Because I have to go to the dentist every six months I read fashion magazines and popular newspapers that are available on the waiting room table. In these papers one often reads the following. Pink is the new black, grey is the new black, blue is the new black. Why not just wear black all the time? I do. Well most of the time. I do not wear pink however. Maybe on Halloween I will make an exception.
Anyway I read some other fascinating stuff while waiting for my dentist about aging for example that are written in these magazines. 40 is the new 20, 50 is the new 30 and 10 is the new -10. Ok the last one you won’t find in those glossy magazines. It is a bit of a joke on my part, but on the other hand it is not. It reveals linear thinking. Most scientists like straights, like in straight lines not like in like curvaceous ones. However, most of us like curvaceous exotic curves. I do.
(Picture of my Jacket that I bought in Vegas dropped on a funky chair in my hotel room in Copenhagen.)
Let us consider a concrete example. Below we show two curves one linear (left) and one curvaceous (right). The challenge is to find the intersection of the curves with the horizontal axis. It goes by the fancy name of root finding in mathematics. Visually it is easy to find the crossings just by visual inspection only. Any kid can do this.
What about solving this problem with a computer? The crossing for the straight one (left) is easy to compute while the crossings for the curvaceous one (right) is harder to compute. This is because the left curve is linear and the right curve is non-linear. Notice that the straight one is guaranteed to have only one crossing while the curvaceous one can have multiple crossings. That is why people who deal with computers like straight curves over curvaceous ones. It makes their job easier. For a linear problem you always get a single solution that is easy to compute. No cigar but you can call it a day.
Here is the catch though. Most concrete problems are curvaceous not straight. Programmers and mathematicians are clever and that is why they can be lazy. In order to deal with the curvaceous curve shown on the right they approximate it locally with a straight line and hope that that will solve their problem globally.
In our example we try to reach the horizontal line using what is called the tangent to the curve. We pick a point on the curve take its tangent at that point and intersect it with the horizontal line. This is just simple boring high school math. It works great when the curve is straight not so much when the curve is curvaceous. The following picture illustrates this.
Here is a more comical drawing stolen from the web: not to be found in my dentist’s office. The following cartoon illustrates the fallacy of the naive linear way of thinking.
So there you have it. Linearity rules, but most of the time it does not get your job done. Beware of the Church of Linearity: simple message, friendly people but not always of too much help in your daily life. This Church is too cozy for my tastes especially when I have to solve a concrete problem. I then sometimes become a heretic. Solving non-linear problems is much harder than solving linear problems.
I claim that non-linear problems should be called normal and that linear problems should be called non-normal. I am not just playing with words. Words are important as they frame our thoughts in whatever language we happen to speak or think in. I think mostly in pictures: yet another language. When looking at some problem that has a non-normal solution one should exclaim: “Cool dude we are lucky it is non-normal!” not: “Ok which fancy linear numerical solver are we going to use on this one.”
Now let us turn to classical physics. I am not talking about quantum physics or some exotica like string theory. I am talking about the physics that explains stuff you can see, feel, touch or smell. The stuff that is directly accessible to our senses. That is the kind of physics I am interested in. Physics provides a great framework to animate natural phenomena: like cloth, ropes, fluids like water, fire or smoke. You name it. These phenomena are all described by non-linear mathematics.
Newton’s laws however are linear. Think of F=ma. What does that mean? According to Newton this law describes the change of the change that affects what we sense. When we try to animate cloth for example we only care about the motion of the positions of the final cloth. The motion in general is very complicated. Think of all the folds, self-collisions, sliding, draping, etc. happening at the same time. No straight lines there: mostly complicated curvaceous surfaces. In standard physics people deal with forces via mass to determine accelerations. These accelerations in turn tell us how the velocities change over time. And finally the velocities move our shapes around. That is what is usually called dynamics. The evolution of the shape is what I and most people care about in the end.
Accelerations and velocities are linear and they are called vectors in mathematics. You know like an arrow that is straight with a pointy end. No one gets hurt however. These vectors all live in linear spaces like for example planes. The slaves called shapes whose daily motions depend on their linear bullies have a secondary status in this world.
I claim that shapes should instead have first class citizen status and the linear vectors should have secondary status. The Linears should be the slaves of the funky shapes not the other way around. Linears can always be derived from the motion of the shapes we really care about.
That is the basic idea behind my Nucleus solver. You can find more information on my Autodesk web page.
This approach is gaining momentum in computer graphics and games. Independently, Matthias Mueller has introduced the same approach. You can find his work at:
An analogy of some sorts follows.
A long time ago some people in computer graphics were obsessed with rendering reality using transport theory (I was and I still am) and claimed they were photorealistic. Those were the glory days. Then some rebels came along and showed us how to render images in a non-photorealistic manner. You get my point. Non-normal is photorealism and normal is non-photorealism. Artists of course knew this for ages even after the camera was invented.
(Lascaux Cave painting created 17,300 years ago. See https://en.wikipedia.org/wiki/Lascaux.)
Photorealism is just one other stylistic way to depict what we see. I love photorealism by the way. Here is an example by one of my favorite artists called Audrey Flack who is using this genre. It is painted using an airbrush one of my favorite brushes after computer code.It is an clever contemporary homage to the "Vanitas" and the "Nature morte" of the Renaissance. They had skulls and here we have a Hollywood Icon.
On the other hand Marcello Barenghi can show you how to create photo-realistic drawings like the following.
Sometimes a “non” is cast into a (void).