> For the complete documentation index, see [llms.txt](https://nkintc.gitbook.io/brainless/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nkintc.gitbook.io/brainless/steam/mathematics/metallic-numbers.md).

# Metallic Numbers

For all the fame and attention that the Golden Ratio ($$\phi \approx 1.618$$) gets in comparison to the rest of the mathematical constants, there is a whole family of constants which get left out of the picture.&#x20;

It is relatively common to know that $$\phi$$ can be represented with an algebraic representation: $$\frac{1 + \sqrt{5}}{2}=\phi.$$ But that expression generalizes to the rest of the integers!&#x20;

$$
\frac{n + \sqrt{n^2+4}}{2} \quad \forall n\in \N
$$

### Laplace&#x20;