TRIBONACCI

 

MY_OLD_NOTEBOOK

ENGLISH_TRANSLATION

MY_POST_TO_HYACINTHOS
 

On Sun, 19 Nov 2000 I wrote:
 

>
> /
> angle w / angle q
> /
> / I_0
> J_0 /
> / I_1
> J_1 / I_2
>--------------------------/-----------------------------
>
>(I_0, r_0), (I_1, r_1), (I_2,r_2),...... is a decreasing sequence of circles
>inscribed in the angle q, and (I_n) touches externally both (I_n-1, I_n+1)
>
>(J_0, R_0), (J_1,R_1), .... is a similar sequence but in the angle w(omega)
>
>Now, if R_0 = r_0, and R_1 = r_2, find the angles.
 

Here is the solution:

We have:
1 - sin(q/2) 1 - sin(w/2)
r_n = r_0 *(-------------)^n, R_n = R_0 * (------------)^n
1 + sin(q/2) 1 + sin(w/2)

If r_0 = R_0, r_2 = R_1 we get:

1 - sin(q/2) (1 - sin(w/2)
(------------)^2 = ------------- (1)
1+ sin(q/2) (1 + sin(w/2)

q + w = Pi ==> sin(w/2) = cos(q/2) := a & cos(w/2) = sin(q/2) := b

a, b in (0,1)

1 - b 1 - a 2b
(1) ==> (-------) ^ 2 = ------- ==> a = -------- (2)
1 + b 1 + a 1 + b^2

We have: a^2 + b^2 = 1 (3)

(2) and (3) ==> b^6 + b^4 + 3b^2 - 1 = 0 ==>

(b^3 + b^2 + b - 1)(b^3 - b^2 + b + 1) = 0

and since 0 < b < 1, ==> b^3 + b^2 + b - 1 = 0

The acceptable root of this equation is that one in (0,1).

This root is 0.5436890126........, a well-known constant, namely:

T_n
lim --------
n -->+oo T_(n+1)

where T_n is the Tribonacci sequence: 0,1,1,2,4,7,13,24,44,81,149,274,504,...
(T_1 = 0, T_1 = 1, T_2 = 1, T_(n+2) = T_n + T_(n+1) + T_(n+2))

cos(w/2) = b ==> w = 114d 7' 47",13....

Steven Finch, in his page "Cubic Variations of the Golden Mean" at:

http://www.mathsoft.com/asolve/constant/gold/cubic.html

T_(n+1)
asks about 1/b = lim --------- = 1.839286.... :
n --> +oo T_n

<quote>
[T]he ratio A/B is 1.8392867552.... Are other geometric interpretations
of this constant possible?
</quote>

Welll... the above is one!

Other references for Tribonacci Constant:

Simon Plouffe's page, containing 2000 digits of Tribonacci constant
http://plouffe.fr/simon/constants/tribo.txt

Brian Hayes: Computing Science. The Vibonacci Numbers
American Scientist July-August, Volume 87, No. 4


Antreas