CP032 – Số Fibonacci, toán học của thiên nhiên
Trong những dãy số nguyên, dãy số được nhiều nhà toán học nghiên cứu nhiều nhứt là
dãy số Fibonacci:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 . . .
Các số hạng của dãy số Fibonacci gọi là số Fibonacci.
Các số Fibonacci, từ số hạng thứ ba, là tổng số của 2 số hạng đi trước chúng, thí dụ
34 = 13 + 21, 377 = 155 + 233
Nếu gọi F(n) là số Fibonacci thứ n, thì dãy số Fibonacci được định nghĩa bằng công thức truy chứng:
F(n) = F(n-1) + F(n-2) với F(1) = F(2) = 1 và n > 2
Số 0 có thể thêm vào dãy số Fibonacci và được xem như có thứ tự 0 và F(0) = 0.
Số Fibonacci thực ra đã được biết đến dưới tên “maatraameru” trong công trình nghiên cứu về nhịp phách của Pingala, nhà văn phạm học về Sanskrit Ấn độ, năm 450 hay 200 BC, nhưng
được Leonard of Pisa, được biết như Fibonacci, giới thiệu cho thế giới phương Tây năm 1202 trong quyển sách Liber Abaci.
Dãy số Fibonacci có nhiều tính chất, đơn giản nhứt là những tính chất sau đây.
a) Cứ cách 3 số là có một số chẳn (2, 8, 34, 144, …), hay nếu nhận xét rằng F(3) = 2 thì
F(6) = 8, F(9) = 34, … đều la bội số của F(3). Một cách tổng quát, cứ cách khoảng k số là
có một số Fibonacci là bội số của F(k), tức là:
Khoảng cách k => F(k), F(2k), F(3k), F(4k), … là bội số của F(k)
b) Tổng số của n số Fibonacci đầu tiên bằng số Fibonacci thứ n+2 trừ đi 1.
Thí dụ: n = 4 => F(1) + F(2) + F(3) + F(4) = 1+1+2+3 = F(6) – 1 = 8 – 1 = 7
c) Tổng số của m số Fibonacci có thứ tự lẻ đầu tiên bằng số Fibonacci thứ 2m.
Thí dụ: m = 4 => F(1) + F(3) + F(5) + F(7) = 1+2+5+13 = F(8) = 21
d) Tổng số của m số Fibonacci có thứ tự chẳn đầu tiên bằng số Fibonacci thứ 2m+1
trừ đi 1.
Thí dụ: m = 3 => F(2) + F(4) + F(6) = 1+3+8 = F(7) – 1 = 13-1 = 12
e) Tỉ số của một số Fibonacci với số Fibonacci đứng trước nó tiến dần đến một hằng
số gọi là tỉ số vàng φ (phi)
377 / 233 = 1.618025751…, 987 / 610 = 1.618032787 … => φ = 1.6180339887…
Tỉ số của một số Fibonacci với số Fibonacci đứng sau nó thoái dần đến 1/ φ.
233 / 377 = 0.618037135…, 610 / 987 = 0.618034447… => 1/φ = 0.6180339887…
Tỉ số vàng φ nghiệm đúng công thức φ = 1+1/φ, suy ra φ2-φ-1= 0 và φ = (1+√5)/2
Hai số đo không bằng nhau có tỉ số vàng (Golden ratio) khi tỉ số của tổng số của 2 số
đo đó với số đo dài hơn bằng với tỉ số của số đo dài hơn với số đo ngắn hơn.
Thí dụ: a, b là 2 số đo với a dài hơn b, nếu (a+b)/a = a/b thì a và b có tỉ số vàng
φ = a/b = 1.6180339887…
Tỉ số vàng còn có nhiều tên khác như Trị số trung bình vàng (Golden mean), Phi,
Lát cắt vàng (Golden cut), Tỉ lệ thần diệu (Divine proportion).
Kim tự tháp lớn ở Gazaxây năm 2560 BC là một
trong nhiều áp dụng sớm nhứt của tỉ số vàng.
Chiều dài mỗi cạnh của đáy tháp đo được 230.43
met, chiều cao 146.61 met. Tỉ số của cạnh của đáy
với chiều cao của tháp bằng
230.43/146.61 = 1.5717 …
Người Hy lạp xưa dùng toán học nghiên cứu tỉ mỉ
tỉ số vàng và áp dụng nó trong các công trình
kiến trúc.
Hình trên cho thấy một kiến trúc cổ có kích thước rất hài hoà, tuân theo tỉ số vàng.
Bài toán Fibonacci hay Những con thỏ của Fibonacci
Bài toán nhuyên thủy mà Fibonacci nghiên cứu vào khoảng năm 1202 là về sự tăng trưởng của thỏ sinh sản được trong những điều kiện lý tưởng như: thỏ không chết, thỏ cái thọ thai sau 1 tháng tuổi và cuối tháng tuổi thứ hai sinh ra được một cặp thỏ con, một đực một cái. Người ta thả một cặp thỏ mới sinh, một đực một cái, trên một cánh đồng. Hỏi vậy có bao nhiêu cặp thỏ tạo được sau một năm?
a) Cuối tháng thứ nhứt, cặp thỏ (1-1) nhảy giống, nhưng cũng vẫn có 1 cặp mà thôi.
b) Cuối tháng thứ hai, thỏ cái (1) sinh được 1 cặp thỏ mới (2-2), vậy là có 2 cặp thỏ trên đồng.
c) Cuối tháng thứ ba, thỏ cái đầu tiên (1) sinh được 1 cặp thỏ mới nữa (3-3), vậy là có 3 cặp
thỏ trên đồng.
d) Cuối tháng thứ tư, thỏ cái đầu tiên (1) lại sinh thêm được 1 cặp thỏ mới nữa (4-4), thỏ cái
sinh 2 tháng trước (2) cũng sinh được 1 cặp thỏ đầu tiên (5-5). Vậy là có 5 cặp thỏ trên
đồng.
Tóm lại, số cặp thỏ tạo được từ tháng thứ nhứt đến tháng thứ tư lần lượt là: 1, 2, 3, 5. Đó là phần khởi đầu của dãy số Fibonacci. Lý luận tương tự, ta sẽ có số cặp thỏ ở cuối tháng 12 là số Fibonacci thứ 13 (để ý rằng số Fibonacci đầu tiên không kể), tức là: số cặp thỏ tạo được sau một năm là 233.
Henry E Dudeney (1857 – 1930), tác giả của nhiều sách về trò chơi, đã thay thỏ bằng bò và tháng bằng năm cho bài toán có vẻ thực tế hơn. Đó là bài toán “Những con bò của Dudeney“ như dưới đây:
Nếu 1 con bò cái (cow) sinh được 1 con bò con cái (she-calf) đầu tiên ở cuối 2 năm tuổi và sau đó cứ 1 năm thì lại sinh thêm được 1 con bò con cái nữa thì có tất cả bao nhiêu bò con cái sau 12 năm?
(Để ý rằng ở năm thứ nhứt, con bò cái đầu tiên cũng kể là 1 con bò con cái)
Dấu vết số Fibonacci trên trái thơm
Nếu quan sát các ‘mắt’ trên vỏ của một trái thơm già, bạn có thể may mắn tìm thấy được số mắt trên 2 đường vòng cung chéo trên vỏ trái thơm là 2 số Fibonacci nào đó, thí dụ 13 và 21.
Rau bơ 4 lá
Thiên hạ nói rằng tìm được rau bơ (clover) 4 lá là gặp may mắn, mua lô-tô thế nào cũng trúng lớn! Nhưng cả đám rau bơ sau vườn nhà của tác giả đều có 3 lá. Thật thất vọng, vì rau bơ tuân theo luật Fibonacci và 4 không phải là một số Fibonacci như số 3! Nhưng nếu bạn thực sự gặp may mắn thì rau bơ 4 lá cũng có thể tìm được vì thiên nhiên cũng có nhiều đột biến lắm!
Chuyện rau bơ 3 lá cũng xảy ra cho nhiều loài hoa trong thiên nhiên với số cánh hoa là những số Fibonacci như:
- hoa lily (huệ), iris (irit) có 3 cánh
- hoa buttercup (mao lương hoa vàng), larspur (phi yến) có 5 cánh
- hoa delphiniu (la lết) có 8 cánh
- hoa ragwort (cỏ lưỡi chó) có 13 cánh
- hoa chicory (rau diếp xoắn) có 21 cánh
- hoa pyrethrum (cúc lá nhỏ) có 34
- …
Một đầu hoa hướng dương (sunflower) có 2 chuỗi đường cong, một hướng bên nầy, một hướng bên kia. Số các đường xoắn ốc không bằng nhau ở mỗi phía, thường là 21, 34 hoặc 34, 55 hoặc 55, 89 hoặc 89, 144.
Sự sắp xếp những cánh hoa chỉ vì nhu cầu về không gian và ánh sáng, các cánh hoa phải có đủ khoảng cách để lớn và hứng được đầy đủ ánh sáng như nhau. Nhưng tại sao lại có sự trùng hợp với những số Fibonacci? Chả trách sao nhiều người gọi số Fibonacci là Toán học của thiên nhiên.
Các tính chất về toán học, kiến trúc, nghệ thuật, thực vật, … có liên quan với số Fibonacci thì còn nhiều. Các nghiên cứu về số Fibonacci và những ngành toán liên quan vẫn còn tiếp tục và kết quả thường được công bố trong tạp chí ‘Fibonacci Quarter’ điều hành bởi hiệp hội ‘Fibonacci Association’ thành lập từ năm 1996.
Trước khi chấm dứt, tác giả xin có một câu hỏi nhờ độc giả giải đáp giùm:
“Có thể nào vẽ được một tam giác có chiều dài 3 cạnh là 3 số Fibonacci liên tiếp
hay không?”
ĐỌC VUI VÀ SUY NGHĨ 
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Thành thật cám ơn ông/bà . ĐV-SN rất vui nếu được độc giả góp ý kiến giúp cho ĐV-SN càng ngày càng phát triển trong công việc phục vụ cộng đồng người Việt ở mọi nơi.
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Thành thật cám ơn Ông/Bà đã có nhã ý muốn giúp chúng tôi làm cho ĐV-SN tốt hơn. Chúng tôi rất hân hạnh nếu nhận được sự giúp đỡ đó. Thật sự là chúng tôì rất bận việc. Có rất nhiều việc liên quan đến ĐV-SN mà chúngtôi phải giải quyết hàng tuần. Chúc Ô/B sức khoẻ.
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Thân kính,
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