The *tribonacci constant *is the real root of the cubic equation,

and is the limiting ratio of the *tribonacci numbers* = {0, 1, 1, 2, 4, 7, 13, 24, …} where each term is the sum of the previous three, analogous to the *Fibonacci numbers*. Let , then,

We’ll see that the tribonacci constant is connected to the *complete elliptic integral of the first kind* . But first, given the golden ratio’s infinite radical representation,

then *T* also has the beautiful infinite radical,

as well as a continued fraction,

where *q* is the negative real number,

Recall that at *elliptic singular values*, the *complete elliptic integral of the first kind* *K*(*k*) satisfies the equation,

or, in the syntax of *Mathematica*,

Interestingly, we can express both and in terms of the tribonacci constant as,

where,

and,

where is the *gamma function*, as well as the infinite series,

With a slight tweak of the formula, we instead get,

Finally, saving the best for last, given the* snub cube*, an Archimedean solid,

then the Cartesian coordinates for its vertices are all the even and odd permutations of,

{± 1, ± 1/*T*, ±*T *}

with an even and odd number of plus signs, respectively, similar to how, for the vertices of the *dodecahedron* — a Platonic solid — one can use the golden ratio.

For more about the tribonacci constant, and the equally fascinating *plastic constant*, kindly refer to “*A Tale of Four Constants *“.

Posted by Tribonacci numbers, Padovan sequence, and more (Part 1) « tpiezas on May 21, 2012 at 5:41 pm

[…] already written about the tribonacci constant before. But I want to include how Lin found that powers of t can be expressed in terms of those three […]

Posted by Alex on July 14, 2015 at 1:22 pm

The intersection (X,Y) of the Bernoulli lemniscate and the line y = 1 – x.

defines the lemniscate cotangent = (X/Y) = T

where X = T/(T + 1)