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Off the top of my head, here's things that greatly influenced me:

Paul Bourke

Pixar

Craig Venter Institute

Ender's Game

Bodyguard of Lies

The Millennium Problems

The Manhattan Project

The Empire Strikes Back

Crossroad

For me this was:

Finding inspiration in formulas

Validation that a company can be both technical and artistic

Exponential advance of technology

The start of my love for reading

Confirmation that I will never have enough information about the history of war

My love for mathematics

A geek making a difference

My first first art influence in Ralph McQuarrie

When music grabbed my attention

Math daemons

So, numbers are talking to me. No big deal uh? Patterns seems to jump out of chaos, begging me to write down a equation defining theme. I can't even talk to the people I know about this stuff, because I and the people around me, didn't studied in math. So, thanks to the web and a few hours here and there, I do some maths alone.


It all started three years ago when my brain decided that it liked math more than before. A lot more. How do you fight something like that? Why would I even fight it? It only cost time, paper and a pencil, so that is one cheap pastime.

One cool thing about loving math is the the sheer quantity of problems that need to be solve. Its' an endless sea of 'fun'. Let the nerd in you free and take a look at the Clay Institute Millennium Problems. If I had time, I'd work on P vs NP and The Navier-Stokes existence and smoothness all day long. Just because it's there, like the mountain for the rock climber.

I bet, if I was in high school, I'd get my ass kicked for saying stuff like that. Good thing I'm 37.

I'll just keep filling pages after pages of notes, hypothesis and calculus about 'why in the world' some points in a non linear dynamical system are not stabilizing when m goes to infinity. I'm so close to the answer I can smell it.

Dimensions... again

In a previous post, I was wondering if it was possible to describe a dimension solely in terms understandable by the previous dimension(s) (i.e. describe (n)D in terms of (n-1)D). While thinking about this, I stumble on something interesting. But first lets start with some basic setup:

The Point
It all starts with the point. We say that the point is in 0D (no dimension).

The Line
To make a line starting from that point we can go in any directions around it. We than stretch that point into that direction and draw a line between the two points. We can then say that we had total freedom of movement to create that line. We would also be drawn to conclude that this freedom of movement was in a 3D volume. (Spherical coordinate system.)

The Square
Making the square is a similar process but this time we have stay perpendicular to the previous line. We then stretch the line anywhere around the original line and draw two extra lines liking the copied points. We had some freedom of movement to create that square and conclude that this freedom of movement was in a 2D plane. (Circular coordinate system)

The Cube
Repeating the procedure again will generate a cube. This time our movement have to stay perpendicular to the squares plane. We then stretch the square anywhere along a perpendicular line from the plane and draw four lines liking the copied points. Again, we had some freedom of movement to create that square. We would conclude that this freedom of movement was on a 1D line. (Linear coordinate system)


Looking back at what happen, we can see that as the dimensions are increasing from 0D to 3D the freedoms of movement decreases from 3D to 1D, leaving no dimension to continue stretching the cube into a possible next dimension. In a way, I was somewhat happy. It looked like a perfect circle, from 1D to 3D and back. I showed those results to my wife and the first thing she thought was “Of course you start describing the points freedom of movement as 3D because that's the max for you, they would not say that in line world”. She was right.

I ask myself: how would 1D and 2D describe the same transformation?

The Line
The line would say that the point as no dimension (0D). It would say that to make a line, you have to stretch the point into the first dimension. It then stretch the point anywhere along the existing Line World (1D) and draw a line between the two points. It can say that It had some freedom of movement to create that line. It would conclude that this freedom of movement was on a 1D line.

The Square
The square would also say that the point as no dimension (0D). It would also agree that to make a line, you have to stretch the point into 1D. It then stretch the point anywhere around the original point, while staying in the squares plane, and draw a line between the two points. It can say that It had some freedom of movement creating that line. It would conclude that this freedom of movement was in a 2D plane.
The square would then be able to go one step further than the 1D Line, by stretching the newly created line into it's 2D world with a freedom on a 1D line.


Since the maximum freedom of movement is bounded by our knowledge we cannot picture stretching beyond that limit. But, we as 3D beings, know that there is a second dimension even though the Line World can't see it.
A 4D being would say that the freedom of movement of the point, when creating the line, is in four dimensions. It would also be able to stretch the cube into the fourth dimension and understand why we can't see it.

Can that scale of dimensions be infinite?

Precise imprecision

Have you ever thought about how many types of intersections exist for two lines? Just two straight lines, either in 2D or 3D, bounded by two points (not infinite). And only counting the intersections that fall on one of the two lines, including their end points. You probably didn't, because it sounds pointless.


Well, I had to think about that one seriously while working on a problem back in 2001. It looks simple at first. You start imagining the type of intersections:

• The two lines crossing in the middle


• One end of a line touching the other one


• One end of both lines touching each other

Then you think about that one:

• The two lines completely overlapping

And that gets you to:

• The two lines partly overlapping
  
• And both overlapping combinations with a short and a long line...

In that completed list, you arrive at 7 types of intersections. That would be OK in a perfect world where everything is on a nice grid and points are either on or off the line.
In the real world, lines don't fall perfectly on each other. Many times, two points will be really, really, really close but not touching. That's when you start working with an allowance for imprecision. Doing that, you round up numbers for which the difference is smaller than the allowed precision. So in a 3D world, you would consider two points that are close enough together as one point. This then creates a perfect world with 7 types of intersections... or does it?

Not really, if you're picky like me. Because, with this approach, the data changes with each correction created by the precision rounding. So I asked myself: what would happened if I didn't move the rounded points? What kind of intersection types would that create? After a lot of work, I finished a new list of intersection types. Starting with the original 7 types of intersections, I identified variations for almost each of them. And because of the imprecision allowance, I also found a whole set of sub-cases with each of them acting as a multiplication factor on the base types. This brought the grand total to 129 types of intersections.

129, Ouch!

So, next time you think something is simple... think twice.

Flatland 2.0

In the novel Flatland, the sphere fails to enlighten the square about the existence of a 3rd dimension. In the end the sphere simply gives up and pushes the square out of its current plane. This make him hover above his flat world and see it from a 3rd dimension point of view.

This never made sense to me for two reasons. First, this is an easy way out of solving the problem of explaining the 3rd dimension to a 2D being. Second, the square does not acquire a new sense of vision because he is outside of his original plane. He would still see the world through a flat input. What he would see would be incomprehensible to him, the same way the sphere was appearing, morphing and disappearing by intersecting with his plane. All 3D shapes would still look like that to him and all 2D shapes (from his plane or any plane) would look like points and, occasionally, lines.

Is it possible to explain the 3rd dimension using only only 2D terms and their relation to 1D and 0D? That's one of the things that I'm working on...

My poor brain

What's the points?

Back in January 2007, I was browsing Paul Bourke's web site, looking at amazing images generated from fractals. Like most people I find that some of them have an artsy side and would fit perfectly in a modern art exhibit.

Then I came upon this one:

$ x_{n+1}=\sin(ay_n)-\cos(bx_n) $
$ y_{n+1}=\sin(cx_n)-\cos(dy_n) $

It was generated with a mathematical attractor named 'Peter de Jong'. It was simple, no colors and the rendering wasn't that great. Even though all the points are in a single 2D flat plane, I couldn't help seeing depth in this image. To me, there's a bunch of translucent surfaces in a 3D space.
Of course, at that point my brain started working on overdrive wondering what it would take to separate the points for real and see the shape expanding into a 3rd dimension. So I've set some basic rules for this experiment:

1) Only use the existing X and Y data to generate the new Z data.
2) Never modify the existing X and Y data.

I called this process delamination, because it's like trying to separate different layers glued on top of each other. So far I've achieve 80% to 90% delamination. I only understand 20% of the process and I'm analyzing my results to figure it out.

Also. If you've worked on something like this, I would really like to talk with you.