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)