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\chapter{Postface: Mathematics, physics, and computer graphics}\label{ch:postface}

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     \textit{All models are wrong, but some are useful.}

          \hfill--- \textit{George E. P. Box}

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This is a personal essay, almost entirely unrelated to the main contents of this thesis. It might be more like a piece of prose, collecting unorganized thoughts.

% The journey of my PhD started (actually didn’t start) when I finished my master’s study in mathematics, with terrible grades and no chance to do a PhD.
The beginning of my PhD was when I finished my master’s study in mathematics with poor grades and no chance of continuing to a PhD. So, it actually didn’t begin.
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Then I spent some time working in the optics and game industries. Around that time, I came across the field of computer graphics. After some time, I started a PhD and now complete it, with research in the union of mathematics, physics, and computer graphics.


Through this journey, I've got some thoughts about mathematics, physics, and computer graphics. I’d like to leave some of them here.
My path had some twists, but looking back, I think my aim has always been to express how I see nature and the world in a mathematical way. In this regard, I’ve been looking at math and CG through their connection to physics. Additionally, math and CG are often regarded as opposites, like the ``pure'' and ``applied'' ends of science, but I think they actually share a lot in common.
I’d like to write a bit about some aspects that I think math and CG share.

\paragraph{No rigid definitions}
First, when it comes to what mathematics and computer graphics actually are, I think there are no rigid definitions, perhaps even more  so than many other fields.
I’ll briefly describe my view of modern computer graphics, because outside of this field, people often have little idea of what this can be. I probably wouldn’t have known it too without doing research in graphics.
Modern graphics is a much broader field than many people imagine. It’s not just about creating gorgeous imageries that appear in movies of Pixar or Disney, although it is an important part.

For instance, research of fluids, like water and smoke, in graphics can involve many directions and tastes.
It is probably easy to imagine that improving numerical methods for existing fluid equations, like the Navier–Stokes equation, can be a research topic. But that’s not all there is to it.
\cite{Yin2023fluidCohomology,YinZhu2025viscousFluids} pointed out that the existing formulation of the Navier-Stokes equations in terms of fluid vorticity was incorrect, and derived the correct formulation that aligns with the velocity-based formulation. As a consequence, they found closed-form expressions for non-stationary solutions of fluids.
Another valid direction is proposing new equations of motion for physical phenomena. For instance, I geometrically formulated the dynamics of soap bubbles and films \cite{ishida2017hyperbolic}. 
Many people would probably view these types of work as mathematics or physics. But in fact, they can stand as computer graphics work.
Basically, if a paper involves either computer or graphics in some way, it can be a computer graphics work. In my opinion, even that condition isn’t strictly necessary.
\footnote{
In practice, however, as of 2025, papers in computer graphics journals seem to me largely limited to work that presents a \emph{method} achieving or solving something. It seems very difficult to publish a paper about pure discoveries that are common in other fields---like finding a new species, observing or predicting interesting physical phenomena (like black holes) through theory, numerical simulations, or physical experiments, or proving a particular underlying law. In the current graphics venues, remarks or proofs are often considered something that should be pushed to appendices rather than the primary contribution. Still, it seems (and I hope) that gradually more people are recognizing such discoveries or interpretations themselves as valid contributions.
}

Similarly, ``mathematics'' is just a vague and undefined jargon. For me mathematics isn't really a field, but rather an activity or a practice.
It sets up axioms then deduces what can happen within them through logic. It also involves describing vaguely perceived concepts in a language that everyone understands in a unique way, or developing that universal language, like Newton and Leibniz introduced \emph{differentiation}, a language which describes infinitesimal changes of quantities. I think all these activities together are what is called mathematics.
Therefore, mathematics appears in many works that aren’t “mathematics” papers. A mathematics paper is just one in which the claimed main contribution is this activity itself. In the first place, whether a piece of work counts as a ``math'' paper or not may hardly matter mathematically. \footnote{In the past, such categories as ``mathematics'' or ``physics'' were quite vague, and there were no jargons like ``pure math'' or ``applied math''. Everything was more mixed, and today’s standard mathematical rigor was not required, depending on the type of work. So one could be more free in what one can publish as a math work—not only rigorous proofs of statements, but also new concepts without rigorous analysis. Many papers by Poincar\'e, who introduced what is now called topology, did not meet today’s standard rigor, and so they likely wouldn’t have places in today's ``mathematics'' papers. I find this narrowing of math sad.}

\paragraph{Lenses for physics}

Since ancient times, physics research has been conducted by observing natural phenomena, making hypotheses to explain them, and verifying them through experiments. Much of modern physics research is, however, less intuitive to me as it often requires interpreting or reasoning the numbers obtained in experiments. In the old physics, on the other hand, the results of experiments were more direct and clearer. For example, Michael Faraday actually \emph{saw} his discovery when a coil was rotated by the invisible magnetic field provoked by an electric field. 
For me, physics simulation in CG can bring back this intuitive aspect of physics research, allowing us to visually see the results of experiments based on the theoretical models we build.

But unfortunately, it seems to me that the recent CG industry is getting trapped by the curse of ``realistic'' or ``physically-based''. Here ``physically-based'' just means, deriving from or combining existing physics equations. Creating new equations out of nowhere, based on observing natural phenomena, is ``not physically accurate''. Yet that is precisely what physics is about---observing nature and extracting its simple essence. \footnote{For example, this sand model paper \cite{nishimori1993formation} practices this spirit and was accepted in Physical Review Letters, a top journal in physics.} I feel like the papers by Newton or Faraday, who modeled natural phenomena as principles via observation rather than deriving from existing equations, would likely be rejected in today’s CG venues as ``physically unplausible''. Being caught by ``physically-based'' or ``realistic'' attitudes appears to me, ironically,  the opposite of physical science.

Okay, let’s get back on track. Like CG is a modern experimental ground for physics, mathematics is a sandbox for physics too. Creating and validating physical models, and deducing phenomena from them, works the same way. It’s just a matter of whether the experiments are theoretical or numerical involving visualization. For example, Einstein tried to validate his theory of gravity by calculating the precession of Mercury’s perihelion. I think such a work should fit as a graphics paper today.


\paragraph{No need to stay in the real world}
In both computer graphics and mathematics, we do not have to be bounded by the laws or phenomena of real-world physics. In game development, for instance, physics can be designed for the sake of entertainment, without following real-world mechanics. Likewise, mathematics allows one to freely explore imaginary objects. So both fields offer the freedom to create and explore fascinating phenomena within fictional worlds governed by their own rules. \footnote{Just as mentioned above, such freedom seems to have been slipping away as the community gets caught up in ``realistic'' or ``physically-accurate''. It's a shame that CG is throwing away its virtue.}

\paragraph{Do not have to be science or technology. Can be just art}

CG and math don't even have to be science or technology. They can be just art. In a sense, one could even say that they are arts of fake.

Obviously, computer graphics is an effort for making fake imagery that mimics reality. Mathematics can also be seen as a form of fake physics. Mathematical results are considered very rigorous, but they are so only mathematically. They may not be rigorous at all in different senses.

Many mathematical models start from drastic simplifications and approximations of physical phenomena, yet humans often do not even know to what extent they are justified.
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They also begin with presumed principles like Newton’s laws, or with hypothetical setups to make them tractable within mathematical frameworks, like elementary particles, Hilbert spaces, or the concept of continuum. But actual natural phenomena do not respect these artificial setups.

The quote by George E. P. Box at the opening of this essay --- \emph{``All models are wrong, but some are useful''} ---  was originally about statistical models, but I feel it applies to many sciences including physics.

And I think this is okay. Models are wrong by definition. They aren't nature itself. They are, like elementary particles for instance, just theoretical accounts that try to explain a class of (but not all) observations seemingly without contradiction. They are never truth, but they are the only and best we can do to interpret how the world looks and behaves.

Different people see the world in different ways. One can start from --- how I see this world --- as their own principle and express that view through (fake) arts of science like mathematics and computer graphics.

I hope modern mathematics and computer graphics are and remain grounds that accommodate and celebrate such diversity of perspectives.

\paragraph{A side note: A personal note on my journey}

Here is a brief note on how I came to have these views.

During my master’s studies, I wanted to do research in mathematics and I had some vague ideas. But I had no clue how to start concretely or what I should learn for this. I think I was preoccupied with mathematical expressions and formulas, and following the correctness of logic, like many math students feel they are forced to do so. And like many other students again, I was not able to develop an intuitive understanding of what these concepts are. For instance, I didn’t even realize that the Fourier transform are waves. Even though I said I liked mathematics, I guess I had lost sight of how I liked it.

After finishing my master’s degree and before starting my PhD, I spent some time in the optics and game industry. There, I learned what mathematical concepts intuitively are, how they look, and how they affect the real world—for example, how the Fourier transform influences pictures seen through a camera lens.

Then, I came across research in computer graphics, in which I could visually perceive how the mathematics I design affects the motion and shapes of objects, and how it builds the laws of a (real and imaginary) world. Through this process, mathematics came to life inside me. Observing motions and shapes of physical phenomena in CG research also brought me many questions that I wanted to investigate. CG turned out not to be just a place to apply fancy mathematics, but a source of mathematical questions and it made me love mathematics again.

Hopefully, this journey goes on, but I wanted to record these thoughts and feelings of mine, at this moment.
