Research Projects are Like Fractals — No Matter How Close One Looks, One Can Always Zoom in More (An Analogy)
When I think about the research process, I often think about fractals.
Fractals are mathematical objects that are beautiful at all resolutions — when zoomed out fully and when zoomed in very, very, very closely. In fact, no matter how far one zooms into a fractal, it is possible to zoom in more, and there is beauty at every level.
The same applies to research — there is beauty in high-level research about a project’s subject, and there is beauty as one dives deeper and deeper into specific aspects of the research subject.
The image below is of a famous fractal: the Mandelbrot set.
First approaching a research project is like looking at a fractal from a distance.
When initiating a research project, I often imagine that we are starting to explore a fractal that we have never seen before. That means that we are zoomed out, looking at the fractal (the project subject) as a whole.
We ask questions about this fractal (this project subject). As we start to answer those questions, we develop a better understanding of the shape of the whole fractal (the shape of the project subject).
Oh, that’s interesting! Diving deeper into a region of a fractal.
When studying a fractal, a particular region might look especially beautiful or interesting. If so, it is possible to zoom in on that region and repeat the above process: Ask and answer / explore the overall shape of that subregion.
Likewise, for research, the answers to the initial research questions almost always lead to more questions. Research can, of course, explore those newly raised questions, too. Doing so is like diving into a region of a fractal.
For a visual deep-dive into the Mandelbrot set, see this animation from Wikipedia.
An observation and a research methodology challenge.
One takeaway is that, for the iterative ask-answer-ask-answer-ask-answer process of research, no matter how close we look, we can always look more closely. The answers to research questions (almost) always create new questions, which we could of course explore and, in doing so, raise new questions along the way, and so on.
A methodology challenge thus arises: As new understandings and questions arise during the fractal exploration process (during the research process), do we stay looking at the high-level structure of the fractal (the initial project and initial research questions), or do we dive deeply into an exploration of the new region of the fractal (the new research questions) at the expense of forgoing a full (even if not deep) exploration of the whole fractal (the full exploration of the initial research questions)?
A recommendation.
My suggestion is simple, at least in words: Keep in mind the level of the fractal that one wishes to view (the depth and scope one wants for a research project).
Either of the following is okay:
- It is okay to explore a project area at a high level and not jump into additional research questions that come up along the way (or, with the fractal analogy, to make a decision to focus on the fractal as a whole).
- It is also okay to begin exploring a research area with some initial research questions in mind, make some discoveries and raise some new questions, and then change the focus of the project to study those new questions (or, with the fractal analogy, to focus on a zoomed-in region of the fractal).
To reiterate: My recommendation is for researchers to know which of the above they are doing. Both are great options! But it is generally better to know which option one is choosing vs unknowingly try to do both or go back and forth between the two. For example, a situation to avoid is iteratively diving deeper and deeper into new questions (deeper and deeper into a fractal) without explicitly considering whether the researchers wish to dive deeply into a region or remain focused on the initial research questions and the initial project as a whole (the fractal as a whole).
Fractals and self-similarity: Insights from from fractal self-similarity.
Some fractals have a level of self-similarity — as one dives deeply into a region of the fractal, the high-level structure of the fractal is again visible in the detailed region.
The same is often true for research. As we dive deeply into new research questions, familiar themes reappear — echoes of themes that we saw when exploring the high-level research questions or echoes of themes we saw in other projects. There is a beauty in these echoes. Moreover, the research of a new region can be strengthened by observing and leveraging the echoes from other regions / other projects.
Changing fractal parameters and switching projects / focus is also okay.
The above discussion focuses on the question of whether to dive more deeply into a study of a particular research subject or to remain focused on the original research questions, i.e., whether to zoom in on a region of a fractal or not.
It is worth acknowledging that, as one gains new knowledge about a research subject, one may for very good reason decide to pause work on that research subject and consider a different research subject instead.
Or, one may formulate new, high-level questions about the research subject.
Continuing the fractal analogy, doing the above is similar to deciding to explore a different fractal — perhaps an entirely different fractal, or a similar fractal in the same family of fractals.
Closing remarks: A reminder on fractals and research.
Fractals are beautiful regardless of what scale we look at them. It is simply a question of what scale we want to look at them. The same applies to research. It is fine and beautiful to look at a fractal from a distance. And, it is also fine and beautiful to zoom in. Most importantly, it is valuable to know where one is looking (whether they are looking at the fractal as a whole or at a zoomed-in region) and/or to thoughtfully choose where to look.
More background on my perspective on the fractal analogy.
Why the fractal analogy? I like the fractal analogy in part because it is related to my interest in fractals as a high school student and, more recently, because of Barrish Sensei’s observation of fractal patterns in aikido — a deep study of aikido reveals spirals within spirals within spirals. Fractals are super interesting, and one might ask: where else do fractal-like structures exist?
Acknowledgements.
I learned so much about the PhD process from my own advisor, Mihir Bellare. After obtaining a faculty position, I continued to develop my own philosophy on advising and the PhD process through the advising my own PhD students. Much of my thoughts on advising and the PhD process have also been shaped through the co-advising of PhD students with UW Security Lab co-director Franziska Roesner. Thank you to all the students and postdocs that I have advised, past and present!