Tribonacci ratio

Template:Short description Template:Infobox non-integer number In mathematics, the tribonacci ratio is a geometrical proportion, given by the unique real solution of the equation Template:Nowrap Its decimal expansion begins with Template:Math Template:OEIS.

The moniker tribonacci was introduced by high-school student Mark Feinberg in an article published in the Fibonacci Quarterly of october 1963.

Definition

Template:Math. With Template:Math the boxes have volumes Template:Math

Three quantities Template:Math are in the tribonacci ratio if $$\frac{a+b+c}{a}=\frac{a}{b}=\frac{b}{c}=\tau$$ This ratio is commonly denoted Template:Tmath

Substituting Template:Tmath and Template:Tmath in the first fraction gives $$\tau =\frac{c({\tau }^2+\tau +1)}{{\tau }^{2}c}.$$ It follows that the tribonacci ratio is the unique real solution of the cubic equation ${\tau }^3={\tau }^2+\tau +1$.

Closed-form expressions for Template:Tmath are found by solving the depressed cubic Template:Tmath, which has real zero Template:Tmath. $$\begin{array}{cc}\tau & =\frac{1}{3}(1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}})\\ & =\frac{1}{3}(1+4\cosh (\frac{1}{3}\mathrm{arcosh}\left(\frac{19}{8}\right))).\end{array}$$

The point of intersection Template:Tmath with Template:Tmath of unit circle $x^2+y^2=1$ and the hyperbola $y=\frac{1}{x+1}$ is Template:Tmath Substituting for Template:Tmath, the equation of the abscissa is $x^3+2x^2-2=0$ or $x^2(x+2)=2.$

Dividing both sides by Template:Tmath results in the fixed point iteration $x_{n+1}\leftarrow \sqrt{\frac{2}{2+x_n}},$ and the continued reciprocal square root $$\tau -1=\sqrt{\frac{2}{2+\sqrt{\frac{2}{2+\sqrt{\frac{2}{2+\ddots }}}}}}$$ An image of the Julia set of the backward iteration $z_{n-1}\leftarrow \frac{2}{z_n^2}-2$ in the complex plane is found below.

Alternative expressions for the tribonacci ratio can be derived from the reciprocal equation, that is, the depressed cubic Template:Tmath with real zero Template:Tmath. $$\begin{array}{cc}{w}_{1,2}& =(1\pm \frac{1}{3}\sqrt{\frac{11}{3}})/4\\ \tau & =1+(\sqrt[3]{w_1}+\sqrt[3]{w_2}{)}^{-1}\\ & =1+\frac{\sqrt{3}}{2}\mathrm{sech}(\frac{1}{3}\mathrm{arcosh}\left(\frac{3\sqrt{3}}{4}\right)).\end{array}$$

Additionally, $2x^3=2x+1$ leads to the infinite radical $$\frac{1}{\tau -1}=\sqrt[3]{\frac{1}{2}+\sqrt[3]{\frac{1}{2}+\sqrt[3]{\frac{1}{2}+\cdots }}}$$

Template:Tmath is the superstable fixed point of the Newton iteration Template:Tmath.

Properties

Rectangles with aspect ratios Template:Math tile the square.

The tribonacci ratio can be written in terms of itself as fractions $$\begin{array}{cc}\tau & =\frac{{\tau }^2+1}{{\tau }^2-1}\\ {\tau }^2& =\frac{\tau +1}{\tau -1}\\ {\tau }^3& =\frac{{\tau }^4+1}{2}.\end{array}$$

Similarly as the infinite geometric series $$\begin{array}{cc}\frac{{\tau }^2+1}{2}& =\sum _{n=0}^{\mathrm{\infty }}{\tau }^{-n}\\ \frac{\tau +1}{2}& =\sum _{n=0}^{\mathrm{\infty }}{\tau }^{-2n}\\ \frac{1}{\tau -1}& =\sum _{n=0}^{\mathrm{\infty }}{\tau }^{-3n}.\end{array}$$

For every integer Template:Tmath one has $$\begin{array}{cc}{\tau }^n& ={\tau }^{n-1}+{\tau }^{n-2}+{\tau }^{n-3}\\ & =2{\tau }^{n-2}+2{\tau }^{n-3}+{\tau }^{n-4}\\ & =3{\tau }^{n-2}+{\tau }^{n-4}+{\tau }^{n-6}\end{array}$$ from this an infinite number of further relations can be found. A notable example is Template:Tmath.

Continued fraction pattern of a few low powers  $$\begin{array}{cc}{\tau }^{-1}& =[0;1,1,5,4,2,305,1,8,2,\mathrm{...}]\approx 0.5437(\frac{31}{57})\\ {\tau }^0& =[1]\\ {\tau }^1& =[1;1,5,4,2,305,1,8,2,1,\mathrm{...}]\approx 1.8393(\frac{103}{56})\\ {\tau }^2& =[3;2,1,1,1,1,2,1,152,2,\mathrm{...}]\approx 3.3830(\frac{159}{47})\\ {\tau }^3& =[6;4,2,305,1,8,2,1,4,6,\mathrm{...}]\approx 6.2223(\frac{56}{9})\\ {\tau }^4& =[11;2,4,152,1,17,1,2,2,\mathrm{...}]\approx 11.4445(\frac{103}{9})\end{array}$$

The tribonacci ratio is the fourth smallest cubic Pisot number. By definition of these numbers, the absolute value $1/\sqrt{\tau }$ of the algebraic conjugates is smaller than 1, thus powers of Template:Tmath generate almost integers. For example: ${\tau }^{18}=\mathrm{58034.99919...}$ Template:Tmath. After 18 rotation steps the phases of the inward spiraling conjugate pair – initially close to Template:Tmath – nearly align with the imaginary axis.

The first implied mention of the tribonacci constant was in the eleventh century, when the Persian poet and polymath Omar Khayyam found the solution Template:Tmath of the cubic $x^3+200x=20x^2+2000$ by considering the intersection of a circle and a rectangular hyperbola.

$W_{11}(x)=x^3-2x^2+2x-2,$ with real zero $\omega =\frac{\tau +1}{\tau }=\tau (\tau -1),$ is the Weber class polynomial associated with discriminant Template:Tmath. Properties of the related Klein j-invariant result in near-identity $\omega \approx (e^{\pi \sqrt{-\mathrm{\Delta }}}+24{)}^{1/24}.$

Construction of the tribonacci ratio with compass and marked ruler. Template:Math and Template:Math.

Argument $\theta =\arccos (\frac{1}{2}\tau )$ satisfies $4\sin (3\theta )-\tan (\theta )=\sqrt{11}$, a result which is related through distance parameter $z=\tau (1-\tau )\cdot 2\cos (\frac{2\pi }{11})$ to the 'miraculous' neusis construction of the hendecagon, found by Benjamin and Snyder.

The reciprocal Template:Tmath of the tribonacci ratio solves the equation $2\arctan (x)=\arccos (x)$. The angle is close to Template:Math radian. Its complement $\arccos (\tau -1)=\arcsin (\frac{1}{\tau })$ figures in the geometric construction of the tribonacci constant found by biologist Xerardo Neira.

The tribonacci ratio is particularly important in the study of the snub cube: all its metrics can be expressed in terms of Template:Tmath.

Tribonacci sequence

The first muddled mention of the tribonacci sequence is in Charles Darwin's On the Origin of Species (1859), illustrating the population growth of elephants on the supposition that during their lifetime each pair of parents produces three pair of young.

The number of compositions of Template:Math into parts 1, 2 and 3 is counted by the nth tribonacci number (Template:Math).

The tribonacci sequence is defined by the third-order recurrence relation $$T_n=T_{n-1}+T_{n-2}+T_{n-3}\text{ for }n>2,$$ with initial values $$T_0=T_1=0,T_2=1.$$

The first few terms are 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927,... Template:OEIS.Template:Br The limit ratio between consecutive terms is the tribonacci constant:$\underset{n\to \mathrm{\infty }}{lim}{T}_{n+1}/T_n=\tau .$Template:Br An alternative relation for Template:Tmath is given by $T_n=2T_{n-1}-T_{n-4}.$

The sequence is extended to negative indices using $$T_n=T_{n+3}-T_{n+2}-T_{n+1},$$ obtaining (0), 1,−1, 0, 2,−3, 1, 4,−8, 5, 7,−20, 18, 9,−47,... Template:OEIS.

The relationship between the negative and positive indexed segments of the sequence is given by $$T_{1-n}=T_n^2-T_{n+1}{T}_{n-1}.$$ This reflection formula holds for all integers Template:Tmath and can be derived from Agronomof's identity  $$T_{n+k}=T_{k+1}{T}_{n+1}+(T_k+T_{k-1})T_n+T_k{T}_{n-1}.$$

The sequence is related to sums of binomial coefficients by $$T_{n+2}=\sum _{i=0}^{\lfloor n/2\rfloor }\sum _{j=0}^i(\genfrac{}{}{0ex}{}{i}{j})(\genfrac{}{}{0ex}{}{n-i-j}{i})$$

Julia set (in black) of the map Template:Math with a single attracting, real five-point limit cycle. Viewport width Template:Math, pole Template:Math. The tiny white circle is centred at repelling fixed point Template:Math.

Powers of the tribonacci ratio can be written with tribonacci numbers as quadratic coefficients $${\tau }^n={\tau }^2{T}_n+\tau (T_{n-1}+T_{n-2})+T_{n-1},$$ which is proved by mathematical induction on Template:Tmath This relation also holds for Template:Tmath The order of the coefficients corresponds to the bottom row of matrix Template:Tmath below.

The generating function of the tribonacci sequence for non-negative Template:Math is given by $$\frac{x^2}{1-x-x^2-x^3}=\sum _{n=0}^{\mathrm{\infty }}{T}_n{x}^n\text{ for }x<\frac{1}{\tau }$$

Let Template:Tmath and complex conjugate pair Template:Tmath and Template:Tmath be the zeros of polynomial Template:Tmath with discriminant Template:Tmath, the tribonacci numbers are then given by the Binet formula $$T_{n+1}=a{\tau }^n+b{\beta }^n+c{\gamma }^n,$$ with real Template:Tmath and conjugates Template:Tmath and Template:Tmath the roots of $44y^3-2y-1=0.$

Since $|b{\beta }^n+c{\gamma }^n|<\frac{2}{5}$, the number Template:Tmath is the nearest integer to Template:Tmath, with Template:Tmath and coefficient $a={\tau }^2/({\tau }^3+\tau +2)=$ Template:Gaps Template:Efn

The tribonacci numbers are obtained as integral powers Template:Tmath of a matrix with real eigenvalue Template:Tmath $$Q=\left(\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),$$

$$Q^n=\left(\begin{array}{ccc}{T}_{n+2}& T_{n+1}+T_n& T_{n+1}\\ T_{n+1}& T_n+T_{n-1}& T_n\\ T_n& T_{n-1}+T_{n-2}& T_{n-1}\end{array}\right)$$

The trace of Template:Tmath gives the tribonacci-Lucas numbers 3, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499, 2757,... satisfying the same recurrence relation. Variously, $L_n=\lfloor {\tau }^n\rceil \text{ for }n>3.$ Template:OEIS

These Lucas numbers have the Fermat property: if p is prime, $L_p\equiv L_{1}modp.$ The converse does not hold, but the small number of tribonacci pseudoprimes $n\mid (L_n-1)$ makes the sequence special. The only composite numbers below Template:Math to pass the test are n = 182, 25201, 233Template:Sup, 63618, 194390, 750890, 804055, 1889041, 2487941, 3542533, 3761251, 6829689. Template:OEIS

The tribonacci word sequence on the alphabet Template:Tmath is defined by the substitution rule $$\{\begin{array}{c}a\mapsto ab\\ b\mapsto ac\\ c\mapsto a\end{array}$$ With initiator Template:Tmath, the first seven words are $$c,a,ab,ab.ac,abac.aba,abacaba.abacab,abacabaabacab.abacabaabac$$ For Template:Tmath, the series of words Template:Tmath produced by iterating the substitution are similarly obtained by concatenating the previous three: $w_n=w_{n-1}+w_{n-2}+w_{n-3}.$ The number of Template:Math in each word is equal to successive tribonacci numbers and the word length Template:Tmath. The tribonacci word sequence is the basis of the construction of the classic Rauzy fractal.

Tribonacci spiral

Tribonacci spirals with different initial radii on a Template:Math− rectangle.

A tribonacci spiral is a logarithmic spiral that gets wider by a factor of Template:Tmath for every quarter turn. It is described by the polar equation $r(\theta )=a\exp (k\theta ),$ with initial radius Template:Tmath and parameter $k=\frac{2}{\pi }\ln (\tau ).$

If drawn on a tribonacci rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio Template:Tmath which are perpendicularly aligned and successively scaled by a factor Template:Tmath. The pole of the spiral divides the main diagonal in ratio Template:Tmath and bisects the diagonal of the lower right rectangle.

See also

Solutions of equations similar to $x^3=x^2+x+1$:

• Golden ratio – the positive solution of the equation $x^2=x+1$
• Plastic ratio – the real solution of the equation $x^3=x+1$
• Supergolden ratio – the real solution of the equation $x^3=x^2+1$