Maths does not need colours

If the world would be in black and white maths was exactly the same that it is.

There is indeed no need of colours in maths at all. The basic fact that justifies this is that we can code maths with only zeros and ones, or in other words,  in black and white.  In this sense, maths is a science that do not “see in colours”.

In practical terms, if we could create a mathematical machine to simulate a human being it should be unable to see colours, however it could distinguish among them. Let me explain this. It is easy to write a program in order to distinguish colours, and surely it will work well in most of the cases. However, distinguish is different from see. If we take a look to what the machine really sees whenever it watches a colour, we should realise that it is indeed an image in black and white of what a colour is. The reason is that the machine understands in mathematical terms. This makes a huge difference between our perception of the word and the one of a mathematically made machine.

Paradoxically, it is usually said that teaching mathematics needs colours. This is because we are not mathematically programmed machines, moreover, colours allows us to incorporate, understand, remember and differentiate mathematical objects (surely a phycologist could say more about this). More in general, we are used to perceive our surrounding using colours, so, it is not very surprising that representing maths with colours do help us to understand them better (this is an interesting fact, that many maths professors could exemplify vasty).

It is a bit sad that a colour is indeed a not definable object in mathematics, so a proper definition of something so trivial is far beyond what we (mathematicians) can define. Defining a colour is a problem that more successfully than we, physicists, artists, phycologists, physicians and writers have accomplished, however, its fully comprehension looks at least intriguing.

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Fire simulation

I was playing the nice video game Ni no Kuni. To be more specific, I was playing a stage called The Mountain of Fire, about a volcano that is going to erupt (in three minutes!). In the scene there are molten rocks, lava and fire. A natural question at this point is: how does the game simulate fire?

A short answer by a mathematician is using the Navier-Stokes Equation, and the one by a ”wikipedian” is using the paper Real-Time Fluid Dynamics for Games by Jos StamHowever, this problem is far more fascinating than this. My goal here is to explain in a few simple words why. I will start exhibiting the Navier-Stokes Equation, then I will comment about how to simulate fire and I will exhibit some simulations. Finally, I will enumerate current applications and possible ones.

The Navier-Stokes Equation

The Navier-Stokes equation is a basic model for incompresible fluids, like the water. We start by considering a velocity field

\begin{aligned}  u:&\mathbb R\times \mathbb R^3& \to &  \mathbb R^3\\  &(t,x)&\mapsto& u(t,x)=(u_1(t,x),u_2(t,x),u_3(t,x))  \end{aligned}

and the pressure

\begin{aligned}  p:&\mathbb R\times \mathbb R^3& \to &  \mathbb R\\  &(t,x)&\mapsto& p(t,x)  \end{aligned}

that depend on the time t and position x. Each fluid has a viscosity constant \nu>0. The Navier-Stokes equation is given by the system of equations

\begin{aligned}  &\frac{\partial}{\partial t} u+ (u \cdot\nabla)u=\nu \Delta u-\nabla p,\\  &\nabla \cdot u=0,\\  &u(0,x)=u_0(x).  \end{aligned}

Given a smooth initial velocity field u_0(x), a global smooth solution of the Navier-Stokes equation is a smooth velocity field u:[0,+\infty)\times\mathbb{R}^3\to\mathbb{R}^3 and a pressure function p:[0,+\infty)\times\mathbb{R}^3\to\mathbb{R} satisfying the system of equations.

The justification of the Navier-Stokes equation will not be discussed here, a note on its more important points can be found here, for example.

A solution u of the Navier-Stokes equation is said to blow up if \sup_{x\in\mathbb{R}^3}|u(t,x)|\to +\infty as t\to +\infty, i.e., at some time there is flow with arbitrarily high velocity. Whether is theoretically possible to always find global solutions that do not blow up is an open problem. You can read an explanation of why it is a complicated problem in Terry weblog.

How to simulate Fire

There is a well developed research field called Computational Fluid dynamics, that in particular, studies algorithms to simulate solutions of the Navier-Stokes equation. Numerical simulations using real data suggest, indeed, that there is always a global solution that do not blow up (as it is expected). If you are interested in understanding how to simulate solutions of the Navier-Stokes equation you can read this paper, for example.

Simulations

The Simulations showed here were made using some examples provided in the software Fire Dynamics Simulator (FDS) and Smokeview (SMV).

hqdefault Circular burner

Smoke_Simulation Smoke Simulation

 

Applications

As I motivated this post, the Navier-Stokes equation can be applied to simulate fire and smoke in modern video games. However, there are many other applications. I can enumerate some natural examples:

  1. Weather forecasting.
  2. Chemical and Energy industries.
  3. Evacuation routes and placement of fire alarms in case of smoke or fire in cities, forest, buildings, etc. See for example

    [RUSHMEIER, H. 1994. Rendering Participating Media: Problems and Solutions from Application Areas. In Proceedings of the 5th Eurographics Workshop on Rendering, 35–56].

  4. Ventilation design of the ocean, cities, buildings, miners, computers, etc.
  5. Design of planes, trains, ships, cars, bikes, etc.  More in general, the design of any device that interact with a fluid in order to accomplish a specific function, like, high stability, high velocity, etc.
  6. Design of anti-tsunami systems.
  7. Mathematical Cardiology, i.e. the modelling and simulation of the circulatory system.
  8. Increasing the velocity of transmission of information about certain properties that depend on fluids.

Artistic representation of fluids

The representation of fluids in arts is typical of an artistic movement called Expressionism. Expressionist artists combine together the representation of certain specific physical phenomena and the representation of daily life situations,  as a results, they create an emotional response to the viewer. Usually, the expressionism is studied from the point of view of the psychology, however, from a mathematical physic point of view we are interested in the specific physical phenomena that the artists use. We show examples of artistic representation of fluids by expressionist artists. 

The_Scream

The Scream (or The Cry) 1893, Edvard Munch.

The Cry is a representation of different fluids interacting together: a human being, a river and the atmosphere.


STARRY NIGHT by Vincent Van Gogh
The Starry Night 1889, Vincent van Gogh.

The Starry Night has been studied by many people, I will limite myself here to mention that the astronomer Charles A. Whitney studied the painting from the astronomic point of view. In particular, the brightest star is Venus and the moon was waning gibbous. From my point of view van Gogh is a step away (in physical terms) from the Navier-Stokes equation model of incompressible flows, because he is more likely representing the superfluidity of the lights. Even though, you can experimentally check (by watching here) that The Starry Night is closer related to the Navier Stokes equation than you may had thought.