Why are there no identities for the golden ratio?
While there are a plethora of identities for Fibonacci and Lucas numbers [see, for example, here and here: S Vajda, Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications, Dover Press (2008)], there are few, if any, for golden ratio beyond it definition $\varphi^2=\varphi+1$.
We have developed a few and hope this will inspire more such activity. Briefly, by utilizing the areas of gnomonic tilings we’ve been able to develop relations between powers of $\varphi$. In a gnomonic tiling you begin with a seed and add (logarithmically growing) tiles in succession that maintain the overall shape of the seed. Dignomonic tiling is much the same, but you add two unique gnomons at each step. Such tilings are possible with the golden ratio specifically because $\varphi^2=\varphi+1$.
The first two figures below show two such tilings of golden ratio squares and rectangles. Taking the overall area as the sum of those of the gnomons, plus the area of the seed, we find that
$$ \varphi^n\varphi^{n+1}=\sum_{k=0}^n(\varphi^k)^2+\frac{1}{\varphi}=\sum_{k=-\infty}^n(\varphi^k)^2\\ (\varphi^n)^2=\sum_{k=0}^{n-1}\varphi^k\varphi^{k+1}+1=\sum_{k=-\infty}^{n-1}\varphi^k\varphi^{k+1} $$
The sums with $k=-\infty$ indicate taking the tiling backward into the seed, thus driving the seed area to zero. Now, these equations are essentially the same and can be converted one to the other. Nevertheless, we can play around with them a bit, For example, subtracting the second from the first we can show that
$$ (\varphi^n)^2=\frac{1}{\varphi}\sum_{k=-\infty}^n(\varphi^k)^2 $$
Which is interesting because $(\varphi^n)^2$ appears on both sides of the equation, or it could be expressed as
$$ (\varphi^n)^2=\frac{1}{\varphi-1}\sum_{k=-\infty}^{n-1}(\varphi^k)^2 $$
Something more interesting occurs, however, when we substitute the first equation into the second one. Here, we find after working through the double summation that
$$ (\varphi^n)^2=\sum_{k=-\infty}^{n}(n-k)(\varphi^k)^2 $$
Of course, we can also readily determine that
$$ \sum_{k=-\infty}^{0}(\varphi^k)^2=\varphi\ \ \text{ or } \ \ \sum_{k=-\infty}^0\varphi^k\varphi^{k-1}=1 $$
Finally, we applied the same idea to the golden triangle gnomonic tiling with the isosceles triangle as the seed and the oblique as the gnomon (seen in the third figure below). The results were the same as the second equation in this posting. That not surprising. However, along the way we discovered what is possibly a new trigonometric identity; it turns out that
$$ \frac{\cos \frac{\pi}{10}}{\cos \frac{3\pi}{10}}=\varphi $$
These are the angles that arise with the areas of the two types of triangles.
The question arises as to whether there are other golden ratio identities yet to be discovered.
NOTE: Just as I am posting this I find an MSE posting by Ed Pegg (here) where he notes that $\sum_{n=1}^{\infty} \varphi^{-n} = \varphi$.
