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# Creative Mathematics: An Application of Category Theory

By on 20/02/2018

Hello everyone!

Here we are again. For those who don’t know us, a very brief introduction on us!

MathIsInTheAir is an italian blog of Applied Maths and surroundings and here are the links: english version and italian version. Go and read the “not so quick introduction” to MathIsInTheAir I wrote two years ago.

Today I just want to present our next english mathematical post, called “Creative Mathematics: An Application of Category Theory” written by Maria Mannone.

No spoiler on this post 😉 .

# Creative Mathematics: An Application of Category Theory

PANGOLIN (Ground Pangolin, manis temminckii)

A pangolin curled up in the defensive position:

When the pangolin is frightened, it curls up, becoming a sort of armored ball that the predatory animals are not able to open, but easy to be caught by poachers.

The pangolin is characterized by a strong scale armor that makes it look almost like a small dinosaur. The pangolin is too little to run and too big to hide, thus nature gave it the armor. The particular armor of the pangolin can guide us to the discovery of some basic concepts of category theory.

Let us consider one scale.
Then, two scales.

The repetition of two scales is defined by a transformation that we call g:

The scales are our “objects,” and the arrows g are our “morphisms.” If we compose several g-arrows, we again obtain scales, as the following image shows.

If we don’t make any repetition, thus, if we move from a scale and we apply the arrow “1” that gives us again the same scale, we just defined what mathematicians call an “identity.” Thus, already the very first observation allows us to define the category “scale,” with its objects, arrows, identity-arrows, and arrow composition.

The whole image of the pangolin in its closed-defensive position is much more complicated. We can try to add new “transformations” to build such a whole image.

After the “horizontal” composition, we add a vertical composition, given by the “vertical” repetition of scales, through the arrow h.

In this way, by combining g and h, we can build several rows of scales, the ones partially superposed to the others. However, by looking at the real image of the pangolin’s armor, we can see that the rows are offset as if we introduced a little shift in the even rows. Let us indicate such a transformation with Sh (where Sh stands for shift and not for ‘s composed with h’):

We obtain all the scales by repeating N-times the action of g and h. To obtain a whole image that is more realistic, we can modify the “general shape” of the image via the arrow I:

Finally, we “close” the obtained figure, to imitate the image of the defensive-position curled-up pangolin’s back. Here, we use letter L which stands for Loop.

Remaining within the category “scales,” we reconstructed a complex image through a sequence of progressive transformations.

If we keep working with a category-typical operation, we can take such a collection of objects (points) and morphisms (arrows), and we can apply this to something else. It is like… to have some bridges, from an initial to a final point (to be opportunely shifted into a new initial and new final point), and we can transform each bridge into another bridge: we would create a “bridge of bridges”! The concept of “transformation between transformations” is, in fact, the “primum movens” that gave birth to Category Theory in the Forties of last Century.

Now, let us try to apply all of that to music. We can move from the category “images” to the category “musical fragments.”

There is an endless number of ways (mappings) to transfer a visual shape into a musical structure (in general, a set of non-sound data into a sound data). We are in the field of “sonification.”