# Is Visualization of Data a Subject of Mathematical Research? [closed]

Please excuse my naive question, but what kind of rôle does the visualization of (especially high-dimensional) data play in mathematical research?

I know, that it plays an important rôle in the analysis and interpretation of data and utilizes advanced mathematics, but, apart from being a servant to mathematical research, are there also examples, where mathematical research is solely dedicated to data-visualisation or, where data-visualisation brought up questions, that initiated important mathematical research?

A prominent example of mathematical research that has been initiated by data visualisation is Guthries observation, that apparently four colors suffice to color every map; hat observation had very fruitful consequences for mathematics.

Questions:

• what are further examples of problems that originated in data-visualization and became the subject of mathematical research?

• are there examples of mathematical research, that is dedicated to improving the visualization of data, i.e. to make features of interest, that are "burried" in the data, more "prominent"?

## closed as too broad by YCor, j.c., Andy Putman, Ben McKay, RP_Jan 14 '18 at 9:44

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• If somebody has an idea to add one or several broader tag it would be useful (being confused by the claim that the 4-color conjecture was based on "data visualization", I have little idea of the scope of the question) – YCor Jan 13 '18 at 17:09
• The 4 color conjecture came from the task to make countries with common border visually distinguishable; the naive solution would be to assign a different color to each country. If also economical considerations come into play, then one might strive for using as few colors as possible and a natural question is then for the minimal number of colors necessary. As it turned out, that question could not be answered with then existing mathematical theorems and dedicated research was necessary. – Manfred Weis Jan 13 '18 at 18:33
• I understand that it comes from visualization. I'm rather puzzled by the use of the word "data". Data means something already encoded in some way. If you're asking whether visualization plays a role mathematical research the answer is obviously yes and if you're asking how, this sounds way too broad. – YCor Jan 13 '18 at 18:58
• @YCor: I concur with your last comment. I also struggled with the scope of the question, but could not resist the temptation to post a constructive response, however imperfect. – Victor Protsak Jan 13 '18 at 19:27
• @YCor: let me explain, why I consider drawing a political map as visualizing data: a political territory is defined via its boundaries, that are documented in field survey, which yields the original data, that can be interpreted and visualized in various way, depending what kind of information one is interested in. Even if coloring a map doesn't resemble mining or filtering of data, it is still the problem of emphasizing that adjacent territories are e.g. controlled by different political authorities. – Manfred Weis Jan 13 '18 at 20:34

Here is a specific illustration already firmly within mathematical setting. The notion of Hasse diagram of a partially ordered set provides an important visualization of this structure that has been used to study its properties. For example, it is easy to see that the diamond lattice $M_3$ and the pentagon lattice $N_5$ are not distributive. In fact, a lattice is distributive if and only if it does not contain a sublattice isomorphic to $M_3$ or $N_5$. On a more rudimentary level, the fundamental concepts such as maximal and minimal elements, chains and maximal chains of a poset can be thought of as visualizations. Some research directions in group and module theory deal with the structure of groups/modules based on their properties of their subgroup/submodule lattice and one can argue that, for example, uniserial modules (where all submodules form a single chain) first became objects of interest because of this visualization aspect. Later, the success of this theory motivated investigating more complicated cases.