Talk:Compound of two snub cubes

reciprocal of the tribonacci constant

The formula for the reciprocal of the tribonacci constant as currently shown on this page is :$\xi ={\frac {1}{3}}\left({\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{-17+3{\sqrt {33}}}}-1\right)$. It has a "17" in two places. Both Generalizations of Fibonacci numbers and MathWorld show the formula as being :$\xi ={\frac {1}{3}}\left({\sqrt[{3}]{19+3{\sqrt {33}}}}-{\sqrt[{3}]{-19+3{\sqrt {33}}}}-1\right)$.

Both of these references show a "19" instead of a "17" as the correct value. In addition, using "19", the result is approximately 0.543689 as all three pages indicate it should be. Given that both other sources use "19", and the fact that the math works out correctly with "19" rather than "17", I assume that the "17" was merely a typo and so I corrected it a few days ago, but now I find that my edit has been undone.

Can anyone explain why the value of "19" should not be retained?

Thanks, David 2601:C2:8300:229B:D515:F39D:1321:D7DB (talk) 02:00, 8 November 2016 (UTC)

What I did is plug in your solution[1] and found ${\sqrt[{3}]{-19+3{\sqrt {33}}}}$ failed, that is $-19+3{\sqrt {33}}<0$, while while the formula $\xi ^{3}+\xi ^{2}+\xi =1$ was true with with the original value, computed as $\xi ={\frac {1}{3}}\left({\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{-17+3{\sqrt {33}}}}-1\right)=0.543689012693$. Tom Ruen (talk) 01:58, 8 November 2016 (UTC)

Thanks for the quick reply Tom. I'll continue the conversation here instead of duplicating it on your personal talk page.

I'm not sure why $-19+3{\sqrt {33}}<0$ is a problem. Did you perhaps take a square root instead of a cube root? The cube root of a negative number is still a valid real number even though it's negative. Taking the reciprocal doesn't change the constant values under the radicals. It just changes :${\frac {1}{3}}{\left(1+{\sqrt[{3}]{19-3{\sqrt {33}}}}+{\sqrt[{3}]{19+3{\sqrt {33}}}}\right)}\approx 1.83929.$ to :$3/{\left(1+{\sqrt[{3}]{19-3{\sqrt {33}}}}+{\sqrt[{3}]{19+3{\sqrt {33}}}}\right)}\approx 0.543689.$

Substituting $17$ for both instances of $19$ in the tribonacci constant results in incorrect estimated values of $\approx 1.210355...$ as calculated by Wolfram Alpha at and $\approx 0.8262...$ for the reciprocal,

The estimated values shown at Wolfram Alpha seem to indicate that $19$, not $17$ is the correct constant in the formula. See for the tribonacci constant calculation, and for the reciprocal.

BTW, also lists the constant as $19$, not $17$. — Preceding unsigned comment added by 2601:C2:8300:229B:D515:F39D:1321:D7DB (talk) 03:36, 8 November 2016 (UTC)

Is this now resolved? It looks like the article has a -1 and the talk page has a +1. Was that the difference? — Preceding unsigned comment added by 2600:1:9253:C5CE:4427:342D:8C75:64B (talk) 23:08, 22 August 2017 (UTC)

References