## Thursday, September 25, 2008

### The Illusion of Center

There really exists a center, or it is an illusion?

The O’Reilly’s rotating grid is an example of an illusion of center, as I will explain. Then, some math skills will allow us to find a grid rotating around two exact centers. The next part will be about the self and the illusion of center.

O'Reilly's rotating grid

Let's take a look at an interesting effect, discovered by David O’Reilly:

As its discoverer noticed, it is an effect of temporal aliasing, but I will add an ingredient - the illusion of center. The temporal aliasing is an effect which usually appears when the moving images are presented as a sequence of frames. Presenting the image by frames creates the illusion of motion, but sometimes the information contained in the frames is ambiguous. As we can see below, if we highlight the parts that seem to rotate, the region delimited by the yellow square is rotating around a central point. The regions delimited by the red squares, although may appear to have their own centers, at a second examination are found not to have a center.

If we look at a real rotating grid, the effect cannot appear. It happens only because the frames present us only some lines. Two rotating orthogonal lines pass through the center. Together with them, other parallel lines rotate around the center. Near every point on the image there are lines, which are parallel to the lines passing through the center. The "approximate centers" are near more lines than the others, but it is only the center that contains lines for all possible directions. We can see this by overlapping all the frames:

Rotation around two centers

If we rotate the grid at each step with an angle of 360/n, the center is the only point containing all the time vertices of the grid (except, of course, the case when n=1, 2 or 4, when no rotation is viewed). But if we are good enough at math, we can modify the animation such that we obtain more than one fixed point (please click to see the full image):

The animation above is based on some properties of the number 65. This number plays the role of the hypotenuse in eight Pythagorean triples:

652=162+632=252+602=332+562=392+522=522+392=562+332=602+252=632+162.

If we allow the hypotenuse not necessarily be integer (for example to square to 325), then the obtained image can be smaller:

and a tile version:

Here is a checker board pattern:

and you can see some dots:

The self and the illusion of center

Why are our minds tricking us into believing that there is a center where in fact it is not? Perhaps this happens because, in a perpetually moving world, we need to believe that there is a fixed point of the Universe. We need to believe that we can see where the phenomena converge, so that we can understand the laws of Nature, so that we can survive and have benefits. In order to want to survive and evolve, we need to believe that we are important, that we have a central self.

At different stages in our lives, we may be so different, that we hardly can say that we are the same person. Our interests and ideals as children may differ very much from those we have as adults. Our priorities and even values at the office or school may be very different than the ones we manifest in family, or with our friends. If we are changing that much during our lifetimes, is there a convergence point of all our facets? Do we really have a self? Or this is just another trick of our mind?

In everything happens to us, sometimes we may feel the need to believe that there is a reason, other than blind chance. We may be tempted to hope that there is a reason behind everything, a central point towards everything converges. Is there such a Center of the World, like the one marked by cross? If there is, how are we, the other beings, like the areas marked with red squares, with no real center? Or are we all centers in the same time, like the vertices marked with red dots? Or there is nothing else but the emptiness of the grid?