Click the link below the picture
.
In mathematics, simple rules can unlock universes of complexity and beauty. Take the famous Fibonacci sequence, which is defined as follows: It begins with 1 and 1, and each subsequent number is the sum of the previous two. The first few numbers are:
1, 1, 2, 3, 5, 8, 13, 21, 34 …
Simple, yes, but this unassuming recipe gives rise to a pattern of far-reaching significance, one that appears to be woven into the very fabric of the natural world. It’s seen in the whorls of nautilus shells, the bones in our fingers, and the arrangement of leaves on tree branches. Its mathematical reach extends to geometry, algebra, and probability, among other areas. Eight centuries since the sequence was introduced to the West — Indian mathematicians studied it long before Fibonacci — the numbers continue to attract the interest of researchers, a testament to how much mathematical depth can underlie even the most elementary number sequence.
In the Fibonacci sequence, every term builds on the ones that came before it. Such recursive sequences can exhibit a wide range of behaviors, some wonderfully counterintuitive. Take, for instance, a curious family of sequences first described in the 1980s by the American mathematician Michael Somos.
Like the Fibonacci sequence, a Somos sequence starts with a series of ones. A Somos-k sequence starts with k of them. Each new term of a Somos-k sequence is defined by pairing off previous terms, multiplying each pair together, adding up the pairs, and then dividing by the term k positions back in the sequence.
The sequences aren’t very interesting if k equals 1, 2 or 3 — they are just a series of repeating ones. But for k = 4, 5, 6 or 7 the sequences have a weird property. Even though there is a lot of division involved, fractions don’t appear.
.
Kristina Armitage/Quanta Magazine
.
.
Click the link below for the article:
.
__________________________________________
James' World 2 
Nov 20, 2023 @ 20:52:28
🙄😲🙄
LikeLike
Nov 21, 2023 @ 06:04:22
Thanks for your comment!
LikeLiked by 1 person
Nov 21, 2023 @ 07:15:06
🩶
LikeLike
Nov 21, 2023 @ 08:17:46
Thank you!
LikeLike