Tribonacci ratio

A tribonacci rectangle contains two scaled copies of itself, τ = ((τ − 1)2 + 2(τ − 1) + 1) / τ
Rationality	irrational algebraic
Symbol	τ
Representations
Decimal	1.83928675521416113255...
Algebraic form	real root of x3 = x2 + x + 1
Continued fraction (linear)	[1;1,5,4,2,305,1,8,2,1,4,6,14,...] [1] not periodic infinite

In mathematics, the tribonacci ratio is a geometrical proportion, given by the unique real solution of the equation x³ = x² + x + 1. Its decimal expansion begins with 1.839286755214161... (sequence A058265 in the OEIS).

The moniker tribonacci was introduced by high-school student Mark Feinberg in an article published in the Fibonacci Quarterly of october 1963.^[2]

Definition

Three quantities a > b > c > 0 are in the tribonacci ratio if $${\displaystyle {\frac {a+b+c}{a}}={\frac {a}{b}}={\frac {b}{c}}=\tau }$$ This ratio is commonly denoted ⁠${\displaystyle \tau .}$⁠

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

Closed-form expressions for ⁠${\displaystyle \tau }$⁠ are found by solving the depressed cubic ⁠${\displaystyle y^{3}-{\tfrac {4}{3}}y-{\tfrac {38}{27}}}$⁠, which has real zero ⁠${\displaystyle \tau -{\tfrac {1}{3}}}$⁠.^[3] $${\displaystyle {\begin{aligned}\tau &={\frac {1}{3}}\left(1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}\right)\\&={\frac {1}{3}}\left(1+4\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {19}{8}}\right)\right)\right).\end{aligned}}}$$

The iteration ${\displaystyle x\gets {\sqrt[{3}]{{\tfrac {1}{2}}+x}}}$ with fixed point ⁠${\displaystyle {\frac {1}{\tau -1}}}$⁠ results in the continued radical $${\displaystyle \tau -1=1/{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+\cdots }}}}}}}$$ Since the iteration derives from ⁠${\displaystyle 2x^{3}=2x+1}$⁠,^[4] alternative expressions for ⁠${\displaystyle \tau }$⁠ are $${\displaystyle {\begin{aligned}w_{1,2}&=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {11}{3}}}\right)/4\\\tau &=1+({\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}})^{-1}\\&=1+{\frac {\sqrt {3}}{2}}\operatorname {sech} \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{4}}\right)\right).\end{aligned}}}$$

⁠${\displaystyle {\frac {1}{\tau -1}}}$⁠ is the superstable fixed point of the Newton iteration ⁠${\displaystyle x\gets (2x^{3}+{\tfrac {1}{2}})/(3x^{2}-1)}$⁠.

Properties

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

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

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

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

The tribonacci ratio is the fourth smallest cubic Pisot number.^[6] By definition of these numbers, the absolute value ${\displaystyle 1/{\sqrt {\tau }}}$ of the algebraic conjugates is smaller than 1, thus powers of ⁠${\displaystyle \tau }$⁠ generate almost integers. For example: ${\displaystyle \tau ^{18}=58034.99919...}$ ⁠${\displaystyle \approx 58035-1/1234}$⁠. After 18 rotation steps the phases of the inward spiraling conjugate pair – initially close to ⁠${\displaystyle \pm 9\pi /13}$⁠ – 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 ⁠${\displaystyle 10{\tfrac {\tau +1}{\tau }}}$⁠ of the cubic ${\displaystyle x^{3}+200x=20x^{2}+2000}$ by considering the intersection of a circle and a rectangular hyperbola.^[7]

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

Argument ${\displaystyle \theta =\arccos({\tfrac {1}{2}}\tau )\,}$ satisfies ${\displaystyle \,4\sin(3\theta )-\tan(\theta )={\sqrt {11}}}$, a result which is related through distance parameter ${\displaystyle \,z=\tau (1-\tau )\cdot 2\cos({\tfrac {2\pi }{11}})}$ to the 'miraculous' neusis construction of the hendecagon, found by Benjamin and Snyder.^[8][9]

The reciprocal ⁠${\displaystyle {\tfrac {1}{\tau }}}$⁠ of the tribonacci ratio solves the equation ${\displaystyle \,2\arctan(x)=\arccos(x)}$.^[10] The angle is close to 1 radian. Its complement ${\displaystyle \,\arccos(\tau -1)=\arcsin({\tfrac {1}{\tau }})\,}$ figures in the geometric construction of the tribonacci constant found by biologist Xerardo Neira.^[11]

The tribonacci ratio is particularly important in the study of the snub cube.

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.^[12]

The number of compositions of n − 2 into parts 1, 2 and 3 is counted by the nth tribonacci number (n > 1).

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

The first few terms are 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927,... (sequence A000073 in the OEIS).
The limit ratio between consecutive terms is the tribonacci constant:${\displaystyle \lim _{n\rightarrow \infty }T_{n+1}/T_{n}=\tau .}$

The sequence can be extended to negative indices using $${\displaystyle T_{n}=T_{n+3}-T_{n+2}-T_{n+1},}$$ obtaining (0), 1,−1, 0, 2,−3, 1, 4,−8, 5, 7,−20, 18, 9,−47,... (sequence A057597 in the OEIS).

The relationship between the negative and positive indexed segments of the sequence is given by $${\displaystyle T_{1-n}=T_{n}^{2}-T_{n+1}T_{n-1}.}$$ This reflection formula holds for all integers ⁠${\displaystyle n}$⁠ and can be derived from Agronomof's identity ^[13][14] $${\displaystyle T_{n+k}=T_{k+1}T_{n+1}+\left(T_{k}+T_{k-1}\right)T_{n}+T_{k}T_{n-1}.}$$

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

Powers of the tribonacci ratio can be written with tribonacci numbers as quadratic coefficients $${\displaystyle \tau ^{n}=\tau ^{2}T_{n}+\tau (T_{n-1}+T_{n-2})+T_{n-1},}$$ which is proved by mathematical induction on ⁠${\displaystyle n.}$⁠ This relation also holds for ⁠${\displaystyle n<0.}$⁠ The order of the coefficients corresponds to the bottom row of matrix ⁠${\displaystyle Q}$⁠ below.

The generating function of the tribonacci sequence for non-negative n is given by $${\displaystyle {\frac {x^{2}}{1-x-x^{2}-x^{3}}}=\sum _{n=0}^{\infty }T_{n}x^{n}{\text{ for }}x<{\tfrac {1}{\tau }}}$$

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

Since ${\displaystyle \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <{\tfrac {2}{5}}}$, the number ⁠${\displaystyle T_{n}}$⁠ is the nearest integer to ⁠${\displaystyle \,a\,\tau ^{n-1}}$⁠, with ⁠${\displaystyle n>0}$⁠ and coefficient ${\displaystyle a=\tau ^{2}/(\tau ^{3}+\tau +2)=}$ 0.3362281169949410942253629... ^[a]

The tribonacci numbers are obtained as integral powers ⁠${\displaystyle n\geq 2}$⁠ of a matrix with real eigenvalue ⁠${\displaystyle \tau }$⁠ ^[15] $${\displaystyle Q={\begin{pmatrix}1&1&1\\1&0&0\\0&1&0\end{pmatrix}},}$$

$${\displaystyle Q^{n}={\begin{pmatrix}T_{n+2}&T_{n+1}+T_{n}&T_{n+1}\\T_{n+1}&T_{n}+T_{n-1}&T_{n}\\T_{n}&T_{n-1}+T_{n-2}&T_{n-1}\end{pmatrix}}}$$

The trace of ⁠${\displaystyle Q^{n}}$⁠ 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, ${\displaystyle L_{n}=\lfloor \tau ^{n}\rceil {\text{ for }}n>3.}$ (sequence A001644 in the OEIS)

These Lucas numbers have the Fermat property: if p is prime, ${\displaystyle L_{p}\equiv L_{1}{\bmod {p}}.}$ The converse does not hold, but the small number of tribonacci pseudoprimes ${\displaystyle \,n\mid (L_{n}-1)}$ makes the sequence special. The only composite numbers below 10⁷ to pass the test are n = 182, 25201, 233², 63618, 194390, 750890, 804055, 1889041, 2487941, 3542533, 3761251, 6829689. (sequence A371805 in the OEIS)

The tribonacci word sequence on the alphabet ⁠${\displaystyle \{a,b,c\}}$⁠ is defined by the substitution rule $${\displaystyle {\begin{cases}a\;\mapsto \;ab\\b\;\mapsto \;ac\\c\;\mapsto \;a\end{cases}}}$$ with initiator ⁠${\displaystyle w_{0}=c}$⁠. The series of words ⁠${\displaystyle w_{n}}$⁠ produced by iterating the substitution have the number of c's, b's and a's equal to successive tribonacci numbers. The length of these words is ${\displaystyle l(w_{n})=T_{n+2}.}$ The tribonacci word sequence is the basis of the construction of the classic Rauzy fractal.^[16]

Tribonacci spiral

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

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 ⁠${\displaystyle \tau (\tau -1)}$⁠ which are perpendicularly aligned and successively scaled by a factor ⁠${\displaystyle \tau ^{-1}}$⁠. The pole of the spiral divides the main diagonal in ratio ⁠${\displaystyle \tau ^{2}}$⁠ and bisects the diagonal of the lower right rectangle.

See also

Solutions of equations similar to ${\displaystyle x^{3}=x^{2}+x+1}$:

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

Notes

References