  The aim of this paper is to present a comprehensive survey of cubic Fibonacci
identities, trying to uncover as many as possible. From the outset, our
rationale for a very careful search on an apparently obscure problem was not
only a matter of mathematical curiosity, but also motivated by a quest for 3D
Fibonacci spirals. As we were not able to find any survey on the particular
topic of cubic Fibonacci identities we decided to try to fill this void. We
started by surveying many Fibonacci identities and recording cubic ones.
Obviously, tracing all Fibonacci identities (for identifying a handful) is a
daunting task. Checking several hundred we have realized that it is not always
clear who the author is. The reason is that in many cases an identity was
stated in one article (sometimes without a proof, e.g., as an open problem, or
a conjecture) while later being proven in another article, or effectively
rediscovered independently by other authors. However, we have done our best to
present the identities chronologically. We have supplied our own proof for one
identity, having tried, but failed, to find a published proof. For all the
other identities, we either proved them on a computer, or else verified by hand
their original published proofs. Somehow unexpectedly, our investigations have
revealed only a rather small number of cubic Fibonacci identities, representing
a tiny fraction of all published Fibonacci identities (most of which are linear
or quadratic). Finally, out of these, only a handful of cubic Fibonacci
identities are homogeneous.
