Looking at nature for answers to math problems I recently watched a video about the Gömböc, a shape with one stable and one unstable point of equilibrium. Dr. Gábor Domokos proved the existence of this shape. Domokos originally looked at pebbles when trying to find the shape, although this approach was ultimately unsuccessful. Later on, some turtles were found to have a similar shape to the Gömböc.
I am looking for some other examples of when researchers have drawn inspiration (or completely copied) from nature in order to find solutions to math/engineering problems. To clarify, I am not looking for examples of explaining nature using math, as there are many examples of that (e.g. the Fibonacci spiral, arrangement of electrons in atoms/molecules, normal distribution, etc), but instead for examples of explaining math/engineering using nature.
I wasn't able to come up with or find any more examples myself, only the Gömböc.
Edit: I feel like there must have been some instance of when researchers have used the natural positioning of electrons to model something else, but I can't think of anything off the top of my head.
I remember reading about a research paper in which the researchers used an amoeba to gain approximate solutions to the traveling salesman problem. Although that is a computer science problem, it is not entirely irrelevant to mathematics.

Amoeba finds approximate solutions to NP-hard problem in linear time.
In the new study, the researchers found that an amoeba can find
reasonable (nearly optimal) solutions to the TSP in an amount of time
that grows only linearly as the number of cities increases from four
to eight.

I also found the following book that might be of interest to you. A look at the table of contents will provide you with a lot of interesting examples.
Nature-Inspired Optimization Algorithms by Xin-She Yang.
Hope this helps.
This is a cool question. One example I know of is that of the wholeness axiom of Paul Corazza. Roughly, the axiom is a large cardinal axiom which states something about the reflection properties of the universe $V$ by way of large cardinals. Paul Corazza is said to have had access to this axiom after meditating. Not exactly looking at nature, but I guess looking inside oneself.
https://en.wikipedia.org/wiki/Wholeness_axiom
More corny is the classical story of Newton and the apple.