions of the continued fraction 1+(1/(1+(1/(1+(1/(1+(1/... [0] https://youtu.be/sj8Sg8qnjOg https://youtu.be/sj8Sg8qnjOg

thaumasiotes 5y ago
> and fibonacci series converges on the golden ratio because they are equivelent expressions of the continued fraction 1+(1/(1+(1/(1+(1/(1+(1/... Just for fun, note that those numerators and denominators reflect the coefficients (1 and 1) in the Fibonacci recurrence equation F_n = 1*F_{n-1} + 1*F_{n-2}.[1] Any sequence obeying this equation will converge to the same ratio between adjacent terms. But the Fibonacci series is "perfect" in that it bottoms out at 1 and 1, never interrupting the pattern even after an infinite number of terms have gone by. In contrast, if you defined a similar series starting at 3 and 4, that series would converge to the continued fraction 1/(1+1/(1+1/1+(1/ ... 1+(3/4) ))), which is exactly equal as long as you can't get across the infinite number of 1s in the expansion of "...". [1] As an example of how this works, if you have the recurrence relation f(k) = 3f(k-1) + 2f(k-2), then the limit of f(n)/f(n-1) as n goes to infinity is the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ...

4 ms·

a mind-rewiring video by numberphile concerning the golden ratio [0] made sense of all the spiral overlay art: yes, plants use it because it is an optimal solu

by jazzyjackson 5y ago

thaumasiotes 5y ago
> and fibonacci series converges on the golden ratio because they are equivelent expressions of the continued fraction 1+(1/(1+(1/(1+(1/(1+(1/... Just for fun, note that those numerators and denominators reflect the coefficients (1 and 1) in the Fibonacci recurrence equation F_n = 1*F_{n-1} + 1*F_{n-2}.[1] Any sequence obeying this equation will converge to the same ratio between adjacent terms. But the Fibonacci series is "perfect" in that it bottoms out at 1 and 1, never interrupting the pattern even after an infinite number of terms have gone by. In contrast, if you defined a similar series starting at 3 and 4, that series would converge to the continued fraction 1/(1+1/(1+1/1+(1/ ... 1+(3/4) ))), which is exactly equal as long as you can't get across the infinite number of 1s in the expansion of "...". [1] As an example of how this works, if you have the recurrence relation f(k) = 3f(k-1) + 2f(k-2), then the limit of f(n)/f(n-1) as n goes to infinity is the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ...