# 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.

• You're welcome! I also only heard about it recently, and was fascinated by it (although I still don't understand the physics behind "unstable equilibrium points"). – Varun Vejalla Aug 8 '20 at 23:26

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.

• Neat - this is an interesting example! Thanks for this. Maybe we'll have living computer chips made out of amoeba one day :) – Varun Vejalla Aug 9 '20 at 0:01
• @VarunVejalla You are very welcome. :$)$ – Khashayar Baghizadeh Aug 9 '20 at 0:02
• @VarunVejalla Check out the edit. – Khashayar Baghizadeh Aug 9 '20 at 18:45
• Wow, there's a lot of examples in the book. I guess a lot of machine learning and artificial intelligence uses nature. – Varun Vejalla Aug 9 '20 at 18:55
• I think we can also learn from the chipmuks about how they solve their TSP. Chipmunks love peanuts so I feed them peanuts so they can survive the Canadian winter. It's really interesting to see how they collect the peanuts you throw at them really fast. The advantage of observing chipmunks instead of amoeba is obvious. Chipmunks are much bigger and can be observed an video-recorded. One can also feed peanuts to the squirrels, the chipmunks cousins to learn how they solve their TSP. But it will cost more to feed a squirrel. – user25406 Sep 6 '20 at 21:53

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.

• I'm not sure the wholeness axiom fits, since "looking inside oneself" could just be considered as thinking intensely (or self-surgery, but that's another matter). Newton and the apple might work, although I think it fits better into the "using math to explain nature" category. – Varun Vejalla Aug 8 '20 at 23:13
• Yes Varun. Maybe another way to look at the question would be to see how the entire beginings of Differential Equations be seen as humans looking at nature and seeing it change. As it changes, humans want to understand what governs that change. We have access to how fast nature changes, hence rate of change. So maybe just looking at a river or leaves shaking under the action of the wind, makes you think about the laws upon which all of this rests. Is that what you had in mind? – Rachid Atmai Aug 8 '20 at 23:21
• As a decades-long meditator and former math professor I assure you that meditating is the exact OPPOSITE of looking to nature as asked in the question. I can't even imagine a method that is more the opposite of what the questioner had in mind. – David G. Stork Aug 8 '20 at 23:25
• @RachidAtmai That's somewhat what I'm looking for, but not completely. I guess it is the "direction of influence" that matters - I'm not looking for when humans have tried explaining nature using math, but explaining math using nature. – Varun Vejalla Aug 8 '20 at 23:28
• @DavidG.Stork You seem pretty riled up for someone who meditates haha... Peace man, hope you find some equanimity. Although aren't we ourselves part of nature though? Think about it... – Rachid Atmai Aug 8 '20 at 23:39