Sunday, January 14, 2018

Studying discrete-time dynamical systems (VI): mathematical models for symmetry and asymmetry and some biological coincidences

Let us do today some experimental mathematics. In my former posts  I have shown some stages of stochastic systems that look very "organic". Indeed they seem to have organic-like symmetrical and asymmetrical properties: while there is a clear axis of symmetry, both sides are not exactly symmetrical. Verifying the content of both sides of the axis it is possible to see that the main structures are symmetrical but there are small sub-structures that are only in one of the sides. It is very similar to the asymmetry that can be found in Nature, e.g. the human body looks externally symmetrical in a global level, but it is not. The human face, limbs, etc. are not symmetrical indeed, they are not exact mirrored copies. 

The mathematical models I was able to find seem to have some of these characteristics, so I am showing in the current post some biological coincidences of some stages of them and real life organic structures at different scale levels. For instance, jellyfish umbrella-shaped bells, sand wasp bodies and tardigrade limbs can be reflected at some extent with this type of complex systems. The pictures I am using at the right side of the images are just for the sake of completeness (they belong to their respective owners, I do not own them, if there is any problem I will remove them, so just please let me know):

1. The mathematical model (left) explained in this previous post shows similarities with (right) jellyfish structures (e.g. umbrella-shaped bell, including bioluminiscence patterns), the sand wasps and other similar insect bodies and the limbs of tardigrades. Click to enlarge:

2. The mathematical model (left) explained in this previous post shows similarities with some families of moths with rounded wings (right). Click to enlarge:

It would be great if these complex systems were able to be humbly used at some extent for some field of biology. For now they are just coincidences, I hope the readers will enjoy the samples.

3. A slight variation of the mathematical model explained at point 1 shows similarities with the bodies of some families of moths (e.g. Acherontia atropos). Click to enlarge:

This is the original image in full version. Click to enlarge:

I will try to expand this chimerical bestiary as much as possible.


  1. These visualizations are mesmerizing! Stunning! Would you please be willing to share your code for them?

    1. Hi sure, I am preparing a paper regarding this findings but it will take some time to be finished. I will make a new post this week with the Python code, stay tuned.

    2. OH MY GOSH THANK YOU SO MUCH!!! I am about to cry tears of joy, this visual is just the most beautiful and inspiring thing I've seen in a long time. I look forward to the paper also! Are there books and other resources you could recommend on this topic (no pressure though!)? Do you have an email by which we could continue this conversation?

    3. Thanks for the words, but no problem really! I use to share the code of my tests, in this case I did not share it because it was a little bit longer. When I publish it, if you post it in other places, please keep the link to this page in the comments so other people can find the original information. The paper will took still some months to be finished. Some questions: do you know how to use Python? and about the references, what topic do you want to learn? mathematical models for bioluminescence? (by your nickname I understand you like that topic) or about visuals of system dynamics? depending on the specific topic I can give you some advises. Do you want to read general information or (mathematics-related) technical papers about the topic? You can let here in the comments your doubts, I will try to help. :)

    4. Ok the code is up, I hope the comments will be self-explanatory... :)